{"kind":"Article","sha256":"4fb4a548798789924715d28d82be99728cd298910bef0a7dc5cae01bd9617f73","slug":"entanglement-1gaussian","location":"/entanglement/4entanglement_1gaussian.md","dependencies":[],"frontmatter":{"title":"Gaussian states","short_title":"Gaussian states","subtitle":"Introduction to Gaussian states","numbering":{"heading_1":{"enabled":true},"heading_2":{"enabled":true}},"authors":[{"nameParsed":{"literal":"Victor Gondret","given":"Victor","family":"Gondret"},"name":"Victor Gondret","orcid":"0009-0005-8468-161X","email":"victor.gondret@normalesup.org","affiliations":["Université Paris-Saclay, CNRS"],"url":"http://www.normalesup.org/~gondret/","id":"contributors-myst-generated-uid-0","corresponding":true}],"license":{"content":{"id":"CC-BY-NC-SA-4.0","name":"Creative Commons Attribution Non Commercial Share Alike 4.0 International","CC":true,"url":"https://creativecommons.org/licenses/by-nc-sa/4.0/"}},"github":"https://github.com/QuantumVictor","keywords":[],"affiliations":[{"id":"Université Paris-Saclay, CNRS","name":"Université Paris-Saclay, CNRS"}],"abbreviations":{"MOT":"Magneto-Optical Trap","BEC":"Bose-Einstein Condensate","MCP":"Micro-Channel Plate","DCE":"Dynamical Casimir Effect","HBT":"Hanbury-Brown and Twiss","CFD":"Constant Fraction Discriminator","TDC":"Time-to-Digital Converter","FPGA":"Field Programmable Gate Array","AOM":"Acousto-Optics Modulator","RF":"Radio-frequency","ODT":"Optical Dipole Trap","IGBT":"Insulated-Gap Bipolar Transistor","MPQ":"Max Planck Institute of Quantum Optics","PPT":"Positive Partial Transpose","SSR":"SuperSelection Rule","LN":"Logarithmic Negativity","UV":"UltraViolet","TOF":"Time-Of-Flight","TF":"Thomas-Fermi","CMB":"Cosmic Background Radiation"},"settings":{"myst_to_tex":{"codeStyle":"minted"}},"thumbnail":"/~gondret/phd_manuscript/build/wigner_functions_exp-c91cc271360d834816496cbf13b7adad.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/wigner_functions_exp-c91cc271360d834816496cbf13b7adad.webp","exports":[{"format":"md","filename":"4entanglement_1gaussian.md","url":"/~gondret/phd_manuscript/build/4entanglement_1gauss-4df6223a8ac0b32c4bcd55d28b86129a.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"The theoretical study of Gaussian states is abundant in the literature: their general and mathematically rigorous introduction is beyond the scope of this thesis. In this section, I introduce the key properties of two-mode Gaussian states. Subsection ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"Ca1j7TogW7"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"1","key":"c8QPrmJTZp"}],"identifier":"density_matrix_quantum_state","label":"density_matrix_quantum_state","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"density-matrix-quantum-state","key":"gqiufLyshM"},{"type":"text","value":" recalls some key properties of the density matrix operator and subsection ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"vKc0faKzyg"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"2","key":"tk6Ww4aHjI"}],"identifier":"wigner_function","label":"wigner_function","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"wigner-function","key":"ZONizpVe7R"},{"type":"text","value":" the Wigner function. Subsection ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"SXSEnNKcXB"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"3","key":"LQb7kAobVj"}],"identifier":"subsection_gaussian_state","label":"subsection_gaussian_state","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"subsection-gaussian-state","key":"YqnKORzzBp"},{"type":"text","value":" introduces the Gaussian state formalism which will be used in the rest of this chapter, especially the second section and the fourth one, which are the main theoretical results from this work. Subsections ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"sFTqXY6Pli"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"4","key":"sn0lG3nem5"}],"identifier":"single_mode_transfo","label":"single_mode_transfo","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"single-mode-transfo","key":"fF3DlGMfL9"},{"type":"text","value":" and ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"VIoZHO1aNW"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"5","key":"xkjEVdyqOq"}],"identifier":"section_bi_partite_gaussian_state_transofrmations","label":"section_bi_partite_gaussian_state_transofrmations","kind":"heading","template":"Section %s","enumerator":"5","resolved":true,"html_id":"section-bi-partite-gaussian-state-transofrmations","key":"xkZIKp772B"},{"type":"text","value":" recalls usual single mode and two-mode states and transformations. The last subsections present usual states and transformations. Subsection ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"h4rD9dTE3X"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"6","key":"hnBZwZ7aQS"}],"identifier":"joint_proba_distrib_section","label":"joint_proba_distrib_section","kind":"heading","template":"Section %s","enumerator":"6","resolved":true,"html_id":"joint-proba-distrib-section","key":"bGcKjRW9M6"},{"type":"text","value":" provides references to compute the ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"yiVjBxIy9x"},{"type":"inlineMath","value":"N","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"TPZOJcHNcx"},{"type":"text","value":"-mode probability distribution of Gaussian states.","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"cRHaVHNyAD"}],"key":"mGp1Q43nm1"},{"type":"comment","value":" The interested reader might therefore refers to the following references for more information.\\ ","key":"FYc7ayVU6e"}],"data":{"part":"abstract"},"key":"vRrtK8xa2i"},{"type":"block","position":{"start":{"line":15,"column":1},"end":{"line":15,"column":1}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"Density matrix of a quantum state","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"RlanVJFYQk"}],"identifier":"density_matrix_quantum_state","label":"density_matrix_quantum_state","html_id":"density-matrix-quantum-state","enumerator":"1","key":"dggRBrWihJ"},{"type":"paragraph","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"children":[{"type":"text","value":"In this first subsection, we review key properties of the density operator ","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"Zfi3qN7IOq"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"hAFmhf7WzH"},{"type":"text","value":" that we will use throughout this chapter. We will use the properties required by a density operator to derive bounds on the covariance matrix of a Gaussian state.","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"DCKFhKFHhv"}],"key":"cbLYpxG7wb"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Density matrix","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"LEUJcTaLLc"}],"key":"YfBQIjDQ0W"},{"type":"paragraph","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"children":[{"type":"text","value":"The density operator ","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"key":"bQVuEe3RMx"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"mtaLuoBK75"},{"type":"text","value":" defines a quantum state. It must be a Hermitian operator, positive semi-definite","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"key":"I5tirq5JAc"},{"type":"footnoteReference","identifier":"positive-semi-definite","label":"positive-semi-definite","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"number":1,"enumerator":"1","key":"xbEMytSB3j"},{"type":"text","value":" with trace 1.","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"key":"VnT5sUVeuR"}],"key":"Q3SWHMuNs7"}],"key":"egdK6oFX4M"},{"type":"paragraph","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"In particular, if a matrix ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"kAdbarMBMU"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"xAmNROVvEw"},{"type":"text","value":" has a negative eigenvalue, it cannot represent a quantum state. A pure state can be written as ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"dxiM4D9xSq"},{"type":"inlineMath","value":"\\hat{\\rho} = \\ket{\\psi}\\bra{\\psi}","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mo>=</mo><mpadded><mi mathvariant=\"normal\">∣</mi><mi>ψ</mi><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mi>ψ</mi><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho} = \\ket{\\psi}\\bra{\\psi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span></span><span class=\"mord\">∣</span></span></span></span></span>","key":"SlziCIlJNh"},{"type":"text","value":": it is a projector. The purity of an arbitrary quantum state is ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"dqg5fkzgBy"},{"type":"inlineMath","value":"p = \\text{Tr}(\\hat{\\rho}^2)","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>=</mo><mtext>Tr</mtext><mo stretchy=\"false\">(</mo><msup><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mn>2</mn></msup><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">p = \\text{Tr}(\\hat{\\rho}^2)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"></span><span class=\"mord text\"><span class=\"mord\">Tr</span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>","key":"bEhnOiI50R"},{"type":"text","value":". It is 1 for a pure state and smaller than 1 for mixed states.","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"UNvoGFdOfQ"}],"key":"xfCF4MUF6X"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Expectation value of an operator","position":{"start":{"line":32,"column":1},"end":{"line":32,"column":1}},"key":"wiyW4LAsWE"}],"key":"zlfZeG6nx6"},{"type":"paragraph","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"children":[{"type":"text","value":"The expectation value of any operator ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"HZWKQOL5FZ"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"lqrr0FEGma"},{"type":"text","value":" is given by","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"tQ1Sz6d8EJ"}],"key":"HbLNxijMwL"},{"type":"math","value":"\\braket{\\hat{A}}_{\\hat{\\rho}} = \\text{Tr}(\\hat{\\rho}\\hat{A}) = \\braket{\\hat{A}}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mpadded><mo stretchy=\"false\">⟨</mo><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo stretchy=\"false\">⟩</mo></mpadded><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></msub><mo>=</mo><mtext>Tr</mtext><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo stretchy=\"false\">)</mo><mo>=</mo><mpadded><mo stretchy=\"false\">⟨</mo><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\braket{\\hat{A}}_{\\hat{\\rho}} = \\text{Tr}(\\hat{\\rho}\\hat{A}) = \\braket{\\hat{A}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.3826em;vertical-align:-0.4358em;\"></span><span class=\"minner\"><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1864em;\"><span style=\"top:-2.4003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-2.7em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"mord mathnormal mtight\">ρ</span></span><span style=\"top:-2.7em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord mtight\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4358em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord text\"><span class=\"mord\">Tr</span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"1","key":"mzi8Q3Tvr3"}],"key":"zWTCkzWXnS"},{"type":"paragraph","position":{"start":{"line":39,"column":1},"end":{"line":39,"column":1}},"children":[{"type":"text","value":"where we will omit the bracket subscript ","position":{"start":{"line":39,"column":1},"end":{"line":39,"column":1}},"key":"ImGn6BFE03"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":39,"column":1},"end":{"line":39,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"H40FDlsWJ2"},{"type":"text","value":" in the following.","position":{"start":{"line":39,"column":1},"end":{"line":39,"column":1}},"key":"Jx7dgKsPbg"}],"key":"Ict5MnJ7Tl"},{"type":"proof","kind":"theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Cauchy-Schwarz inequality","position":{"start":{"line":41,"column":1},"end":{"line":41,"column":1}},"key":"TDyIWemJc1"}],"key":"FCxMWakzWl"},{"type":"paragraph","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"children":[{"type":"text","value":"Because the density matrix of a quantum state is semipositive definite, it implies that for any operator ","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"key":"lVqDQ4Vhnv"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"r1ANbCSX95"},{"type":"text","value":" and ","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"key":"nNvMjD02jh"},{"type":"inlineMath","value":"\\hat{B}","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{B}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"PZXKslP16I"},{"type":"text","value":", it is always true that the following Cauchy-Schwarz inequality is satisfied ","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"key":"Pt5g6fE6R6"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"children":[{"type":"cite","identifier":"horn_cauchy_schwarz_1990","label":"horn_cauchy_schwarz_1990","kind":"parenthetical","position":{"start":{"line":44,"column":208},"end":{"line":44,"column":233}},"children":[{"type":"text","value":"Horn & Mathias, 1990","key":"jDpyN85XSi"}],"enumerator":"1","key":"tGeC5fyQaF"},{"type":"cite","identifier":"robertson_notes_2021","label":"robertson_notes_2021","kind":"parenthetical","position":{"start":{"line":44,"column":234},"end":{"line":44,"column":255}},"children":[{"type":"text","value":"Robertson, 2021","key":"zJKyqJTL4k"}],"enumerator":"2","key":"bqRt1JbUYk"}],"key":"e9FiaHWEQA"}],"key":"A3X2TURCw3"},{"type":"math","identifier":"cauchy_schwarz_inequality_always","label":"cauchy_schwarz_inequality_always","value":"|\\text{Tr}(\\hat{\\rho}\\hat{A}^\\dagger\\hat{B})|^2 = |\\braket{\\hat{A}^\\dagger \\hat{B}}|^2 \\leq \\braket{\\hat{A}^\\dagger\\hat{A}}\\braket{\\hat{B}^\\dagger\\hat{B}}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">∣</mi><mtext>Tr</mtext><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><msup><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo>†</mo></msup><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover><mo stretchy=\"false\">)</mo><msup><mi mathvariant=\"normal\">∣</mi><mn>2</mn></msup><mo>=</mo><mi mathvariant=\"normal\">∣</mi><mpadded><mo stretchy=\"false\">⟨</mo><mrow><msup><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo>†</mo></msup><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover></mrow><mo stretchy=\"false\">⟩</mo></mpadded><msup><mi mathvariant=\"normal\">∣</mi><mn>2</mn></msup><mo>≤</mo><mpadded><mo stretchy=\"false\">⟨</mo><mrow><msup><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo>†</mo></msup><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mrow><msup><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover><mo>†</mo></msup><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover></mrow><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">|\\text{Tr}(\\hat{\\rho}\\hat{A}^\\dagger\\hat{B})|^2 = |\\braket{\\hat{A}^\\dagger \\hat{B}}|^2 \\leq \\braket{\\hat{A}^\\dagger\\hat{A}}\\braket{\\hat{B}^\\dagger\\hat{B}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord\">∣</span><span class=\"mord text\"><span class=\"mord\">Tr</span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord\">∣</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord\">∣</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\">∣</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≤</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"2","html_id":"cauchy-schwarz-inequality-always","key":"rNGsJVYhBG"}],"key":"SvOAZbeTVZ"},{"type":"paragraph","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"children":[{"type":"text","value":"Note that this inequality is ","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"yKpb1RP0Lb"},{"type":"strong","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"children":[{"type":"text","value":"always true","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"vxVV7ETPqC"}],"key":"iZx2v2CIsW"},{"type":"text","value":" and is never violated. As we shall see after, the so-called “violation of the classical Cauchy-Schwarz inequality” as an entanglement witness is different from this inequality. It refers to the violation of the latter ","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"BDbMOJcgGg"},{"type":"emphasis","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"children":[{"type":"text","value":"normally ordered","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"B2lvcCk6zz"}],"key":"wBTY5kw1hT"},{"type":"text","value":" inequality.","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"dV8PuOmdkD"}],"key":"RcHUamZm0O"},{"type":"heading","depth":2,"position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"children":[{"type":"text","value":"The Wigner function","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"key":"OUs9ObSnkV"}],"identifier":"wigner_function","label":"wigner_function","html_id":"wigner-function","enumerator":"2","key":"eNGJadW3nH"},{"type":"paragraph","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"children":[{"type":"text","value":"The Wigner function ","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"isk46o3pOr"},{"type":"inlineMath","value":"W(x,p)","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>W</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">W(x,p)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span></span></span></span>","key":"tQ3akaY1e5"},{"type":"text","value":", introduced by ","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"lBNBIy9WL1"},{"type":"cite","identifier":"wigner_quantum_1932","label":"wigner_quantum_1932","kind":"narrative","position":{"start":{"line":56,"column":45},"end":{"line":56,"column":65}},"children":[{"type":"text","value":"Wigner (1932)","key":"UNMKWiyqNY"}],"enumerator":"3","key":"Z8moNHnZJf"},{"type":"text","value":", represents the quasiprobability distributions of the state in the phase space ","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"uHksE5GRi3"},{"type":"inlineMath","value":"(x,p)","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(x,p)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span></span></span></span>","key":"jDRzCHwpI3"},{"type":"text","value":".","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"HzjIUjOA61"}],"key":"uhQf3QnFL2"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Wigner function","position":{"start":{"line":57,"column":1},"end":{"line":57,"column":1}},"key":"fWJFr7I5nS"}],"key":"GaFx77pcOZ"},{"type":"paragraph","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"children":[{"type":"text","value":"The Wigner function of a state ","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"key":"W3Q4Y2etj5"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"zRY36djMf8"},{"type":"text","value":" is defined as ","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"key":"eSBzz5FOVA"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"children":[{"type":"cite","identifier":"leonhardt_essential_2010","label":"leonhardt_essential_2010","kind":"parenthetical","position":{"start":{"line":59,"column":60},"end":{"line":59,"column":85}},"children":[{"type":"text","value":"Leonhardt, 2010","key":"X5R07RjGl0"}],"enumerator":"4","key":"TFCYeoWOzu"}],"key":"FS2ds1Q0yn"}],"key":"hYFjiEbuKc"},{"type":"math","identifier":"wigner_function_def","label":"wigner_function_def","value":"W(x,p) =\\frac{1}{2\\pi} \\int  e^{ipy/\\hbar}\\braket{x-y/2|\\hat{\\rho}|x+y/2}\\text{d}y.","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>W</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mo>∫</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>p</mi><mi>y</mi><mi mathvariant=\"normal\">/</mi><mi mathvariant=\"normal\">ℏ</mi></mrow></msup><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mi mathvariant=\"normal\">∣</mi><mi>x</mi><mo>+</mo><mi>y</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mtext>d</mtext><mi>y</mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">W(x,p) =\\frac{1}{2\\pi} \\int  e^{ipy/\\hbar}\\braket{x-y/2|\\hat{\\rho}|x+y/2}\\text{d}y.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2222em;vertical-align:-0.8622em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">π</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.938em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mathnormal mtight\">p</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord mtight\">/ℏ</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">/2∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">/2</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"3","html_id":"wigner-function-def","key":"WTCFgIspbv"}],"key":"AJ3TGFed9Q"},{"type":"proof","kind":"remark","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"The Wigner function: a quasiprobability distribution function","position":{"start":{"line":66,"column":1},"end":{"line":66,"column":1}},"key":"ypo4vHQW5T"}],"key":"lfU6KWsLSy"},{"type":"paragraph","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"children":[{"type":"text","value":"The Wigner function is quasiprobability function as its integral over ","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"key":"UoBTFYCUhB"},{"type":"inlineMath","value":"x","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"O9DtLbBTOz"},{"type":"text","value":" (resp ","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"key":"BzVfwJ0kbO"},{"type":"inlineMath","value":"p","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"B3cRQJ05oM"},{"type":"text","value":") gives the probability distribution of the state in ","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"key":"DC1lDbGekD"},{"type":"inlineMath","value":"p","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"LkxWIpvd2d"},{"type":"text","value":" (resp ","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"key":"bhwv0h8YVW"},{"type":"inlineMath","value":"x","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"WypctGSLTD"},{"type":"text","value":")","position":{"start":{"line":69,"column":1},"end":{"line":69,"column":1}},"key":"SslOYqkb4z"}],"key":"vtV5wiD7mm"},{"type":"math","value":"\\begin{split}\n\\braket{x | \\hat{\\rho}|x} =& \\int W(x,p) \\text{d}p \\\\\n\\braket{ p| \\hat{\\rho}|p} =& \\int W(x,p) \\text{d}x  \\\\\n\\end{split}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mi>x</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mi mathvariant=\"normal\">∣</mi><mi>x</mi></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mo>=</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow></mrow><mo>∫</mo><mi>W</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mtext>d</mtext><mi>p</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mi>p</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mi mathvariant=\"normal\">∣</mi><mi>p</mi></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mo>=</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow></mrow><mo>∫</mo><mi>W</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mtext>d</mtext><mi>x</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding=\"application/x-tex\">\\begin{split}\n\\braket{x | \\hat{\\rho}|x} =&amp; \\int W(x,p) \\text{d}p \\\\\n\\braket{ p| \\hat{\\rho}|p} =&amp; \\int W(x,p) \\text{d}x  \\\\\n\\end{split}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:5.0445em;vertical-align:-2.2722em;\"></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7723em;\"><span style=\"top:-4.7723em;\"><span class=\"pstrut\" style=\"height:3.36em;\"></span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\">x</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span></span></span><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.36em;\"></span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\">p</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.2722em;\"><span></span></span></span></span></span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.7723em;\"><span style=\"top:-4.7723em;\"><span class=\"pstrut\" style=\"height:3.36em;\"></span><span class=\"mord\"><span class=\"mord\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord mathnormal\">p</span></span></span><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.36em;\"></span><span class=\"mord\"><span class=\"mord\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord mathnormal\">x</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.2722em;\"><span></span></span></span></span></span></span></span></span></span></span></span>","enumerator":"4","key":"R3lLm0ankX"}],"key":"aLXKEufbcK"},{"type":"comment","value":"where $x$ and $p$ are the quadrature operators $\\hat{x} = (\\hat{a}+\\hat{a}^{\\dagger})/\\sqrt{2}$ and $\\hat{p} = i(\\hat{a}^\\dagger-\\hat{a})/\\sqrt{2}$.","position":{"start":{"line":77,"column":1},"end":{"line":77,"column":1}},"key":"h4RWcCWc4O"},{"type":"paragraph","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"and is often referred to as a quasi-probability distribution. The reason is that the projection along any axis (","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"ujVzaRrQ1c"},{"type":"inlineMath","value":"x,p","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">x,p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"oZxBh6A3ri"},{"type":"text","value":") gives the probability distribution along this axis. Wigner functions can be measured using homodyne detection measurement as proposed by ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"cwFRHqlvf6"},{"type":"cite","identifier":"vogel_determination_1989","label":"vogel_determination_1989","kind":"narrative","position":{"start":{"line":80,"column":257},"end":{"line":80,"column":282}},"children":[{"type":"text","value":"Vogel & Risken (1989)","key":"SaxVfM5KAC"}],"enumerator":"5","key":"auoE5X7kQR"},{"type":"text","value":". The first experimental measurement of a Wigner function was realized a few years later by ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"HzmyW8Bsat"},{"type":"cite","identifier":"smithey_measurement_1993","label":"smithey_measurement_1993","kind":"narrative","position":{"start":{"line":80,"column":374},"end":{"line":80,"column":399}},"children":[{"type":"text","value":"Smithey ","key":"aaA3AEHt6Q"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"RvX00zFmF5"}],"key":"gCUY6r00Ep"},{"type":"text","value":" (1993)","key":"NFaPS8hUnr"}],"enumerator":"6","key":"oVAKpiTvqo"},{"type":"text","value":", who measured the vacuum state ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"SoaQdVU27F"},{"type":"inlineMath","value":"\\ket{0}","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mn>0</mn><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{0}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\">0</span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"EqQ0i5aT74"},{"type":"text","value":" and a squeezed vacuum state","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"umpXOy9PZL"},{"type":"footnoteReference","identifier":"note_wigner_helium","label":"note_wigner_helium","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"number":2,"enumerator":"2","key":"H55DbAGqy7"},{"type":"text","value":". Their experimental measurement is reproduced in the first subplot of ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"t6PCtXDlH2"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"Figure ","key":"QWyo0TQBBj"},{"type":"text","value":"1","key":"uW2aJ9MEw8"}],"identifier":"wigner_experimental","label":"wigner_experimental","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"wigner-experimental","key":"hm2MaWm60L"},{"type":"text","value":". After this pioneering measurement, more and more complex states were measured: ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"cUeeUM6Ytw"},{"type":"cite","identifier":"lvovsky_quantum_2001","label":"lvovsky_quantum_2001","kind":"narrative","position":{"start":{"line":80,"column":665},"end":{"line":80,"column":686}},"children":[{"type":"text","value":"Lvovsky ","key":"zIlQutcfqM"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"YD38vxgFOS"}],"key":"RbVwdpaFMk"},{"type":"text","value":" (2001)","key":"W3ov8VuHl1"}],"enumerator":"7","key":"rJCB0qeauB"},{"type":"text","value":", ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"vvdIXG9Oyp"},{"type":"cite","identifier":"ourjoumtsev_quantum_2006","label":"ourjoumtsev_quantum_2006","kind":"narrative","position":{"start":{"line":80,"column":688},"end":{"line":80,"column":713}},"children":[{"type":"text","value":"Ourjoumtsev ","key":"b5XJjfhh3p"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"PSISulnQA3"}],"key":"XjImu9ds0h"},{"type":"text","value":" (2006)","key":"yG00GvjpJR"}],"enumerator":"8","key":"Gpm3owudbu"},{"type":"text","value":", and ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"kiOWNB5cId"},{"type":"cite","identifier":"cooper_experimental_2013","label":"cooper_experimental_2013","kind":"narrative","position":{"start":{"line":80,"column":719},"end":{"line":80,"column":744}},"children":[{"type":"text","value":"Cooper ","key":"qgDFftMJOd"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"MxPTusYDdg"}],"key":"BQ4dBZd2AO"},{"type":"text","value":" (2013)","key":"Cs9QpyS67f"}],"enumerator":"9","key":"MGy9fpN6g3"},{"type":"text","value":" respectively measured the one, two, and three photon Fock states before producing and measuring ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"wOe1EpERZO"},{"type":"link","url":"https://en.wikipedia.org/wiki/Cat_state","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"cat states","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"B2oAiBk2bW"}],"urlSource":"https://en.wikipedia.org/wiki/Cat_state","data":{"page":"Cat_state","wiki":"https://en.wikipedia.org/","lang":"en"},"internal":false,"protocol":"wiki","key":"OfuRfRWgvf"},{"type":"text","value":" ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"MatxV5puxe"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"cite","identifier":"ourjoumtsev_generating_2006","label":"ourjoumtsev_generating_2006","kind":"parenthetical","position":{"start":{"line":80,"column":896},"end":{"line":80,"column":924}},"children":[{"type":"text","value":"Ourjoumtsev ","key":"iSGdf92U0X"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"YUPCEgQNjQ"}],"key":"O5D7k31cMj"},{"type":"text","value":", 2006","key":"ECmxWY08dW"}],"enumerator":"10","key":"nIOonkq6sL"}],"key":"amLrM7suMX"},{"type":"text","value":" ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"Q8sTt9dvRk"},{"type":"footnoteReference","identifier":"recent_developments","label":"recent_developments","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"number":3,"enumerator":"3","key":"B2fOeVheyz"},{"type":"text","value":".","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"FTDVTE61aZ"}],"key":"WA8aSnAN8z"},{"type":"comment","value":" \nReconstruction of the Wigner function of four quantum states: the vacuum $\\ket{0}$, a squeezed vacuum and two Fock states with occupation number 1 and 2. More details about homodyne detection can be found in the fifth chapter of @leonhardt_essential_2010.\\ \nThe Wigner function of the vacuum state $\\ket{0}$ is Gaussian $W(x,p)=\\frac{1}{\\pi}e^{-x^2 - p^2}$ [^vacuum_wigner_function].\nThe vacuum $\\ket{0}$ being defined such that $\\hat{a}\\ket{0}=0$, one can deduce the vacuum wave function $\\psi_0(q)$ using  The Wigner function of vacuum and a quadrature squeezed state of a mode of an electromagnetical field @smithey_measurement_1993\n\n[^vacuum_wigner_function]: To derive the vacuum Wigner function, one can use its wave-function $\\psi_0(x)$. The vacuum $\\ket{0}$ is defined such that $\\hat{a}\\ket{0}=0$ where the annihilation operator is $\\hat{a} = (\\hat{x}+i\\hat{p})/\\sqrt{2}$. Given that in units where $m=\\hbar=1$, the definition of the annihilation operator leads to $(x+\\partial_x)\\psi_0 = 0$, *i.e.* a Gaussian wave-function. Inserting $\\psi_0$ in [](#wigner_function_def) leads to the Wigner distribution of vacuum. \n","key":"CaVEGecDwz"},{"type":"container","kind":"figure","identifier":"wigner_experimental","label":"wigner_experimental","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/wigner_functions_exp-c91cc271360d834816496cbf13b7adad.png","alt":"Wigner functions experimentally measured.","width":"100%","align":"center","key":"SBHw8Gj5TY","urlSource":"images/wigner_functions_experimental.png","urlOptimized":"/~gondret/phd_manuscript/build/wigner_functions_exp-c91cc271360d834816496cbf13b7adad.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"wigner_experimental","identifier":"wigner_experimental","html_id":"wigner-experimental","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"g6dFIkZPAJ"},{"type":"text","value":"1","key":"iAINncPX8b"},{"type":"text","value":":","key":"JfP1ALmxtv"}],"template":"Figure %s:","key":"CMeWVAx2sZ"},{"type":"text","value":"Wigner function reconstructed from experimental data. Sub-figure (1) represents the first measured Wigner distribution by ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"OnIdCF8A1b"},{"type":"cite","identifier":"smithey_measurement_1993","label":"smithey_measurement_1993","kind":"narrative","position":{"start":{"line":100,"column":123},"end":{"line":100,"column":148}},"children":[{"type":"text","value":"Smithey ","key":"DMafPiclpd"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"tnCHG1qrbf"}],"key":"m3wu8udM8J"},{"type":"text","value":" (1993)","key":"FpuKAZN27O"}],"enumerator":"6","key":"nhDTbSffyM"},{"type":"text","value":" of a squeezed state (left, a-c) and of a vacuum state (right, b-d). The squeezed state is elliptical (squeezed in the X quadrature) compared to the vacuum state whose shape is circular. Sub-figure (2): Wigner distribution of a  Be","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"nhP03WQEav"},{"type":"superscript","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"children":[{"type":"text","value":"+","key":"yiPVH53IAe"}],"key":"jRmpJwHK3Z"},{"type":"text","value":" ion in the Fock state ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"vORAPMwb1r"},{"type":"inlineMath","value":"\\ket{1}","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mn>1</mn><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{1}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\">1</span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"zR6W89kM1Y"},{"type":"text","value":" measured by ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"g8lSEAFD4x"},{"type":"cite","identifier":"leibfried_experimental_1996","label":"leibfried_experimental_1996","kind":"narrative","position":{"start":{"line":100,"column":428},"end":{"line":100,"column":456}},"children":[{"type":"text","value":"Leibfried ","key":"kq0IOc7FGJ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OZSDRO8FOb"}],"key":"eEitqRnGbT"},{"type":"text","value":" (1996)","key":"SGcmMFTCsj"}],"enumerator":"11","key":"xdTeeH5QTP"},{"type":"text","value":" and (3) of free-propagating photons in the Fock state ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"G02IpgKkRc"},{"type":"inlineMath","value":"\\ket{2}","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mn>2</mn><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\">2</span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"Vfs4sSBxe3"},{"type":"text","value":" by ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"VgoWiX8b6s"},{"type":"cite","identifier":"ourjoumtsev_quantum_2006","label":"ourjoumtsev_quantum_2006","kind":"narrative","position":{"start":{"line":100,"column":524},"end":{"line":100,"column":549}},"children":[{"type":"text","value":"Ourjoumtsev ","key":"pQjbbd99Iw"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"mA8IkS3jBJ"}],"key":"pGoLrEEFIa"},{"type":"text","value":" (2006)","key":"DcKyQmtSX3"}],"enumerator":"8","key":"AnmGqYjqBf"},{"type":"text","value":". The Wigner functions of (1) are Gaussian while  (2) and (3) are obviously not Gaussian. The negative value taken by the Wigner function is a signature of the quantumness of the state, that cannot be mimic by any classical-like state.  ©Figure from ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"IJPNU2ypJS"},{"type":"cite","identifier":"smithey_measurement_1993","label":"smithey_measurement_1993","kind":"narrative","position":{"start":{"line":100,"column":799},"end":{"line":100,"column":824}},"children":[{"type":"text","value":"Smithey ","key":"f4MdjfNoNU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"GLGT5J0xIx"}],"key":"UDWWblZNn6"},{"type":"text","value":" (1993)","key":"ElVn3NsXWm"}],"enumerator":"6","key":"WWgr3BXwQm"},{"type":"text","value":", ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"anuB02WfcM"},{"type":"cite","identifier":"leibfried_experimental_1996","label":"leibfried_experimental_1996","kind":"narrative","position":{"start":{"line":100,"column":826},"end":{"line":100,"column":854}},"children":[{"type":"text","value":"Leibfried ","key":"R5gFId27wc"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"bFaLKnd3Xp"}],"key":"G17r3tGCdx"},{"type":"text","value":" (1996)","key":"VXQoaLkUdU"}],"enumerator":"11","key":"KKUHyHNAZ5"},{"type":"text","value":" and ","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"eSihuIaG4B"},{"type":"cite","identifier":"ourjoumtsev_etude_2007","label":"ourjoumtsev_etude_2007","kind":"narrative","position":{"start":{"line":100,"column":859},"end":{"line":100,"column":882}},"children":[{"type":"text","value":"Ourjoumtsev (2007)","key":"qpSoGFLwF0"}],"enumerator":"12","key":"sC5Li0Nnl3"},{"type":"text","value":".","position":{"start":{"line":100,"column":1},"end":{"line":100,"column":1}},"key":"Yl5RI6PWOv"}],"key":"ooEgvFy34k"}],"key":"TQtXS9rw7r"}],"enumerator":"1","html_id":"wigner-experimental","key":"qBcyFQu7Cq"},{"type":"paragraph","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"children":[{"type":"emphasis","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"children":[{"type":"text","value":"What is the Wigner function of the vacuum ?","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"wlaBaYKLPr"}],"key":"jvIcSxfRFI"},{"type":"text","value":" The Wigner function definition ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"OoyYEvA6cd"},{"type":"crossReference","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"children":[{"type":"text","value":"(","key":"YfCvW8btDE"},{"type":"text","value":"3","key":"G0mQHakGBh"},{"type":"text","value":")","key":"gg8xr8t20i"}],"identifier":"wigner_function_def","label":"wigner_function_def","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"wigner-function-def","key":"eWJHHgS34R"},{"type":"text","value":" involves the wave-function of the vacuum state that can be deduced by the definition of the annihilation ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"jMql6PkZQU"},{"type":"inlineMath","value":"\\hat{a} = (\\hat{x}+i\\hat{p})/\\sqrt{2}","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mo>=</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mo>+</mo><mi>i</mi><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mo stretchy=\"false\">)</mo><mi mathvariant=\"normal\">/</mi><msqrt><mn>2</mn></msqrt></mrow><annotation encoding=\"application/x-tex\">\\hat{a} = (\\hat{x}+i\\hat{p})/\\sqrt{2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1572em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">/</span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9072em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span></span></span><span style=\"top:-2.8672em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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Indeed, the action of the latter on the vacuum gives 0 and noting that ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"rYr4u8I4KH"},{"type":"inlineMath","value":"\\hat{p}=i\\partial_x","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mo>=</mo><mi>i</mi><msub><mi mathvariant=\"normal\">∂</mi><mi>x</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\hat{p}=i\\partial_x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord\" style=\"margin-right:0.05556em;\">∂</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"uUycmloRMh"},{"type":"text","value":" in the ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"lKwLNOG9yv"},{"type":"inlineMath","value":"x","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"r9rLhVhO5O"},{"type":"text","value":" representation, the vacuum wave-function ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"LerfMf2cfr"},{"type":"inlineMath","value":"\\psi_0(x)","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ψ</mi><mn>0</mn></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\psi_0(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span></span></span></span>","key":"CZDihagpGl"},{"type":"text","value":" satisfies ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"gK7w0RCOFL"},{"type":"inlineMath","value":"(x+\\partial_x)\\psi_0 = 0","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo>+</mo><msub><mi mathvariant=\"normal\">∂</mi><mi>x</mi></msub><mo stretchy=\"false\">)</mo><msub><mi>ψ</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">(x+\\partial_x)\\psi_0 = 0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\" style=\"margin-right:0.05556em;\">∂</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0</span></span></span></span>","key":"nuTQcafrun"},{"type":"text","value":", ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"BDk9P5jutR"},{"type":"emphasis","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"QTovaf5Rba"}],"key":"ORCMtiS5pI"},{"type":"text","value":" it is Gaussian.  Inserting ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"or5wy39eN6"},{"type":"inlineMath","value":"\\psi_0","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ψ</mi><mn>0</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\psi_0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"niAarrGa3p"},{"type":"text","value":" in equation ","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"ECb0zD9ghI"},{"type":"crossReference","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"children":[{"type":"text","value":"(","key":"HSqI2nUReN"},{"type":"text","value":"3","key":"TH3WBFBo7f"},{"type":"text","value":")","key":"WcvsYf9pKy"}],"identifier":"wigner_function_def","label":"wigner_function_def","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"wigner-function-def","key":"g8NdWobGvz"},{"type":"text","value":" leads to the Wigner distribution of the vacuum state","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"nTxLNJMkK9"}],"key":"lvrW3E6oYR"},{"type":"math","identifier":"wigner_function_vacuum","label":"wigner_function_vacuum","value":"W_0(x,p) =\\frac{1}{\\pi}\\exp\\left(-x^2 - p^2\\right)","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi>W</mi><mn>0</mn></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">(</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>p</mi><mn>2</mn></msup><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">W_0(x,p) =\\frac{1}{\\pi}\\exp\\left(-x^2 - p^2\\right)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">π</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(</span></span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)</span></span></span></span></span></span></span>","enumerator":"5","html_id":"wigner-function-vacuum","key":"w5HHMglSol"},{"type":"paragraph","position":{"start":{"line":109,"column":1},"end":{"line":109,"column":1}},"children":[{"type":"text","value":"that is Gaussian. As we shall see, Gaussian states are transformed into other Gaussian states by second order transformations, hence their simple Wigner function simplifies their study. In fact, the Wigner function of a state is just a way to describe this state, and we can recover the expectation value of any operator from its Wigner function. To do so, we need to introduce the Weyl transform of an operator.","position":{"start":{"line":109,"column":1},"end":{"line":109,"column":1}},"key":"a0PIb0cxWq"}],"key":"lhnVmNWQxh"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Weyl transform","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"lIqlv9cwvm"}],"key":"eIg4IxcRd4"},{"type":"paragraph","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"children":[{"type":"text","value":"The Weyl transform of an operator ","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"key":"AsU6ofJhAV"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"cGDtdeyalz"},{"type":"text","value":" is defined by ","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"key":"AJMN5agfBe"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"children":[{"type":"cite","identifier":"weyl_theory_1950","label":"weyl_theory_1950","kind":"parenthetical","position":{"start":{"line":112,"column":60},"end":{"line":112,"column":77}},"children":[{"type":"text","value":"Weyl, 1950","key":"FY9mN9kVZ4"}],"enumerator":"13","key":"HcbF820BF9"}],"key":"AsOLjqIPFp"}],"key":"ZJpqrTsy5f"},{"type":"math","identifier":"weyl_function_def","label":"weyl_function_def","value":"\\tilde{A}(x,p) = \\int e^{-iyp/\\hbar}\\bra{x+y/2}\\hat{A}\\ket{x-y/2}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>~</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>∫</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>y</mi><mi>p</mi><mi mathvariant=\"normal\">/</mi><mi mathvariant=\"normal\">ℏ</mi></mrow></msup><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><mi mathvariant=\"normal\">∣</mi></mpadded><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mpadded><mi mathvariant=\"normal\">∣</mi><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\tilde{A}(x,p) = \\int e^{-iyp/\\hbar}\\bra{x+y/2}\\hat{A}\\ket{x-y/2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1702em;vertical-align:-0.25em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9202em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.6023em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">~</span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2222em;vertical-align:-0.8622em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.938em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord mathnormal mtight\">p</span><span class=\"mord mtight\">/ℏ</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">/2</span></span><span class=\"mord\">∣</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">/2</span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"6","html_id":"weyl-function-def","key":"kkxwE2uBkd"}],"key":"v0jbAcCwst"},{"type":"paragraph","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"children":[{"type":"text","value":"where the similarity with the definition of the Wigner ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"tmBma8J4I3"},{"type":"crossReference","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"children":[{"type":"text","value":"(","key":"c2Cbb3MXEO"},{"type":"text","value":"3","key":"MiwQ1nesPB"},{"type":"text","value":")","key":"LANmkoPBwO"}],"identifier":"wigner_function_def","label":"wigner_function_def","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"wigner-function-def","key":"YE81N4EMLA"},{"type":"text","value":" appears clearly. Here, we expressed the operator ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"jIYNcSQa4M"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"vXhIB0s43Y"},{"type":"text","value":" in the ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"cVXpTHW8Bk"},{"type":"inlineMath","value":"x","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"frcLxIFUff"},{"type":"text","value":" basis but it is possible also to express it in the ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"qlinpvnqv6"},{"type":"inlineMath","value":"p","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"I789yqCHNr"},{"type":"text","value":" basis, simply by flipping the role of ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"t5S6wvnuV3"},{"type":"inlineMath","value":"x ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x </annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"vK5BluJBCo"},{"type":"text","value":" and ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"u9JovKQSVi"},{"type":"inlineMath","value":" p","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\"> p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"LykT1IJbF1"},{"type":"text","value":" and the sign in the exponential ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"QHqicTHIob"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"children":[{"type":"cite","identifier":"case_wigner_2008","label":"case_wigner_2008","kind":"parenthetical","position":{"start":{"line":118,"column":291},"end":{"line":118,"column":308}},"children":[{"type":"text","value":"Case, 2008","key":"Wq06cEwx6i"}],"enumerator":"14","key":"lSLxv3Fb72"}],"key":"ROjdbkXFES"},{"type":"text","value":". The average value of any operator ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"oSV74AXolU"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"x3aIGakOyW"},{"type":"text","value":" over a state ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"dIWnmAQ3N9"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"yRwU0RHBYf"},{"type":"text","value":" is then given by the phase-space integral of the product of the Wigner function ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"gcRzBDF5ft"},{"type":"inlineMath","value":"W_\\rho","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>W</mi><mi>ρ</mi></msub></mrow><annotation encoding=\"application/x-tex\">W_\\rho</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">ρ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span></span>","key":"Q5T5vkNmAk"},{"type":"text","value":" of this state and the Weyl transform of the operator ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"fdxAAUWiKn"},{"type":"inlineMath","value":"\\hat{A}","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"hUmrGt0WPD"}],"key":"RfWM4WTFUv"},{"type":"math","identifier":"weyl_transform_average_value_operator","label":"weyl_transform_average_value_operator","value":"\\braket{\\hat{A}} = \\int W_\\rho (x,p)\\tilde{A}(x,p) \\text{d}x\\text{d}p.","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mpadded><mo stretchy=\"false\">⟨</mo><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo stretchy=\"false\">⟩</mo></mpadded><mo>=</mo><mo>∫</mo><msub><mi>W</mi><mi>ρ</mi></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mover accent=\"true\"><mi>A</mi><mo>~</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mtext>d</mtext><mi>x</mi><mtext>d</mtext><mi>p</mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">\\braket{\\hat{A}} = \\int W_\\rho (x,p)\\tilde{A}(x,p) \\text{d}x\\text{d}p.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2222em;vertical-align:-0.8622em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">ρ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9202em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.6023em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">~</span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord mathnormal\">x</span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord mathnormal\">p</span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"7","html_id":"weyl-transform-average-value-operator","key":"BCivqj2YD5"},{"type":"paragraph","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"children":[{"type":"text","value":"For example, the Weyl transform of the projector ","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"key":"GMjNmPPIbs"},{"type":"inlineMath","value":"\\ket{x}\\bra{x}","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mi>x</mi><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mi>x</mi><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{x}\\bra{x}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span><span class=\"mord\">∣</span></span></span></span></span>","key":"IRw9JOmF2v"},{"type":"text","value":" is simply ","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"key":"KlO8k9EYXM"},{"type":"inlineMath","value":"x","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"Sli1Rmf8hn"},{"type":"text","value":". An other example is the Weyl transform of the projector on the Fock basis ","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"key":"YEJyGCfW8X"},{"type":"inlineMath","value":"\\ket{j}\\bra{j}","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mi>j</mi><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mi>j</mi><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{j}\\bra{j}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05724em;\">j</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05724em;\">j</span></span><span class=\"mord\">∣</span></span></span></span></span>","key":"Q5XFqf2ZO1"},{"type":"text","value":" ","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"key":"OVOcSSAfF1"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":123,"column":1},"end":{"line":123,"column":1}},"children":[{"type":"cite","identifier":"leonhardt_essential_2010","label":"leonhardt_essential_2010","kind":"parenthetical","position":{"start":{"line":123,"column":174},"end":{"line":123,"column":199}},"children":[{"type":"text","value":"Leonhardt, 2010","key":"j3fNnOnPky"}],"enumerator":"4","key":"LsTxhEr0MY"}],"key":"KNwtbrEbmk"}],"key":"jlWMfvEZcd"},{"type":"math","identifier":"weyl_fock","label":"weyl_fock","value":"W_j(x, p) = \\frac{(-1)^j}{\\pi}e^{-x^2-p^2}\\mathcal{L}_j(2x^2+2p^2)","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi>W</mi><mi>j</mi></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>p</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mrow><mo stretchy=\"false\">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy=\"false\">)</mo><mi>j</mi></msup></mrow><mi>π</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>p</mi><mn>2</mn></msup></mrow></msup><msub><mi mathvariant=\"script\">L</mi><mi>j</mi></msub><mo stretchy=\"false\">(</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">W_j(x, p) = \\frac{(-1)^j}{\\pi}e^{-x^2-p^2}\\mathcal{L}_j(2x^2+2p^2)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.1877em;vertical-align:-0.686em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5017em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">π</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\">1</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8247em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0369em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913em;\"><span style=\"top:-2.931em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">p</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913em;\"><span style=\"top:-2.931em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathcal\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>","enumerator":"8","html_id":"weyl-fock","key":"LBVgZRQnNT"},{"type":"paragraph","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"key":"rhsdPuyCZ2"},{"type":"inlineMath","value":"\\mathcal{L}_j","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"script\">L</mi><mi>j</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\mathcal{L}_j</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathcal\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span></span>","key":"vrvfGTZSEn"},{"type":"text","value":" are the ","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"key":"Mm4wGNuGXx"},{"type":"link","url":"https://en.wikipedia.org/wiki/Laguerre_polynomials","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"children":[{"type":"text","value":"Laguerre polynomials","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"key":"htvYnwe7i5"}],"urlSource":"https://en.wikipedia.org/wiki/Laguerre_polynomials","data":{"page":"Laguerre_polynomials","wiki":"https://en.wikipedia.org/","lang":"en"},"internal":false,"protocol":"wiki","key":"t3yS30SVLE"},{"type":"text","value":".","position":{"start":{"line":128,"column":1},"end":{"line":128,"column":1}},"key":"T7J9XALUuf"}],"key":"GG2sMIgrWI"},{"type":"comment","value":" Depending on the symmetry of the operator *i.e.* if the operator is normally or anti-normally ordered, it exists also  ","key":"mDgIN97nPb"},{"type":"heading","depth":2,"position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"children":[{"type":"text","value":"Gaussian states","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"key":"eJjLiLo0ZH"}],"identifier":"subsection_gaussian_state","label":"subsection_gaussian_state","html_id":"subsection-gaussian-state","enumerator":"3","key":"SyeiCSr0cw"},{"type":"paragraph","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"children":[{"type":"text","value":"Hamiltonians of second order in creation and annihilation operators are central in physics. It is for example the case of the BCS","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"xwIPaZVaQo"},{"type":"footnoteReference","identifier":"footnote_bcs","label":"footnote_bcs","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"number":4,"enumerator":"4","key":"cFa7KaF5oX"},{"type":"text","value":" model for superconductivity or the Bogoliubov theory discussed in the last ","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"QFcBZGp38A"},{"type":"crossReference","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"children":[{"type":"text","value":"chapter","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"rnPGk88h9Z"}],"identifier":"bogoliubov_approx_section","label":"bogoliubov_approx_section","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"bogoliubov-approx-section","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"gQ8i9JJGg5"},{"type":"text","value":" (see ","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"EYc1m2BRVx"},{"type":"crossReference","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"children":[{"type":"text","value":"Section","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"vmC5zxF9U1"}],"identifier":"bogoliubov_approx_section","label":"bogoliubov_approx_section","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"bogoliubov-approx-section","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"JBGmCtabAc"},{"type":"text","value":" ","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"KHBqwpe7dY"},{"type":"crossReference","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"children":[{"type":"text","value":"1","key":"iCkMjg3Zw9"}],"identifier":"bogoliubov_approx_section","label":"bogoliubov_approx_section","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"bogoliubov-approx-section","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"cQps0p0Bb2"},{"type":"text","value":"). A Gaussian state that evolves under such Hamiltonian remains Gaussian. The vacuum, thermal states and coherent states are Gaussian: this means that the evolution of those states under second order hamiltonians are also Gaussian. This motivated the detailed study of ","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"IU4478VeKB"},{"type":"emphasis","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"children":[{"type":"text","value":"Gaussian states","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"vVEUpk63JL"}],"key":"EOKaSDwgIQ"},{"type":"text","value":" and the description of their evolution.","position":{"start":{"line":135,"column":1},"end":{"line":135,"column":1}},"key":"z1NdLqSTMs"}],"key":"fnFXLzvMDV"},{"type":"comment","value":"Indeed, in the phase space, the dimension is $2N$, where $N$ is the number of considered modes Dimension f Sp(2n, R) = n(2n+1) + n","position":{"start":{"line":136,"column":1},"end":{"line":136,"column":1}},"key":"zMBbIE95Au"},{"type":"paragraph","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"children":[{"type":"text","value":"Gaussian states are only defined by their first and second moment: a state of ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"fWw2NizdKX"},{"type":"inlineMath","value":"N","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"JCnie4vPbH"},{"type":"text","value":" Gaussian modes is characterized by its first moment vector ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"nGkx6iFsdd"},{"type":"inlineMath","value":"\\boldsymbol{\\mu}","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span></span></span></span>","key":"f6uDhf2vdQ"},{"type":"text","value":" and its covariance matrix ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"rZcU6ACFij"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma}","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span>","key":"cPOU04Ntbd"},{"type":"text","value":", defined in ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"OP08TJYAVq"},{"type":"crossReference","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"children":[{"type":"text","value":"(","key":"ZFEwPzCgSW"},{"type":"text","value":"9","key":"Iu4x39CeuP"},{"type":"text","value":")","key":"bHYmSLLMpp"}],"identifier":"covariance_matrix_def","label":"covariance_matrix_def","kind":"equation","template":"(%s)","enumerator":"9","resolved":true,"html_id":"covariance-matrix-def","key":"xGDb7yaDqd"},{"type":"text","value":". The size of the vector is ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"qWMSlzF4yw"},{"type":"inlineMath","value":"2N","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">2N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"EqVWW2iWTh"},{"type":"text","value":" and the covariance matrix is ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"YU4JL3LK9y"},{"type":"inlineMath","value":"N\\times N","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">N\\times N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">×</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"keTvUS3ZH5"},{"type":"text","value":". However, because of the canonical commutation rules and the Heisenberg uncertainty relation, this matrix belongs to the so-called symplectic group ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"CeWXKchrkF"},{"type":"inlineMath","value":"Sp(2N, \\mathbb{R})","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>S</mi><mi>p</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo separator=\"true\">,</mo><mi mathvariant=\"double-struck\">R</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">Sp(2N, \\mathbb{R})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">Sp</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathbb\">R</span><span class=\"mclose\">)</span></span></span></span>","key":"OlzpACMTv4"},{"type":"text","value":" with size ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"GmfRdzAwRb"},{"type":"inlineMath","value":"N(2N+1)","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">N(2N+1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span>","key":"vYOTwl62AT"},{"type":"text","value":" ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"Q9yzhXZmv7"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"children":[{"type":"cite","identifier":"arvind_real_1995","label":"arvind_real_1995","kind":"parenthetical","position":{"start":{"line":137,"column":511},"end":{"line":137,"column":528}},"children":[{"type":"text","value":"Arvind ","key":"N1JzrdIoxn"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"NMuWBZmFyV"}],"key":"vW8KvIgZTe"},{"type":"text","value":", 1995","key":"qyy26rNaml"}],"enumerator":"15","key":"h8U6imWNNc"}],"key":"uAoQQnOMLC"},{"type":"text","value":". The Gaussian state formalism is therefore quite practical as it avoids working with an infinite Hilbert space (for example the Fock basis ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"uSinved8t2"},{"type":"inlineMath","value":"\\{\\ket{n}\\}","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">{</mo><mpadded><mi mathvariant=\"normal\">∣</mi><mi>n</mi><mo stretchy=\"false\">⟩</mo></mpadded><mo stretchy=\"false\">}</mo></mrow><annotation encoding=\"application/x-tex\">\\{\\ket{n}\\}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">{</span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">n</span></span><span class=\"mclose\">⟩</span></span><span class=\"mclose\">}</span></span></span></span>","key":"LWO91hZ0XZ"},{"type":"text","value":") but rather with ","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"QjOjC46yoG"},{"type":"inlineMath","value":"N(2N+3)","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>3</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">N(2N+3)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">3</span><span class=\"mclose\">)</span></span></span></span>","key":"vTdcTcrOG1"},{"type":"text","value":" parameters (degree of freedom for the covariance matrix and the mean vector).","position":{"start":{"line":137,"column":1},"end":{"line":137,"column":1}},"key":"r6jiXM9dgV"}],"key":"V40VMAein9"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Literature references","position":{"start":{"line":143,"column":1},"end":{"line":143,"column":1}},"key":"DdhsgYoPPN"}],"key":"o2Oti2wKgy"},{"type":"paragraph","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"children":[{"type":"text","value":"This section is an aggregate of many references. Among them, the comprehensive description of Gaussian states offered in the book by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"KUfMu03BK6"},{"type":"cite","identifier":"serafini_quantum_2017","label":"serafini_quantum_2017","kind":"narrative","position":{"start":{"line":144,"column":134},"end":{"line":144,"column":156}},"children":[{"type":"text","value":"Serafini (2017)","key":"rRnFOAfvLb"}],"enumerator":"16","key":"BXwvXGCHMU"},{"type":"text","value":" was highly utilized. The ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"vXICcLeF64"},{"type":"emphasis","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"children":[{"type":"text","value":"Analog Gravity in Benasque","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"ezj0jhv0KO"}],"key":"BW5JAMk7aY"},{"type":"text","value":" lecture notes by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"mVoFSlCVAl"},{"type":"cite","identifier":"brady_gaussian_2023","label":"brady_gaussian_2023","kind":"narrative","position":{"start":{"line":144,"column":228},"end":{"line":144,"column":248}},"children":[{"type":"text","value":"Brady (2023)","key":"FhvOQYyQBa"}],"enumerator":"17","key":"dbH50jOoFm"},{"type":"text","value":" were also fundamental in my comprehension of Gaussian formulism. In the same vein but more concise, the review articles by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"YclIucJd98"},{"type":"cite","identifier":"weedbrook_gaussian_2012","label":"weedbrook_gaussian_2012","kind":"narrative","position":{"start":{"line":144,"column":372},"end":{"line":144,"column":396}},"children":[{"type":"text","value":"Weedbrook ","key":"BuDuEXTuDm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OB0mPwMB62"}],"key":"SIIoyo469o"},{"type":"text","value":" (2012)","key":"WGXyhtDDJK"}],"enumerator":"18","key":"bYhpPxYGGS"},{"type":"text","value":", ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"QxQyI7K6FV"},{"type":"cite","identifier":"adesso_continuous_2014","label":"adesso_continuous_2014","kind":"narrative","position":{"start":{"line":144,"column":398},"end":{"line":144,"column":421}},"children":[{"type":"text","value":"Adesso ","key":"Ob0SfHmoo3"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"xViG2B3Zl2"}],"key":"y5TvaAbEvr"},{"type":"text","value":" (2014)","key":"b7Spe6vQMS"}],"enumerator":"19","key":"RNtdOKVC9x"},{"type":"text","value":", and ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"hp2HHSzFZI"},{"type":"cite","identifier":"braunstein_quantum_2005","label":"braunstein_quantum_2005","kind":"narrative","position":{"start":{"line":144,"column":427},"end":{"line":144,"column":451}},"children":[{"type":"text","value":"Braunstein & Loock (2005)","key":"P8bhQ4U3o6"}],"enumerator":"20","key":"gpf8piRqBu"},{"type":"text","value":" provide a really good overview of the literature. I also used and recommend the pedagogical article by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"vgzXQLAJB4"},{"type":"cite","identifier":"case_wigner_2008","label":"case_wigner_2008","kind":"narrative","position":{"start":{"line":144,"column":555},"end":{"line":144,"column":572}},"children":[{"type":"text","value":"Case (2008)","key":"QZc9bs5GwR"}],"enumerator":"14","key":"c7HHOwijHu"},{"type":"text","value":", which introduces ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"McL7EBAJfl"},{"type":"emphasis","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"children":[{"type":"text","value":"Wigner functions and Weyl transforms for pedestrians","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"hyI99G2B2z"}],"key":"QE6G6kmyK1"},{"type":"text","value":". Note finally the excellent note by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"FEMaqZ0mht"},{"type":"cite","identifier":"brask_gaussian_2022","label":"brask_gaussian_2022","kind":"narrative","position":{"start":{"line":144,"column":682},"end":{"line":144,"column":702}},"children":[{"type":"text","value":"Brask (2022)","key":"AyoE5UHPgm"}],"enumerator":"21","key":"r2ky16adiH"},{"type":"text","value":" that summarizes well the topic. From a mathematical perspective, the review article by ","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"b7Wqc00QvC"},{"type":"cite","identifier":"arvind_real_1995","label":"arvind_real_1995","kind":"narrative","position":{"start":{"line":144,"column":790},"end":{"line":144,"column":807}},"children":[{"type":"text","value":"Arvind ","key":"b14AqXIQGY"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"dv3PfA6dTV"}],"key":"THKF8D3oJg"},{"type":"text","value":" (1995)","key":"nS5OL6Wo7E"}],"enumerator":"15","key":"CmFI4nJhjy"},{"type":"text","value":" provides a complete and rigorous overview of the key properties of the real symplectic group.","position":{"start":{"line":144,"column":1},"end":{"line":144,"column":1}},"key":"CfwPunX5mf"}],"key":"doVW4pSXG8"},{"type":"paragraph","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"children":[{"type":"text","value":"For shorter references, many articles provide nice summaries. Among them, ","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"key":"RkQsJzRHEn"},{"type":"cite","identifier":"serafini_entanglement_2004","label":"serafini_entanglement_2004","kind":"narrative","position":{"start":{"line":146,"column":75},"end":{"line":146,"column":102}},"children":[{"type":"text","value":"Serafini ","key":"WoDtXrCndn"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"qLNybuE7No"}],"key":"gX1Qf8fDll"},{"type":"text","value":" (2004)","key":"rCIIGDdUvB"}],"enumerator":"22","key":"FVUHpWdIET"},{"type":"text","value":" and ","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"key":"WyZhh9zCqm"},{"type":"cite","identifier":"pirandola_correlation_2009","label":"pirandola_correlation_2009","kind":"narrative","position":{"start":{"line":146,"column":107},"end":{"line":146,"column":134}},"children":[{"type":"text","value":"Pirandola ","key":"THZZ606WFF"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"a7Qb7kLOa2"}],"key":"Qq0kii95qw"},{"type":"text","value":" (2009)","key":"PAvJNIsepw"}],"enumerator":"23","key":"fNpDm2RBPt"},{"type":"text","value":" introduce well the progress made in the early 2000s. The article by ","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"key":"awghOCE1Kc"},{"type":"cite","identifier":"brady_symplectic_2022","label":"brady_symplectic_2022","kind":"narrative","position":{"start":{"line":146,"column":203},"end":{"line":146,"column":225}},"children":[{"type":"text","value":"Brady ","key":"PUwrp74Krh"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"P51VPArawK"}],"key":"AZFxaxTcJd"},{"type":"text","value":" (2022)","key":"y1g8Q81Onu"}],"enumerator":"24","key":"KH4wfP8NHl"},{"type":"text","value":" pedagogically introduces symplectic circuits as a tool to model multi-mode scattering events in analog gravity. ","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"key":"b3w3r8dQy8"},{"type":"cite","identifier":"martin_comparing_2023","label":"martin_comparing_2023","kind":"narrative","position":{"start":{"line":146,"column":338},"end":{"line":146,"column":360}},"children":[{"type":"text","value":"Martin ","key":"c8pjEmAkGL"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ahVCiGYkl9"}],"key":"KdD6W7K6cx"},{"type":"text","value":" (2023)","key":"AbB55m1lkO"}],"enumerator":"25","key":"BStZLlsWlu"},{"type":"text","value":" focuses on the comparison of entanglement criteria and derives analytic expressions for the effective squeezing parameter and purity for a noisy or lossy two-mode squeezed thermal state.","position":{"start":{"line":146,"column":1},"end":{"line":146,"column":1}},"key":"E3lKH28dW6"}],"key":"c5pLSrzo0l"}],"key":"z9cNpZFBFA"},{"type":"paragraph","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"children":[{"type":"text","value":"In the following, we will group the canonical operator as ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"AB8LlxbeRu"},{"type":"inlineMath","value":"\\boldsymbol{\\hat{r}}=(\\hat{x}_1, \\hat{p}_1,..., \\hat{x}_N, \\hat{p}_N)","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi><mover accent=\"true\"><mi mathvariant=\"bold-italic\">r</mi><mo>^</mo></mover></mi><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mn>1</mn></msub><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">.</mi><mi mathvariant=\"normal\">.</mi><mi mathvariant=\"normal\">.</mi><mo separator=\"true\">,</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mi>N</mi></msub><mo separator=\"true\">,</mo><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mi>N</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\hat{r}}=(\\hat{x}_1, \\hat{p}_1,..., \\hat{x}_N, \\hat{p}_N)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7079em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7079em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span><span style=\"top:-3.0134em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2875em;\"><span class=\"mord mathbf\">^</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">...</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>","key":"KYrJ4hFGJf"},{"type":"text","value":", with a special focus on ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"MxoNGoiH2X"},{"type":"inlineMath","value":"N=2","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">N=2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">2</span></span></span></span>","key":"tf1yOKTGE2"},{"type":"text","value":"-mode systems. Here, ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"S0qHnCgh0m"},{"type":"inlineMath","value":"\\hat{x}","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{x}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"KzFEuaFZpt"},{"type":"text","value":" and ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"ZlcVn6nhUM"},{"type":"inlineMath","value":"\\hat{p}","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{p}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span></span></span></span>","key":"sGNnXvvFSv"},{"type":"text","value":" are related to the creation and annihilation operators ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"YvzcZ8pjlO"},{"type":"inlineMath","value":"\\hat{a}_j = (\\hat{x}_j+i\\hat{p}_j)/\\sqrt{2}","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mi>j</mi></msub><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mi>j</mi></msub><mo>+</mo><mi>i</mi><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mi>j</mi></msub><mo stretchy=\"false\">)</mo><mi mathvariant=\"normal\">/</mi><msqrt><mn>2</mn></msqrt></mrow><annotation encoding=\"application/x-tex\">\\hat{a}_j = (\\hat{x}_j+i\\hat{p}_j)/\\sqrt{2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1933em;vertical-align:-0.2861em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">/</span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9072em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span></span></span><span style=\"top:-2.8672em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1328em;\"><span></span></span></span></span></span></span></span></span>","key":"IDyLN10MwH"},{"type":"text","value":". The first moment vector and the covariance matrix of the state read ","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"key":"E3AgGjPPbR"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":148,"column":1},"end":{"line":148,"column":1}},"children":[{"type":"cite","identifier":"serafini_quantum_2017","label":"serafini_quantum_2017","kind":"parenthetical","position":{"start":{"line":148,"column":377},"end":{"line":148,"column":399}},"children":[{"type":"text","value":"Serafini, 2017","key":"GibyQx2smp"}],"enumerator":"16","key":"K3sp7CZROz"}],"key":"pGhATHOo7P"}],"key":"W24PbyUaZp"},{"type":"math","identifier":"covariance_matrix_def","label":"covariance_matrix_def","value":"\\mu_i= \\braket{\\hat{r}_i} \\quad ;\\quad \n\\sigma_{i,j} = \\braket{\\{\\Delta\\hat{r}_i,\\Delta\\hat{r}_j\\}}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi>μ</mi><mi>i</mi></msub><mo>=</mo><mpadded><mo stretchy=\"false\">⟨</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo stretchy=\"false\">⟩</mo></mpadded><mspace width=\"1em\"/><mo separator=\"true\">;</mo><mspace width=\"1em\"/><msub><mi>σ</mi><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mo stretchy=\"false\">{</mo><mi mathvariant=\"normal\">Δ</mi><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Δ</mi><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>j</mi></msub><mo stretchy=\"false\">}</mo></mrow><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\mu_i= \\braket{\\hat{r}_i} \\quad ;\\quad \n\\sigma_{i,j} = \\braket{\\{\\Delta\\hat{r}_i,\\Delta\\hat{r}_j\\}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mopen\">{</span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mclose\">}</span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"9","html_id":"covariance-matrix-def","key":"wrzsBRvAFF"},{"type":"paragraph","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"YdhnGSM0kn"},{"type":"inlineMath","value":"\\Delta\\hat{r}_i = \\hat{r}_i - \\braket{\\hat{r}_i}","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">Δ</mi><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo>=</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo>−</mo><mpadded><mo stretchy=\"false\">⟨</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\Delta\\hat{r}_i = \\hat{r}_i - \\braket{\\hat{r}_i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"></span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"RJfg69mZRk"},{"type":"text","value":" and the Poissonian bracket ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"rdmXqA1rR2"},{"type":"inlineMath","value":"\\{,\\}","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">{</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">}</mo></mrow><annotation encoding=\"application/x-tex\">\\{,\\}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">{</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mclose\">}</span></span></span></span>","key":"mUk7by9phZ"},{"type":"text","value":" was used as a shortcut to define the anti-commutator ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"MtUnCgJ2cB"},{"type":"inlineMath","value":"\\{\\hat{A}, \\hat{B}\\} := \\hat{A}\\hat{B} + \\hat{B}\\hat{A}","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">{</mo><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover><mo stretchy=\"false\">}</mo><mo>:</mo><mo>=</mo><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover><mo>+</mo><mover accent=\"true\"><mi>B</mi><mo>^</mo></mover><mover accent=\"true\"><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\{\\hat{A}, \\hat{B}\\} := \\hat{A}\\hat{B} + \\hat{B}\\hat{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mopen\">{</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mclose\">}</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">:=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0301em;vertical-align:-0.0833em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">A</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1111em;\"><span class=\"mord\">^</span></span></span></span></span></span></span></span></span></span>","key":"a4T1lYiydc"},{"type":"text","value":". Note that in the community, the definition of the covariance matrix differs depending on the value of ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"oT4eRBPlxE"},{"type":"inlineMath","value":"\\hbar","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">ℏ</mi></mrow><annotation encoding=\"application/x-tex\">\\hbar</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6889em;\"></span><span class=\"mord\">ℏ</span></span></span></span>","key":"q4RaxB9KEz"},{"type":"text","value":" : here we follow the notation of ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"ZyXLZeTTzE"},{"type":"cite","identifier":"serafini_quantum_2017","label":"serafini_quantum_2017","kind":"narrative","position":{"start":{"line":156,"column":348},"end":{"line":156,"column":370}},"children":[{"type":"text","value":"Serafini (2017)","key":"n8G1ld9L7d"}],"enumerator":"16","key":"QbViC355LX"},{"type":"text","value":", but one should be careful when using formulae from the literature. As explained in the introduction of the ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"smv9iz4Agb"},{"type":"link","url":"https://arxiv.org/pdf/1110.3234","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"arxiv","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"uMeWfs8XMY"}],"urlSource":"https://arxiv.org/pdf/1110.3234","key":"U0eX5dtAJH"},{"type":"text","value":" version of  ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"fq5CTQTgQj"},{"type":"cite","identifier":"weedbrook_gaussian_2012","label":"weedbrook_gaussian_2012","kind":"narrative","position":{"start":{"line":156,"column":532},"end":{"line":156,"column":556}},"children":[{"type":"text","value":"Weedbrook ","key":"SiEfBym73G"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"d61RVoH2as"}],"key":"erAieye993"},{"type":"text","value":" (2012)","key":"hplWaIhJHk"}],"enumerator":"18","key":"oIZiUGBqwX"},{"type":"text","value":", ‘‘there is no consensus about the value of the variance of the vacuum, with common choices being either 1/4 (","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"UKPovZ56lk"},{"type":"inlineMath","value":"\\hbar = 1/2","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">ℏ</mi><mo>=</mo><mn>1</mn><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\hbar = 1/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6889em;\"></span><span class=\"mord\">ℏ</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1/2</span></span></span></span>","key":"l8H890wSGz"},{"type":"text","value":"), 1/2 (","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"Z29rVJMc6n"},{"type":"inlineMath","value":"\\hbar = 1","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">ℏ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\hbar = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6889em;\"></span><span class=\"mord\">ℏ</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"B9JCi4sR5y"},{"type":"text","value":") or 1’’. This means that depending on the notation, the covariance matrix might be defined with a factor ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"rjkiyvXEam"},{"type":"inlineMath","value":"1/2","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>1</mn><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">1/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1/2</span></span></span></span>","key":"c6oOcalydO"},{"type":"text","value":". Throughout this chapter, we have in particular","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"KNrI9M3YLD"}],"key":"r6OvNJp6Ky"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":157,"column":1},"end":{"line":160,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":157,"column":1},"end":{"line":157,"column":1}},"children":[{"type":"inlineMath","value":"\\hbar = 1","position":{"start":{"line":157,"column":1},"end":{"line":157,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">ℏ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\hbar = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6889em;\"></span><span class=\"mord\">ℏ</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"Xbx97h2MqD"},{"type":"text","value":",","position":{"start":{"line":157,"column":1},"end":{"line":157,"column":1}},"key":"fcLAlQrgGF"}],"key":"EUDVOEbaD6"},{"type":"listItem","spread":true,"position":{"start":{"line":158,"column":1},"end":{"line":158,"column":1}},"children":[{"type":"inlineMath","value":"\\hat{a}_j = (\\hat{x}_j+i\\hat{p}_j)/\\sqrt{2}","position":{"start":{"line":158,"column":1},"end":{"line":158,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mi>j</mi></msub><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mi>j</mi></msub><mo>+</mo><mi>i</mi><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mi>j</mi></msub><mo stretchy=\"false\">)</mo><mi mathvariant=\"normal\">/</mi><msqrt><mn>2</mn></msqrt></mrow><annotation encoding=\"application/x-tex\">\\hat{a}_j = (\\hat{x}_j+i\\hat{p}_j)/\\sqrt{2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1933em;vertical-align:-0.2861em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">/</span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9072em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span></span></span><span style=\"top:-2.8672em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1328em;\"><span></span></span></span></span></span></span></span></span>","key":"zbvMZY41vQ"}],"key":"QXgjG4Y6YZ"},{"type":"listItem","spread":true,"position":{"start":{"line":159,"column":1},"end":{"line":160,"column":1}},"children":[{"type":"inlineMath","value":"\\sigma_{i,j} = \\braket{\\{\\Delta\\hat{r}_i,\\Delta\\hat{r}_j\\}}","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mo stretchy=\"false\">{</mo><mi mathvariant=\"normal\">Δ</mi><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Δ</mi><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>j</mi></msub><mo stretchy=\"false\">}</mo></mrow><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\sigma_{i,j} = \\braket{\\{\\Delta\\hat{r}_i,\\Delta\\hat{r}_j\\}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mopen\">{</span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mclose\">}</span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"xlzqRyvp5V"},{"type":"text","value":".","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"key":"e32ObBJAd1"}],"key":"CUDv9wjd4K"}],"key":"bYghJpqpGV"},{"type":"paragraph","position":{"start":{"line":161,"column":1},"end":{"line":161,"column":1}},"children":[{"type":"text","value":"With these conventions, the Wigner function of a Gaussian state is simply given by ","position":{"start":{"line":161,"column":1},"end":{"line":161,"column":1}},"key":"HLfxslMQ2P"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":161,"column":1},"end":{"line":161,"column":1}},"children":[{"type":"cite","identifier":"brady_gaussian_2023","label":"brady_gaussian_2023","kind":"parenthetical","position":{"start":{"line":161,"column":85},"end":{"line":161,"column":105}},"children":[{"type":"text","value":"Brady, 2023","key":"o0ystxFzoq"}],"enumerator":"17","key":"wZZgYLUIhh"}],"key":"SEXC64F6cl"}],"key":"AoI2WZk8fB"},{"type":"math","identifier":"wigner_brady","label":"wigner_brady","value":"W(\\boldsymbol{r}) = \\frac{1}{\\pi^N \\sqrt{\\text{Det}[\\boldsymbol{\\sigma}]}}\\exp\\left[ -(\\boldsymbol{r-\\mu})^\\intercal\\boldsymbol{\\sigma}^{-1}(\\boldsymbol{r-\\mu}) \\right].","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>W</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-italic\">r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>π</mi><mi>N</mi></msup><msqrt><mrow><mtext>Det</mtext><mo stretchy=\"false\">[</mo><mi mathvariant=\"bold-italic\">σ</mi><mo stretchy=\"false\">]</mo></mrow></msqrt></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">[</mo><mo>−</mo><mo stretchy=\"false\">(</mo><mi><mrow><mi mathvariant=\"bold-italic\">r</mi><mo mathvariant=\"bold-italic\">−</mo><mi mathvariant=\"bold-italic\">μ</mi></mrow></mi><msup><mo stretchy=\"false\">)</mo><mo>⊺</mo></msup><msup><mi mathvariant=\"bold-italic\">σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi><mrow><mi mathvariant=\"bold-italic\">r</mi><mo mathvariant=\"bold-italic\">−</mo><mi mathvariant=\"bold-italic\">μ</mi></mrow></mi><mo stretchy=\"false\">)</mo><mo fence=\"true\">]</mo></mrow><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">W(\\boldsymbol{r}) = \\frac{1}{\\pi^N \\sqrt{\\text{Det}[\\boldsymbol{\\sigma}]}}\\exp\\left[ -(\\boldsymbol{r-\\mu})^\\intercal\\boldsymbol{\\sigma}^{-1}(\\boldsymbol{r-\\mu}) \\right].</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4514em;vertical-align:-1.13em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.175em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">π</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7673em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span></span></span></span></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.935em;\"><span class=\"svg-align\" style=\"top:-3.2em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mord text\"><span class=\"mord\">Det</span></span><span class=\"mopen\">[</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mclose\">]</span></span></span><span style=\"top:-2.895em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.305em;\"><span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.13em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">[</span></span><span class=\"mord\">−</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin mathbf\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7144em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin mathbf\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mclose\">)</span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">]</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"10","html_id":"wigner-brady","key":"zwPP6jVYFa"},{"type":"paragraph","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"and the bosonic commutation relations are","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"AKLGXaFXm3"}],"key":"XKBe53iEZD"},{"type":"math","value":"\\left[\\hat{r}_i, \\hat{r}_j\\right] = i\\Omega_{i,j} \n\\quad ,\\quad \n\\boldsymbol{\\Omega}= \\bigoplus_{k=1}^N \\boldsymbol{\\Omega}_1 = \\begin{pmatrix}\\boldsymbol{\\Omega}_1 & & \\\\ &  \\ddots  &\\\\ &  & \\boldsymbol{\\Omega}_1  \\end{pmatrix}\\quad ,\\quad \\boldsymbol{\\Omega}_1 = \\begin{pmatrix}0 & 1\\\\ -1 & 0\\end{pmatrix},","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mrow><mo fence=\"true\">[</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>i</mi></msub><mo separator=\"true\">,</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>j</mi></msub><mo fence=\"true\">]</mo></mrow><mo>=</mo><mi>i</mi><msub><mi mathvariant=\"normal\">Ω</mi><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>j</mi></mrow></msub><mspace width=\"1em\"/><mo separator=\"true\">,</mo><mspace width=\"1em\"/><mi mathvariant=\"bold\">Ω</mi><mo>=</mo><munderover><mo>⨁</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi mathvariant=\"bold\">Ω</mi><mn>1</mn></msub><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi mathvariant=\"bold\">Ω</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mo lspace=\"0em\" rspace=\"0em\">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi mathvariant=\"bold\">Ω</mi><mn>1</mn></msub></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow><mspace width=\"1em\"/><mo separator=\"true\">,</mo><mspace width=\"1em\"/><msub><mi mathvariant=\"bold\">Ω</mi><mn>1</mn></msub><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow><mo separator=\"true\">,</mo></mrow><annotation encoding=\"application/x-tex\">\\left[\\hat{r}_i, \\hat{r}_j\\right] = i\\Omega_{i,j} \n\\quad ,\\quad \n\\boldsymbol{\\Omega}= \\bigoplus_{k=1}^N \\boldsymbol{\\Omega}_1 = \\begin{pmatrix}\\boldsymbol{\\Omega}_1 &amp; &amp; \\\\ &amp;  \\ddots  &amp;\\\\ &amp;  &amp; \\boldsymbol{\\Omega}_1  \\end{pmatrix}\\quad ,\\quad \\boldsymbol{\\Omega}_1 = \\begin{pmatrix}0 &amp; 1\\\\ -1 &amp; 0\\end{pmatrix},</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\">[</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\">]</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.9722em;vertical-align:-0.2861em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord\">Ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.1304em;vertical-align:-1.3021em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8283em;\"><span style=\"top:-1.8479em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">⨁</span></span></span><span style=\"top:-4.3em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3021em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"></span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:5.6em;\"></span><span style=\"width:0.875em;height:3.600em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-4.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-3.01em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span><span style=\"top:-1.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-4.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span><span style=\"top:-3.01em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"minner\">⋱</span></span></span><span style=\"top:-1.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-4.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span><span style=\"top:-3.01em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"></span></span><span style=\"top:-1.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:5.6em;\"></span><span style=\"width:0.875em;height:3.600em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">−</span><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span></span></span></span></span>","enumerator":"11","key":"Yi1auH2C07"},{"type":"paragraph","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"key":"VoKLUTSXHv"},{"type":"inlineMath","value":"\\boldsymbol{\\Omega}","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold\">Ω</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\Omega}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span></span></span></span>","key":"QHYjjntg6k"},{"type":"text","value":" is called the (N-mode) symplectic form. A Gaussian transformation is a quantum transformation that preserves Gaussianity of the state. ","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"key":"uxgIFgybro"},{"type":"cite","identifier":"simon_gaussian_wigner_1987","label":"simon_gaussian_wigner_1987","kind":"narrative","position":{"start":{"line":172,"column":164},"end":{"line":172,"column":191}},"children":[{"type":"text","value":"Simon ","key":"A3K4Rz5SLh"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"bYTvWygv2V"}],"key":"rASdBa57xr"},{"type":"text","value":" (1987)","key":"TGeCY1D0mz"}],"enumerator":"26","key":"gVsF3nf614"},{"type":"text","value":" showed that Gaussian unitary transformations can be written as a combination of a displacement ","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"key":"XzbQvDRrbo"},{"type":"inlineMath","value":"\\boldsymbol{d}\\in \\mathbb{R}^4","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">d</mi><mo>∈</mo><msup><mi mathvariant=\"double-struck\">R</mi><mn>4</mn></msup></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{d}\\in \\mathbb{R}^4</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">d</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141em;\"></span><span class=\"mord\"><span class=\"mord mathbb\">R</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4</span></span></span></span></span></span></span></span></span></span></span>","key":"DYD2C2Dcux"},{"type":"text","value":" and a symplectic transformation ","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"key":"F2nsShr3rm"},{"type":"inlineMath","value":"\\boldsymbol{S}\\in \\mathcal{S}p(2N, \\mathbb{R})","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">S</mi><mo>∈</mo><mi mathvariant=\"script\">S</mi><mi>p</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo separator=\"true\">,</mo><mi mathvariant=\"double-struck\">R</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{S}\\in \\mathcal{S}p(2N, \\mathbb{R})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7252em;vertical-align:-0.0391em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathcal\" style=\"margin-right:0.075em;\">S</span><span class=\"mord mathnormal\">p</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathbb\">R</span><span class=\"mclose\">)</span></span></span></span>","key":"ufRVKOSF9P"},{"type":"text","value":". The mean and the covariance matrix of the state is thus transformed as","position":{"start":{"line":172,"column":1},"end":{"line":172,"column":1}},"key":"crunIMVVpq"}],"key":"hPDFPiQVbq"},{"type":"math","value":"\\boldsymbol{\\mu}\\rightarrow \\boldsymbol{S\\mu} +\\boldsymbol{d},\\quad  \\quad \\boldsymbol{\\sigma} \\rightarrow \\boldsymbol{S\\sigma S^\\intercal}.","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>→</mo><mi><mrow><mi mathvariant=\"bold-italic\">S</mi><mi mathvariant=\"bold-italic\">μ</mi></mrow></mi><mo>+</mo><mi mathvariant=\"bold-italic\">d</mi><mo separator=\"true\">,</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>→</mo><mi><mrow><mi mathvariant=\"bold-italic\">S</mi><mi mathvariant=\"bold-italic\">σ</mi><msup><mi mathvariant=\"bold-italic\">S</mi><mo mathvariant=\"bold-italic\">⊺</mo></msup></mrow></mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu}\\rightarrow \\boldsymbol{S\\mu} +\\boldsymbol{d},\\quad  \\quad \\boldsymbol{\\sigma} \\rightarrow \\boldsymbol{S\\sigma S^\\intercal}.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8805em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">Sμ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">d</span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7144em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">Sσ</span><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7144em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"12","key":"ZAqSdmpe6V"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Symplectic group and transformations","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"key":"UqWE7of9I5"}],"key":"n1RlYtxQ0O"},{"type":"paragraph","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"children":[{"type":"text","value":"The group of the real symplectic matrices ","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"key":"koxiVCzaLX"},{"type":"inlineMath","value":"\\mathcal{S}p(2N, \\mathbb{R}) ","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"script\">S</mi><mi>p</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo separator=\"true\">,</mo><mi mathvariant=\"double-struck\">R</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\mathcal{S}p(2N, \\mathbb{R}) </annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathcal\" style=\"margin-right:0.075em;\">S</span><span class=\"mord mathnormal\">p</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathbb\">R</span><span class=\"mclose\">)</span></span></span></span>","key":"P0r5BnhUmH"},{"type":"text","value":" is defined as the real matrices of size ","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"key":"l8406Vpw9x"},{"type":"inlineMath","value":"2N\\times 2N","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mi>N</mi><mo>×</mo><mn>2</mn><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">2N\\times 2N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">×</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"VFHi981e3T"},{"type":"text","value":" that satisfy ","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"key":"AxlZAPNS3y"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":178,"column":1},"end":{"line":178,"column":1}},"children":[{"type":"cite","identifier":"arvind_real_1995","label":"arvind_real_1995","kind":"parenthetical","position":{"start":{"line":178,"column":143},"end":{"line":178,"column":160}},"children":[{"type":"text","value":"Arvind ","key":"WKQDPg6Qds"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"KCPMV3csxK"}],"key":"HhjLoRITC1"},{"type":"text","value":", 1995","key":"jgPBxfqqfO"}],"enumerator":"15","key":"QfLjdJADTR"}],"key":"XTnQSFXsXR"}],"key":"FVmJtgfHc6"},{"type":"math","value":"\\boldsymbol{S\\Omega S^\\intercal} = \\boldsymbol{\\Omega}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi><mrow><mi mathvariant=\"bold-italic\">S</mi><mi mathvariant=\"bold\">Ω</mi><msup><mi mathvariant=\"bold-italic\">S</mi><mo mathvariant=\"bold-italic\">⊺</mo></msup></mrow></mi><mo>=</mo><mi mathvariant=\"bold\">Ω</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{S\\Omega S^\\intercal} = \\boldsymbol{\\Omega}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7144em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span><span class=\"mord mathbf\">Ω</span><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7144em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span></span></span></span></span>","enumerator":"13","key":"fgAuDbh7vD"}],"key":"Qh2bjZiw1U"},{"type":"paragraph","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"children":[{"type":"text","value":"As we shall see, we can bring the covariance matrix to a normal form (which is diagonal, see Williamson decomposition) using symplectic transformations.","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"nTC8MtN9DG"}],"key":"lMOMoajttL"},{"type":"paragraph","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"children":[{"type":"text","value":"Not all symmetric matrices can represent a quantum state. Indeed, a quantum state must respect the canonical commutation relations and be semi-positive definite. ","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"key":"bjz14tIupo"},{"type":"cite","identifier":"simon_quantum_noise_1994","label":"simon_quantum_noise_1994","kind":"narrative","position":{"start":{"line":187,"column":163},"end":{"line":187,"column":188}},"children":[{"type":"text","value":"Simon ","key":"pVfWNFNnJi"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Fxr2vBqh6d"}],"key":"I2sGjlrnqY"},{"type":"text","value":" (1994)","key":"y1QBqmePde"}],"enumerator":"27","key":"mf9GmcNTcT"},{"type":"text","value":" showed that those two conditions can be recast in the following compact form, called the Schrödinger-Robertson inequality or the ","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"key":"YhB6AVTVuu"},{"type":"emphasis","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"key":"qYajmnZdX8"}],"key":"paYvSKYaqS"},{"type":"text","value":" condition.","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"key":"YigOiUaKcS"}],"key":"lvRIo5J59F"},{"type":"proof","kind":"theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Schrödinger-Robertson inequality or bona fide condition","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"AD66VvL8XE"}],"key":"t8uFgyMJTb"},{"type":"paragraph","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"text","value":"Any covariance matrix ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"I0ti4ZI7KM"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma}","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span>","key":"a4oUBtEbYz"},{"type":"text","value":" that represents a positive bosonic quantum state must respect the following ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"lRlkNxfHEz"},{"type":"emphasis","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"OGvRHZIMkv"}],"key":"gILqmAwdl2"},{"type":"text","value":" condition","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"cXuWCt38EM"}],"key":"V9dXFSg5aJ"},{"type":"math","identifier":"shrodingerrobertsonequation","label":"ShrodingerRobertsonEquation","value":"\\boldsymbol{\\sigma} + i\\boldsymbol{\\Omega} \\geq 0.","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi><mo>+</mo><mi>i</mi><mi mathvariant=\"bold\">Ω</mi><mo>≥</mo><mn>0.</mn></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma} + i\\boldsymbol{\\Omega} \\geq 0.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8221em;vertical-align:-0.136em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0.</span></span></span></span></span>","enumerator":"14","html_id":"shrodingerrobertsonequation","key":"mW1v7v0o0v"}],"key":"VRX0SwVZST"},{"type":"paragraph","position":{"start":{"line":198,"column":1},"end":{"line":198,"column":1}},"children":[{"type":"text","value":"The purity ","position":{"start":{"line":198,"column":1},"end":{"line":198,"column":1}},"key":"VmGwRBEdon"},{"type":"inlineMath","value":"p","position":{"start":{"line":198,"column":1},"end":{"line":198,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"mjypaoLxVX"},{"type":"text","value":" of the state can also be computed from the covariance matrix","position":{"start":{"line":198,"column":1},"end":{"line":198,"column":1}},"key":"QJJQ3tAlEE"}],"key":"RoynN8yfwp"},{"type":"math","identifier":"purity","label":"purity","value":"p = \\text{Tr}(\\hat{\\rho}^2) = \\frac{1}{\\sqrt{\\text{det} \\boldsymbol{\\sigma}}}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>p</mi><mo>=</mo><mtext>Tr</mtext><mo stretchy=\"false\">(</mo><msup><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mn>2</mn></msup><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mtext>det</mtext><mi mathvariant=\"bold-italic\">σ</mi></mrow></msqrt></mfrac></mrow><annotation encoding=\"application/x-tex\">p = \\text{Tr}(\\hat{\\rho}^2) = \\frac{1}{\\sqrt{\\text{det} \\boldsymbol{\\sigma}}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"></span><span class=\"mord text\"><span class=\"mord\">Tr</span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.1778em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9322em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span><span style=\"top:-2.8922em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1078em;\"><span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>","enumerator":"15","html_id":"purity","key":"ggptjbVvAU"},{"type":"paragraph","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"uBRy4KdcK5"},{"type":"inlineMath","value":"p=1","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">p=1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"L53feErQ6E"},{"type":"text","value":", the state is pure, and it is said ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"s6BRICgC8e"},{"type":"emphasis","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"children":[{"type":"text","value":"mixed","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"D3QvNquxpd"}],"key":"TpSflk9uUV"},{"type":"text","value":" when ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"UtTwagUnOL"},{"type":"inlineMath","value":"p<1","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">p&lt;1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">&lt;</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"DZ8BbOb95m"},{"type":"text","value":".","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"zkNTig1w5F"}],"key":"IdQLO1c41V"},{"type":"paragraph","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"text","value":"It is possible to decompose the covariance matrix on ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"jGjy0IFrWV"},{"type":"emphasis","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"text","value":"normal modes","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"FYbWb6Owwd"}],"key":"GX54KxmBvq"},{"type":"text","value":", that is a basis in which the system is split into ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"M0LZ0jcy5P"},{"type":"inlineMath","value":"N","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>N</mi></mrow><annotation encoding=\"application/x-tex\">N</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span></span></span></span>","key":"KKrhYllo77"},{"type":"text","value":" decoupled degrees of freedom. Such a transform is often referred as Williamson decomposition.","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"SQ3P0hqLTH"}],"key":"cCL7lMU36i"},{"type":"proof","kind":"theorem","label":"williamson","identifier":"williamson","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Williamson decomposition","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"XDGYcH6hpm"}],"key":"ka6pNWvSig"},{"type":"paragraph","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"children":[{"type":"cite","identifier":"williamson_algebraic_1936","label":"williamson_algebraic_1936","kind":"narrative","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":27}},"children":[{"type":"text","value":"Williamson (1936)","key":"GDzIUlh9S2"}],"enumerator":"28","key":"ZzrxsLIihD"},{"type":"text","value":" showed that, for any symmetric positive-definite matrix ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"YUGN6lYxV0"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma}","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span>","key":"J1gCoWw4pc"},{"type":"text","value":", there exists a symplectic transformation ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"HwRZ6VfYIV"},{"type":"inlineMath","value":"\\boldsymbol{S}\\in \\mathcal{S}p(2N, \\mathbb{R})","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">S</mi><mo>∈</mo><mi mathvariant=\"script\">S</mi><mi>p</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>N</mi><mo separator=\"true\">,</mo><mi mathvariant=\"double-struck\">R</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{S}\\in \\mathcal{S}p(2N, \\mathbb{R})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7252em;vertical-align:-0.0391em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathcal\" style=\"margin-right:0.075em;\">S</span><span class=\"mord mathnormal\">p</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathbb\">R</span><span class=\"mclose\">)</span></span></span></span>","key":"b5J623jhwy"},{"type":"text","value":" such that","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"kfh8uKDCOs"}],"key":"MTcm6PJBdi"},{"type":"math","identifier":"williamson_decomposition_eq","label":"williamson_decomposition_eq","value":"\\boldsymbol{\\sigma} = \\boldsymbol{S}\\bigoplus_{j=1}^N \\begin{pmatrix}\\nu_j & 0 \\\\0 & \\nu_j\\end{pmatrix} \\boldsymbol{S}^\\intercal","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mi mathvariant=\"bold-italic\">S</mi><munderover><mo>⨁</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>ν</mi><mi>j</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>ν</mi><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow><msup><mi mathvariant=\"bold-italic\">S</mi><mo>⊺</mo></msup></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma} = \\boldsymbol{S}\\bigoplus_{j=1}^N \\begin{pmatrix}\\nu_j &amp; 0 \\\\0 &amp; \\nu_j\\end{pmatrix} \\boldsymbol{S}^\\intercal</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.2421em;vertical-align:-1.4138em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8283em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">⨁</span></span></span><span style=\"top:-4.3em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4138em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7404em;\"><span style=\"top:-3.139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span></span></span>","enumerator":"16","html_id":"williamson-decomposition-eq","key":"gp1AqH04PL"},{"type":"paragraph","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"kWLHO3uEON"},{"type":"inlineMath","value":"\\nu_j>0","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ν</mi><mi>j</mi></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\nu_j&gt;0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8252em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">&gt;</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0</span></span></span></span>","key":"XX9CSLi9yn"},{"type":"text","value":" are the ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"cQbWb2jVvl"},{"type":"emphasis","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"children":[{"type":"text","value":"symplectic eigenvalues","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"CUkQjhkniu"}],"key":"JmsTfyIhBw"},{"type":"text","value":" of ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"eJDh0VYqCN"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma}","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span>","key":"G1IUUXeRRw"},{"type":"text","value":".","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"M8zIAzJMj0"}],"key":"eTdk7FPSvB"}],"html_id":"williamson","key":"AyUT6Q1VCw"},{"type":"paragraph","position":{"start":{"line":216,"column":1},"end":{"line":217,"column":1}},"children":[{"type":"text","value":"Note here that the matrix is not diagonalized in the sense that it is not a change of basis ","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"yJzHpy2Bu9"},{"type":"emphasis","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"Kr93qKeL85"}],"key":"kVYKlFbMN1"},{"type":"text","value":" ","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"OPn9Kfgzd9"},{"type":"inlineMath","value":"\\boldsymbol{S}^{-1}\\neq \\boldsymbol{S}^\\intercal","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi mathvariant=\"bold-italic\">S</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo mathvariant=\"normal\">≠</mo><msup><mi mathvariant=\"bold-italic\">S</mi><mo>⊺</mo></msup></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{S}^{-1}\\neq \\boldsymbol{S}^\\intercal</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0846em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8901em;\"><span style=\"top:-3.139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\"></span></span></span><span class=\"fix\"></span></span></span></span></span><span class=\"mrel\">=</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7404em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7404em;\"><span style=\"top:-3.139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span></span>","key":"rGCuF9mdXH"},{"type":"text","value":".\nFurthermore, the ","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"Z8G4AZ0YqJ"},{"type":"emphasis","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"jLenptpatv"}],"key":"nk3Xe9QI9U"},{"type":"text","value":" condition ","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"AK4Gc6xg0R"},{"type":"crossReference","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"children":[{"type":"text","value":"(","key":"bMIHKRA7h7"},{"type":"text","value":"14","key":"OthbyZM0th"},{"type":"text","value":")","key":"c3LiKUefXo"}],"identifier":"shrodingerrobertsonequation","label":"ShrodingerRobertsonEquation","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"shrodingerrobertsonequation","key":"tUIXaLBdFo"},{"type":"text","value":" applied to the normal mode decomposed covariance matrix ","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"E0C4LX1bzM"},{"type":"inlineMath","value":"\\boldsymbol{\\nu}","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">ν</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\nu}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.06898em;\">ν</span></span></span></span></span></span>","key":"QEsaIuxjF8"},{"type":"text","value":" implies","position":{"start":{"line":216,"column":1},"end":{"line":216,"column":1}},"key":"sfQfV8JhgI"}],"key":"gjJa5qHoYb"},{"type":"math","value":"\\boldsymbol{\\nu}+i\\boldsymbol{\\Omega} = \\bigoplus_{j=1}^{N}\\begin{pmatrix}\\nu_j & i\\\\ -i & \\nu_j\\end{pmatrix} \\geq 0","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">ν</mi><mo>+</mo><mi>i</mi><mi mathvariant=\"bold\">Ω</mi><mo>=</mo><munderover><mo>⨁</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>ν</mi><mi>j</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mi>i</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>−</mo><mi>i</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>ν</mi><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow><mo>≥</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\nu}+i\\boldsymbol{\\Omega} = \\bigoplus_{j=1}^{N}\\begin{pmatrix}\\nu_j &amp; i\\\\ -i &amp; \\nu_j\\end{pmatrix} \\geq 0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.06898em;\">ν</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Ω</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.2421em;vertical-align:-1.4138em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8283em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">⨁</span></span></span><span style=\"top:-4.3em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4138em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">−</span><span class=\"mord mathnormal\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0</span></span></span></span></span>","enumerator":"17","key":"IFYlqz6jmy"},{"type":"paragraph","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"children":[{"type":"text","value":"and therefore that ","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"key":"iWF0OWGjUU"},{"type":"emphasis","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"children":[{"type":"text","value":"all symplectic eigenvalues of the covariance matrix of a quantum state must be greater than 1.","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"key":"Mei2nbKTRa"}],"key":"wbK8WGOVJc"}],"key":"hYo2mZpp49"},{"type":"container","kind":"figure","identifier":"table_adesso","label":"table_adesso","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/table_adesso-0168dc7f28eb03b774299df610be3371.png","alt":"Schematic comparison between Hilbert space and phase space pictures for N-mode Gaussian states.","width":"70%","align":"center","key":"fUX8jyYDzh","urlSource":"images/table_adesso.png","urlOptimized":"/~gondret/phd_manuscript/build/table_adesso-0168dc7f28eb03b774299df610be3371.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"table_adesso","identifier":"table_adesso","html_id":"table-adesso","enumerator":"2","children":[{"type":"text","value":"Figure ","key":"yzvcZ4LTwJ"},{"type":"text","value":"2","key":"S9tLD5pIrA"},{"type":"text","value":":","key":"yvjRAkrl7Q"}],"template":"Figure %s:","key":"LKLryrD5F7"},{"type":"text","value":"Schematic comparison between Hilbert space and phase space pictures for N -mode Gaussian states. ©Table from ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"I6RUJevb37"},{"type":"cite","identifier":"adesso_continuous_2014","label":"adesso_continuous_2014","kind":"narrative","position":{"start":{"line":228,"column":110},"end":{"line":228,"column":133}},"children":[{"type":"text","value":"Adesso ","key":"DiJX1FwBnl"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Tgpcqqpi8G"}],"key":"eOXx3VEAXn"},{"type":"text","value":" (2014)","key":"t7xwGw4kRh"}],"enumerator":"19","key":"nf5O3sgnmB"}],"key":"f7Nxu49D11"}],"key":"Q7ubo6RQdc"}],"enumerator":"2","html_id":"table-adesso","key":"TgYggVyusk"},{"type":"paragraph","position":{"start":{"line":231,"column":1},"end":{"line":231,"column":1}},"children":[{"type":"text","value":"For a two-mode Gaussian state ","position":{"start":{"line":231,"column":1},"end":{"line":231,"column":1}},"key":"alzVdMi74z"},{"type":"inlineMath","value":"\\hat{\\rho}_{AB}","position":{"start":{"line":231,"column":1},"end":{"line":231,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}_{AB}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">A</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"j8GxyzAkiJ"},{"type":"text","value":" whose covariance matrix is","position":{"start":{"line":231,"column":1},"end":{"line":231,"column":1}},"key":"urb5qGmP6r"}],"key":"FM8kfAe5PL"},{"type":"math","identifier":"covariance_matrix_twomodes","label":"covariance_matrix_twomodes","value":"\\boldsymbol{\\sigma} = \\begin{pmatrix} \n\\boldsymbol{A} & \\boldsymbol{C}\\\\ \\boldsymbol{C}^\\intercal & \\boldsymbol{B} \\end{pmatrix}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mi mathvariant=\"bold-italic\">A</mi></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mi mathvariant=\"bold-italic\">C</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msup><mi mathvariant=\"bold-italic\">C</mi><mo>⊺</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mi mathvariant=\"bold-italic\">B</mi></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma} = \\begin{pmatrix} \n\\boldsymbol{A} &amp; \\boldsymbol{C}\\\\ \\boldsymbol{C}^\\intercal &amp; \\boldsymbol{B} \\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">A</span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.06979em;\">C</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7404em;\"><span style=\"top:-3.139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.06979em;\">C</span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.04835em;\">B</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"18","html_id":"covariance-matrix-twomodes","key":"jdWSxj5rDS"},{"type":"paragraph","position":{"start":{"line":237,"column":1},"end":{"line":237,"column":1}},"children":[{"type":"text","value":"the Williamson form is ","position":{"start":{"line":237,"column":1},"end":{"line":237,"column":1}},"key":"iULSg9GSA9"},{"type":"inlineMath","value":"\\nu_-\\mathbb{I}_2 \\oplus \\nu_+\\mathbb{I}_2 ","position":{"start":{"line":237,"column":1},"end":{"line":237,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ν</mi><mo>−</mo></msub><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub><mo>⊕</mo><msub><mi>ν</mi><mo>+</mo></msub><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\nu_-\\mathbb{I}_2 \\oplus \\nu_+\\mathbb{I}_2 </annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8972em;vertical-align:-0.2083em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2583em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">−</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">⊕</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8972em;vertical-align:-0.2083em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2583em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">+</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"NITXfJo6EP"},{"type":"text","value":" where the symplectic spectrum was shown by ","position":{"start":{"line":237,"column":1},"end":{"line":237,"column":1}},"key":"wIqlBas54q"},{"type":"cite","identifier":"serafini_symplectic_2004","label":"serafini_symplectic_2004","kind":"narrative","position":{"start":{"line":237,"column":113},"end":{"line":237,"column":138}},"children":[{"type":"text","value":"Serafini ","key":"iFvMIBEQIm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"EawgujtT7T"}],"key":"q9QcJ7bcxl"},{"type":"text","value":" (2004)","key":"tL7wEsJ2cj"}],"enumerator":"29","key":"GLfK5ZO2wj"},{"type":"text","value":" to be given by","position":{"start":{"line":237,"column":1},"end":{"line":237,"column":1}},"key":"Wg0Sw7mVSO"}],"key":"r79EM6TnTk"},{"type":"math","value":"\\nu_\\pm=\\sqrt{\\frac{\\Delta\\pm \\sqrt{\\Delta^2 - 4\\text{det}\\boldsymbol{\\sigma}}}{2}}\\quad \\quad  \\Delta:= \\text{det}\\boldsymbol{A} + \\text{det}\\boldsymbol{B} +2\\text{det}\\boldsymbol{C}.","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi>ν</mi><mo>±</mo></msub><mo>=</mo><msqrt><mfrac><mrow><mi mathvariant=\"normal\">Δ</mi><mo>±</mo><msqrt><mrow><msup><mi mathvariant=\"normal\">Δ</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mtext>det</mtext><mi mathvariant=\"bold-italic\">σ</mi></mrow></msqrt></mrow><mn>2</mn></mfrac></msqrt><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"normal\">Δ</mi><mo>:</mo><mo>=</mo><mtext>det</mtext><mi mathvariant=\"bold-italic\">A</mi><mo>+</mo><mtext>det</mtext><mi mathvariant=\"bold-italic\">B</mi><mo>+</mo><mn>2</mn><mtext>det</mtext><mi mathvariant=\"bold-italic\">C</mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">\\nu_\\pm=\\sqrt{\\frac{\\Delta\\pm \\sqrt{\\Delta^2 - 4\\text{det}\\boldsymbol{\\sigma}}}{2}}\\quad \\quad  \\Delta:= \\text{det}\\boldsymbol{A} + \\text{det}\\boldsymbol{B} +2\\text{det}\\boldsymbol{C}.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.2083em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.06366em;\">ν</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2583em;\"><span style=\"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">±</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.04em;vertical-align:-0.9539em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.0861em;\"><span class=\"svg-align\" style=\"top:-5em;\"><span class=\"pstrut\" style=\"height:5em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5904em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">Δ</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">±</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9134em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\"><span class=\"mord\">Δ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7401em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\">4</span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span><span style=\"top:-2.8734em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1266em;\"><span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span><span style=\"top:-4.0461em;\"><span class=\"pstrut\" style=\"height:5em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:3.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9539em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\">Δ</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">:=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"></span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">A</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"></span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.04835em;\">B</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord\">2</span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.06979em;\">C</span></span></span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"19","key":"tXav5fab8u"},{"type":"paragraph","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"text","value":"In terms of those quantities, the ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"qW6lcOrBd1"},{"type":"emphasis","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"Y6xusdIpDF"}],"key":"AHjxlGNskP"},{"type":"text","value":" condition ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"mQr7FkRX2T"},{"type":"crossReference","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"text","value":"(","key":"sFiAxIZF0g"},{"type":"text","value":"14","key":"jZXjlvKNmX"},{"type":"text","value":")","key":"IB4eTsbJNo"}],"identifier":"shrodingerrobertsonequation","label":"ShrodingerRobertsonEquation","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"shrodingerrobertsonequation","key":"agVLbll62V"},{"type":"text","value":" can be rewritten as ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"LmY0pTjcnt"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"cite","identifier":"pirandola_correlation_2009","label":"pirandola_correlation_2009","kind":"parenthetical","position":{"start":{"line":241,"column":111},"end":{"line":241,"column":138}},"children":[{"type":"text","value":"Pirandola ","key":"sCLdr1x9HD"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"aQBjQkaCqJ"}],"key":"cjgjAs1LSz"},{"type":"text","value":", 2009","key":"pmCajdS0ho"}],"enumerator":"23","key":"iNHNjV6rv1"}],"key":"CiAKbl9j5R"}],"key":"SivUKcVlmv"},{"type":"math","identifier":"bonafideconditionequationpirandola","label":"bonafideconditionequationpirandola","value":"\\boldsymbol{\\sigma}>0 \\quad \\quad\\quad \\text{det}\\boldsymbol{\\sigma} \\geq 1 \\quad \\quad \\quad\\Delta \\leq 1+\\text{det}\\boldsymbol{\\sigma}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi><mo>&gt;</mo><mn>0</mn><mspace width=\"1em\"/><mspace width=\"1em\"/><mspace width=\"1em\"/><mtext>det</mtext><mi mathvariant=\"bold-italic\">σ</mi><mo>≥</mo><mn>1</mn><mspace width=\"1em\"/><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"normal\">Δ</mi><mo>≤</mo><mn>1</mn><mo>+</mo><mtext>det</mtext><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}&gt;0 \\quad \\quad\\quad \\text{det}\\boldsymbol{\\sigma} \\geq 1 \\quad \\quad \\quad\\Delta \\leq 1+\\text{det}\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">&gt;</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8304em;vertical-align:-0.136em;\"></span><span class=\"mord\">0</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8193em;vertical-align:-0.136em;\"></span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\">Δ</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≤</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span></span>","enumerator":"20","html_id":"bonafideconditionequationpirandola","key":"yQrvlxBJxa"},{"type":"paragraph","position":{"start":{"line":246,"column":1},"end":{"line":246,"column":1}},"children":[{"type":"text","value":"Last but not least, ","position":{"start":{"line":246,"column":1},"end":{"line":246,"column":1}},"key":"NGSW5P4JvT"},{"type":"cite","identifier":"brask_gaussian_2022","label":"brask_gaussian_2022","kind":"narrative","position":{"start":{"line":246,"column":21},"end":{"line":246,"column":41}},"children":[{"type":"text","value":"Brask (2022)","key":"yOc5Irk1EC"}],"enumerator":"21","key":"q1E5sYDcqB"},{"type":"text","value":" provides us that by tracing out the system to focus on a single mode, we can assess the properties of the local displacement and covariance matrix. The following statement is extracted from his paper:","position":{"start":{"line":246,"column":1},"end":{"line":246,"column":1}},"key":"StWxcV4yB5"}],"key":"Ym2SfCN1Jc"},{"type":"proof","kind":"theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Tracing out","position":{"start":{"line":248,"column":1},"end":{"line":248,"column":1}},"key":"KdvayOLTzu"}],"key":"pfoWkTwhCf"},{"type":"paragraph","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"children":[{"type":"text","value":"For the two-mode Gaussian state ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"iWUTC03Y8o"},{"type":"inlineMath","value":"\\hat{\\rho}_{AB}","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}_{AB}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">A</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"zNuO7ABXJC"},{"type":"text","value":" with mean vector ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"hKJLcrKgyv"},{"type":"inlineMath","value":"(\\boldsymbol{r}_A,\\boldsymbol{r}_B)","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><msub><mi mathvariant=\"bold-italic\">r</mi><mi>A</mi></msub><mo separator=\"true\">,</mo><msub><mi mathvariant=\"bold-italic\">r</mi><mi>B</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(\\boldsymbol{r}_A,\\boldsymbol{r}_B)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>","key":"ilsLTJ2H8d"},{"type":"text","value":" and covariance matrix ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"yB6R4TcRGY"},{"type":"crossReference","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"children":[{"type":"text","value":"(","key":"uUVGz9htsO"},{"type":"text","value":"18","key":"QR0MQY2Jw2"},{"type":"text","value":")","key":"bZEP2yLI8R"}],"identifier":"covariance_matrix_twomodes","label":"covariance_matrix_twomodes","kind":"equation","template":"(%s)","enumerator":"18","resolved":true,"html_id":"covariance-matrix-twomodes","key":"V7uSv9kWoq"},{"type":"text","value":", the reduced state ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"NrzMvk46Ss"},{"type":"inlineMath","value":"\\hat{\\rho}_A","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mi>A</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}_A</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"TSWmOQzOaH"},{"type":"text","value":" (respectively ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"jPfiPNvk27"},{"type":"inlineMath","value":"\\hat{\\rho}_B","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mi>B</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\hat{\\rho}_B</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"e14LgOuVPH"},{"type":"text","value":") of subsystem ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"P56psoGVz1"},{"type":"inlineMath","value":"A","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>A</mi></mrow><annotation encoding=\"application/x-tex\">A</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">A</span></span></span></span>","key":"FuhlsZSw3m"},{"type":"text","value":" (resp. ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"KnCN01dA3s"},{"type":"inlineMath","value":"B","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>B</mi></mrow><annotation encoding=\"application/x-tex\">B</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span></span></span>","key":"ycjqtQUEn3"},{"type":"text","value":") is also Gaussian with displacement ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"r9mtYNlB4a"},{"type":"inlineMath","value":"\\boldsymbol{r}_A  ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"bold-italic\">r</mi><mi>A</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{r}_A  </annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5944em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"viyod5Uy2F"},{"type":"text","value":" and covariance matrix ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"LRjgH92svB"},{"type":"inlineMath","value":"\\boldsymbol{A}","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">A</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{A}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">A</span></span></span></span></span></span>","key":"wCIzqCrzaE"},{"type":"text","value":" (resp. ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"hmsR2hApDM"},{"type":"inlineMath","value":"\\boldsymbol{r}_B","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"bold-italic\">r</mi><mi>B</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{r}_B</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5944em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"P5R5hlhkUl"},{"type":"text","value":" and ","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"jXbAdaaYI9"},{"type":"inlineMath","value":"\\boldsymbol{B}","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">B</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{B}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.04835em;\">B</span></span></span></span></span></span>","key":"cididbhjoX"},{"type":"text","value":").","position":{"start":{"line":250,"column":1},"end":{"line":250,"column":1}},"key":"OXKSqFJUqc"}],"key":"hCYMV8g20M"}],"key":"WfIxmjreIQ"},{"type":"heading","depth":2,"position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"children":[{"type":"text","value":"Single mode Gaussian states and single mode transformations","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"ec5o4iyZ42"}],"identifier":"single_mode_transfo","label":"single_mode_transfo","html_id":"single-mode-transfo","enumerator":"4","key":"kFA8v40Ib8"},{"type":"paragraph","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"children":[{"type":"text","value":"The expression of usual Gaussian operations can be found in many reviews article and textbooks, see ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"eYE30anqRV"},{"type":"cite","identifier":"weedbrook_gaussian_2012","label":"weedbrook_gaussian_2012","kind":"narrative","position":{"start":{"line":256,"column":101},"end":{"line":256,"column":125}},"children":[{"type":"text","value":"Weedbrook ","key":"zOZkxTFf7K"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"a17mItzdwv"}],"key":"GWwHc50pB3"},{"type":"text","value":" (2012)","key":"zNal6ZvZZX"}],"enumerator":"18","key":"iwdWpVIPww"},{"type":"text","value":" for example. They can be divided into ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"Ns7HGM8uMf"},{"type":"emphasis","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"children":[{"type":"text","value":"passive","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"Z78gXYWxYE"}],"key":"a7zdh2ZA2D"},{"type":"text","value":" and ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"kSoDGKdJam"},{"type":"emphasis","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"children":[{"type":"text","value":"active","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"rR4eoOVHfw"}],"key":"c6f20SMBdn"},{"type":"text","value":" transformations. Passive transformations preserve the value of Tr(","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"v7KpE4uj4H"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma}","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">σ</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span></span>","key":"FJpoyJbb6O"},{"type":"text","value":"), ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"gEKjpjCPPE"},{"type":"emphasis","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"NCXio51wLw"}],"key":"cIpMPKpS6s"},{"type":"text","value":" they preserve the mean energy of the system while ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"q88wZE0yiB"},{"type":"emphasis","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"children":[{"type":"text","value":"active","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"ADMT9EkEib"}],"key":"kB8uRmuyAP"},{"type":"text","value":" transformations do not preserve it. The interested reader might refer to ","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"W1K7hG2rXd"},{"type":"cite","identifier":"adesso_continuous_2014","label":"adesso_continuous_2014","kind":"narrative","position":{"start":{"line":256,"column":416},"end":{"line":256,"column":439}},"children":[{"type":"text","value":"Adesso ","key":"iOi4lF42H0"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wN5LZHnSU9"}],"key":"GrA6wPFupS"},{"type":"text","value":" (2014)","key":"bBFrSeAm51"}],"enumerator":"19","key":"q5dR0xP1cN"},{"type":"text","value":" for a more mathematical description of the difference between passive and active transformations.","position":{"start":{"line":256,"column":1},"end":{"line":256,"column":1}},"key":"xdlbqqmGxu"}],"key":"snVFSHsOVv"},{"type":"comment","value":"@brask_gaussian_2022 provides here again a nice back-up sheet for practi","position":{"start":{"line":257,"column":1},"end":{"line":257,"column":1}},"key":"HAzf5eU0If"},{"type":"comment","value":" Here I will introduce the one I will use in the next section to quantify and study entanglement.  ","key":"pWaZzEfZvb"},{"type":"comment","value":" Such transformations are classified between single-mode operations $\\boldsymbol{S} =\\boldsymbol{S}_1\\oplus \\boldsymbol{S}_2$, where $\\boldsymbol{S}_1\\in \\text{Sp}(2, \\real) $ and mutli-mode operations.  ","key":"Yzvd6s0znP"},{"type":"paragraph","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"children":[{"type":"strong","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"children":[{"type":"text","value":"The vacuum state","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"key":"lHDhnN2mmi"}],"key":"l2YIKPBAhJ"},{"type":"text","value":": the covariance matrix of the vacuum is the identity, and it is centered on the origin of the phase space (","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"key":"gdAUnECyXM"},{"type":"inlineMath","value":"\\boldsymbol{r}=0","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">r</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{r}=0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0</span></span></span></span>","key":"EyM8cUqaDl"},{"type":"text","value":").","position":{"start":{"line":260,"column":1},"end":{"line":260,"column":1}},"key":"PDGzbpVowf"}],"key":"eiPY4G2yIK"},{"type":"math","value":"\\boldsymbol{\\mu} = (0,0)\\quad \\quad \\boldsymbol{\\sigma} = \\mathbb{I}_2 \\quad \\quad \\hat{\\rho}_{vac}= \\ket{0}\\bra{0}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub><mspace width=\"1em\"/><mspace width=\"1em\"/><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mrow><mi>v</mi><mi>a</mi><mi>c</mi></mrow></msub><mo>=</mo><mpadded><mi mathvariant=\"normal\">∣</mi><mn>0</mn><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mn>0</mn><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = (0,0)\\quad \\quad \\boldsymbol{\\sigma} = \\mathbb{I}_2 \\quad \\quad \\hat{\\rho}_{vac}= \\ket{0}\\bra{0}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\">c</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\">0</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\">0</span></span><span class=\"mord\">∣</span></span></span></span></span></span>","enumerator":"21","key":"tTEKBYD1S3"},{"type":"paragraph","position":{"start":{"line":264,"column":1},"end":{"line":264,"column":1}},"children":[{"type":"strong","position":{"start":{"line":264,"column":1},"end":{"line":264,"column":1}},"children":[{"type":"text","value":"Coherent states:","position":{"start":{"line":264,"column":1},"end":{"line":264,"column":1}},"key":"FTuiNPzw4B"}],"key":"Eq10gUXnsq"},{"type":"text","value":" the covariance matrix of a coherent state (displaced vacuum state) is also the identity but is not centered on the phase space origin.","position":{"start":{"line":264,"column":1},"end":{"line":264,"column":1}},"key":"NQZ8GoGrop"}],"key":"WuDad6ia2j"},{"type":"math","value":"\\hat{D}(\\alpha) = \\exp(\\alpha \\hat{a}^\\dagger - \\alpha^*\\hat{a}) \\quad \\Longleftrightarrow \\quad \\boldsymbol{\\mu}\\rightarrow\\sqrt{2} (\\Re (\\alpha),\\Im(\\alpha))+\\boldsymbol{\\mu}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mover accent=\"true\"><mi>D</mi><mo>^</mo></mover><mo stretchy=\"false\">(</mo><mi>α</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>α</mi><msup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mo>†</mo></msup><mo>−</mo><msup><mi>α</mi><mo>∗</mo></msup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mo>⟺</mo><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">μ</mi><mo>→</mo><msqrt><mn>2</mn></msqrt><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">ℜ</mi><mo stretchy=\"false\">(</mo><mi>α</mi><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">ℑ</mi><mo stretchy=\"false\">(</mo><mi>α</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>+</mo><mi mathvariant=\"bold-italic\">μ</mi></mrow><annotation encoding=\"application/x-tex\">\\hat{D}(\\alpha) = \\exp(\\alpha \\hat{a}^\\dagger - \\alpha^*\\hat{a}) \\quad \\Longleftrightarrow \\quad \\boldsymbol{\\mu}\\rightarrow\\sqrt{2} (\\Re (\\alpha),\\Im(\\alpha))+\\boldsymbol{\\mu}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1944em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1491em;vertical-align:-0.25em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7387em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">⟺</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2061em;vertical-align:-0.25em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9561em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span></span></span><span style=\"top:-2.9161em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.0839em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\">ℜ</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">ℑ</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">))</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span></span></span></span></span>","enumerator":"22","key":"Var8mmRUn0"},{"type":"paragraph","position":{"start":{"line":268,"column":1},"end":{"line":269,"column":1}},"children":[{"type":"text","value":"leaving the covariance matrix unchanged.\nIf the average population of a coherent state is ","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"yAeUnhoxXu"},{"type":"inlineMath","value":"\\bar{n}","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\bar{n}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5678em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span></span></span></span>","key":"lhmZjdApl8"},{"type":"text","value":", a coherent state lies on a circle in the phase space of radius ","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"Mch652CCJr"},{"type":"inlineMath","value":"\\sqrt{2\\bar{n}}=\\sqrt{2}|\\alpha|","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msqrt><mrow><mn>2</mn><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover></mrow></msqrt><mo>=</mo><msqrt><mn>2</mn></msqrt><mi mathvariant=\"normal\">∣</mi><mi>α</mi><mi mathvariant=\"normal\">∣</mi></mrow><annotation encoding=\"application/x-tex\">\\sqrt{2\\bar{n}}=\\sqrt{2}|\\alpha|</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9072em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span></span></span><span style=\"top:-2.8672em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1328em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1572em;vertical-align:-0.25em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9072em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span></span></span><span style=\"top:-2.8672em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1328em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">∣</span></span></span></span>","key":"wnxrKDnz34"}],"key":"Nng2sSaDUI"},{"type":"math","value":"\\boldsymbol{\\mu} = \\sqrt{2\\bar{n}}(\\cos\\theta, \\sin\\theta)\\quad \\quad \\boldsymbol{\\sigma} = \\mathbb{I}_2 \\quad \\quad \\ket{\\alpha} = \\exp\\left(-|\\alpha|^2/2\\right)\\sum_i \\frac{\\alpha^i}{\\sqrt{i!}}\\ket{i}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><msqrt><mrow><mn>2</mn><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover></mrow></msqrt><mo stretchy=\"false\">(</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo separator=\"true\">,</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub><mspace width=\"1em\"/><mspace width=\"1em\"/><mpadded><mi mathvariant=\"normal\">∣</mi><mi>α</mi><mo stretchy=\"false\">⟩</mo></mpadded><mo>=</mo><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">(</mo><mo>−</mo><mi mathvariant=\"normal\">∣</mi><mi>α</mi><msup><mi mathvariant=\"normal\">∣</mi><mn>2</mn></msup><mi mathvariant=\"normal\">/</mi><mn>2</mn><mo fence=\"true\">)</mo></mrow><munder><mo>∑</mo><mi>i</mi></munder><mfrac><msup><mi>α</mi><mi>i</mi></msup><msqrt><mrow><mi>i</mi><mo stretchy=\"false\">!</mo></mrow></msqrt></mfrac><mpadded><mi mathvariant=\"normal\">∣</mi><mi>i</mi><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = \\sqrt{2\\bar{n}}(\\cos\\theta, \\sin\\theta)\\quad \\quad \\boldsymbol{\\sigma} = \\mathbb{I}_2 \\quad \\quad \\ket{\\alpha} = \\exp\\left(-|\\alpha|^2/2\\right)\\sum_i \\frac{\\alpha^i}{\\sqrt{i!}}\\ket{i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2061em;vertical-align:-0.25em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9561em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span></span></span><span style=\"top:-2.9161em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.0839em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.7793em;vertical-align:-1.2777em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">(</span></span><span class=\"mord\">−</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\"><span class=\"mord\">∣</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mord\">/2</span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">)</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2777em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5017em;\"><span style=\"top:-2.1778em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9322em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\">i</span><span class=\"mclose\">!</span></span></span><span style=\"top:-2.8922em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1078em;\"><span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8247em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"23","key":"rXZ3kdHyGi"},{"type":"paragraph","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"children":[{"type":"strong","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"children":[{"type":"text","value":"Thermal states:","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"key":"lRFvXtkhXo"}],"key":"nKEUEAUsPp"},{"type":"text","value":" thermal states are not pure states, their purity is ","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"key":"l6iuCYtdPN"},{"type":"inlineMath","value":"p=1/(2\\bar{n}+1)","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mi mathvariant=\"normal\">/</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">p=1/(2\\bar{n}+1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1/</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span>","key":"UIAXQxMeXO"},{"type":"text","value":" where ","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"key":"xlmry6991j"},{"type":"inlineMath","value":"\\bar{n}","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\bar{n}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5678em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span></span></span></span>","key":"ldLMGrxSiH"},{"type":"text","value":" is the mean photon population. They are centered on the phase space origin, but their covariance matrix is the identity multiplied by ","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"key":"Gbd0YryxHO"},{"type":"inlineMath","value":"2\\bar{n}+1","position":{"start":{"line":273,"column":1},"end":{"line":273,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mo>+</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">2\\bar{n}+1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"eycwXACt0q"}],"key":"PbrFlryjk3"},{"type":"math","identifier":"thermal_state_equation","label":"thermal_state_equation","value":"\\boldsymbol{\\mu} = (0,0) \\quad \\quad \\boldsymbol{\\sigma} = (2\\bar{n}+1)\\mathbb{I}_2 \\quad \\quad \\hat{\\rho}_{th}(\\bar{n}) = \\sum_i \\frac{\\bar{n}^i}{(\\bar{n}+1)^{i+1}}\\ket{i}\\bra{i}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mn>2</mn><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub><mspace width=\"1em\"/><mspace width=\"1em\"/><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mrow><mi>t</mi><mi>h</mi></mrow></msub><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><mfrac><msup><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mi>i</mi></msup><mrow><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mo>+</mo><mn>1</mn><msup><mo stretchy=\"false\">)</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mpadded><mi mathvariant=\"normal\">∣</mi><mi>i</mi><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mi>i</mi><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = (0,0) \\quad \\quad \\boldsymbol{\\sigma} = (2\\bar{n}+1)\\mathbb{I}_2 \\quad \\quad \\hat{\\rho}_{th}(\\bar{n}) = \\sum_i \\frac{\\bar{n}^i}{(\\bar{n}+1)^{i+1}}\\ket{i}\\bra{i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t</span><span class=\"mord mathnormal mtight\">h</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.7793em;vertical-align:-1.2777em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2777em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5017em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\">1</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7507em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8247em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.936em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span><span class=\"mord\">∣</span></span></span></span></span></span>","enumerator":"24","html_id":"thermal-state-equation","key":"izq0Qa1GUu"},{"type":"paragraph","position":{"start":{"line":278,"column":1},"end":{"line":278,"column":1}},"children":[{"type":"strong","position":{"start":{"line":278,"column":1},"end":{"line":278,"column":1}},"children":[{"type":"text","value":"Single-mode squeezed states:","position":{"start":{"line":278,"column":1},"end":{"line":278,"column":1}},"key":"TrBjxBF1jY"}],"key":"yzUGa6MFVe"},{"type":"text","value":" A single mode squeezing operator is an active transformation and is parametrized by the squeezing parameter ","position":{"start":{"line":278,"column":1},"end":{"line":278,"column":1}},"key":"UL28DQVibz"},{"type":"inlineMath","value":"r","position":{"start":{"line":278,"column":1},"end":{"line":278,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi></mrow><annotation encoding=\"application/x-tex\">r</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span></span></span>","key":"BpUNvHhCY4"}],"key":"R9OwHLWV4u"},{"type":"math","value":"\\hat{S}= \\exp\\left[ r(\\hat{a}^2-\\hat{a}^{\\dagger 2})/2\\right]\\quad \\leftrightarrow \\quad \\boldsymbol{S}(r) = \\begin{pmatrix}e^{-r} & 0\\\\ 0 & e^{r}\\end{pmatrix}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mover accent=\"true\"><mi>S</mi><mo>^</mo></mover><mo>=</mo><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">[</mo><mi>r</mi><mo stretchy=\"false\">(</mo><msup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mn>2</mn></msup><mo>−</mo><msup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mrow><mo>†</mo><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo><mi mathvariant=\"normal\">/</mi><mn>2</mn><mo fence=\"true\">]</mo></mrow><mspace width=\"1em\"/><mo>↔</mo><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">S</mi><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msup><mi>e</mi><mrow><mo>−</mo><mi>r</mi></mrow></msup></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msup><mi>e</mi><mi>r</mi></msup></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\hat{S}= \\exp\\left[ r(\\hat{a}^2-\\hat{a}^{\\dagger 2})/2\\right]\\quad \\leftrightarrow \\quad \\boldsymbol{S}(r) = \\begin{pmatrix}e^{-r} &amp; 0\\\\ 0 &amp; e^{r}\\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2491em;vertical-align:-0.35em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">[</span></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">†</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">/2</span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">]</span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">↔</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7713em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6644em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"25","key":"jXnEBm526u"},{"type":"paragraph","position":{"start":{"line":282,"column":1},"end":{"line":282,"column":1}},"children":[{"type":"text","value":"generates a squeezed vacuum state when acting on the vacuum ","position":{"start":{"line":282,"column":1},"end":{"line":282,"column":1}},"key":"aaL7qGJpaP"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":282,"column":1},"end":{"line":282,"column":1}},"children":[{"type":"cite","identifier":"yuen_two_photon_1976","label":"yuen_two_photon_1976","kind":"parenthetical","position":{"start":{"line":282,"column":62},"end":{"line":282,"column":83}},"children":[{"type":"text","value":"Yuen, 1976","key":"L7Li206W62"}],"enumerator":"30","key":"SVq9qOAK7c"}],"key":"sSkMlNkyrb"}],"key":"mO86EjCWQv"},{"type":"math","value":"\\boldsymbol{\\mu} = (0,0) \\quad \\quad \\boldsymbol{\\sigma} = \\boldsymbol{S}(2r) \\quad \\quad \\hat{\\rho}_{0}(r) = \\frac{1}{\\sqrt{\\cosh r}}\\sum_i \\frac{\\sqrt{(2i)!}}{2^i \\, i!}\\tanh r^i\\ket{2i}\\bra{2i}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mi mathvariant=\"bold-italic\">S</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>r</mi><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><msub><mover accent=\"true\"><mi>ρ</mi><mo>^</mo></mover><mn>0</mn></msub><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mi>cosh</mi><mo>⁡</mo><mi>r</mi></mrow></msqrt></mfrac><munder><mo>∑</mo><mi>i</mi></munder><mfrac><msqrt><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mi>i</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">!</mo></mrow></msqrt><mrow><msup><mn>2</mn><mi>i</mi></msup><mtext> </mtext><mi>i</mi><mo stretchy=\"false\">!</mo></mrow></mfrac><mi>tanh</mi><mo>⁡</mo><msup><mi>r</mi><mi>i</mi></msup><mpadded><mi mathvariant=\"normal\">∣</mi><mrow><mn>2</mn><mi>i</mi></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mrow><mn>2</mn><mi>i</mi></mrow><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = (0,0) \\quad \\quad \\boldsymbol{\\sigma} = \\boldsymbol{S}(2r) \\quad \\quad \\hat{\\rho}_{0}(r) = \\frac{1}{\\sqrt{\\cosh r}}\\sum_i \\frac{\\sqrt{(2i)!}}{2^i \\, i!}\\tanh r^i\\ket{2i}\\bra{2i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">ρ</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.9077em;vertical-align:-1.2777em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.1778em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9322em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mop\">cosh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span></span><span style=\"top:-2.8922em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1078em;\"><span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2777em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.63em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\">2</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7507em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">!</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.695em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.935em;\"><span class=\"svg-align\" style=\"top:-3.2em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">)!</span></span></span><span style=\"top:-2.895em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.305em;\"><span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">tanh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8747em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\">2</span><span class=\"mord mathnormal\">i</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\">2</span><span class=\"mord mathnormal\">i</span></span><span class=\"mord\">∣</span></span></span></span></span></span>","enumerator":"26","key":"UHhFPRHQ0b"},{"type":"paragraph","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"children":[{"type":"strong","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"children":[{"type":"text","value":"Phase shift:","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"key":"gqbDQ4Ei7B"}],"key":"kMXW4Q8TYo"},{"type":"text","value":" a single mode rotation by an angle ","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"key":"vjay52Ot2D"},{"type":"inlineMath","value":"\\varphi/2","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>φ</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\varphi/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mord\">/2</span></span></span></span>","key":"pNTljsw8EA"},{"type":"text","value":" in phase space is a passive transformation and reads","position":{"start":{"line":287,"column":1},"end":{"line":287,"column":1}},"key":"IPy8QzQahv"}],"key":"lauNTDf2sD"},{"type":"math","value":"\\hat{U} = \\exp (i\\varphi \\hat{a}_i^\\dagger\\hat{a}_i)\\quad \\leftrightarrow \\quad \\boldsymbol{R}(\\varphi) = \\begin{pmatrix}\\cos \\varphi/2 & -\\sin  \\varphi/2 \\\\ \\sin  \\varphi/2 & \\cos  \\varphi/2\\end{pmatrix}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mover accent=\"true\"><mi>U</mi><mo>^</mo></mover><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy=\"false\">(</mo><mi>i</mi><mi>φ</mi><msubsup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mi>i</mi><mo>†</mo></msubsup><msub><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mi>i</mi></msub><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mo>↔</mo><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">R</mi><mo stretchy=\"false\">(</mo><mi>φ</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\hat{U} = \\exp (i\\varphi \\hat{a}_i^\\dagger\\hat{a}_i)\\quad \\leftrightarrow \\quad \\boldsymbol{R}(\\varphi) = \\begin{pmatrix}\\cos \\varphi/2 &amp; -\\sin  \\varphi/2 \\\\ \\sin  \\varphi/2 &amp; \\cos  \\varphi/2\\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9468em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">U</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2439em;vertical-align:-0.2769em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">φ</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.967em;\"><span style=\"top:-2.4231em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.1809em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2769em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">↔</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.00421em;\">R</span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">φ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mord\">/2</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mord\">/2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mord\">/2</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mord\">/2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"27","key":"SFCMR0xPIY"},{"type":"container","kind":"figure","identifier":"distribution_one_mode_gaussian_states","label":"distribution_one_mode_gaussian_states","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/distribution_one_mod-811a3eeedaec681596db4054d5b8a265.png","alt":"Wigner function and Fock distribution of some Gaussian states","width":"100%","align":"center","key":"AGgHuGTosj","urlSource":"images/distribution_one_mode_gaussian_states.png","urlOptimized":"/~gondret/phd_manuscript/build/distribution_one_mod-811a3eeedaec681596db4054d5b8a265.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"distribution_one_mode_gaussian_states","identifier":"distribution_one_mode_gaussian_states","html_id":"distribution-one-mode-gaussian-states","enumerator":"3","children":[{"type":"text","value":"Figure ","key":"ldot5WOFom"},{"type":"text","value":"3","key":"HlexJid4iT"},{"type":"text","value":":","key":"ac8jInL1of"}],"template":"Figure %s:","key":"k7AfQoWMNE"},{"type":"text","value":"Wigner function (top raw) and photon number distribution (second raw) of one mode Gaussian states. From left to right: the vacuum state has 0 mean population while the coherent state, the thermal state and the single-mode squeezed vacuum state were chosen so that their mean population is 2. The last column represents a general squeezed (","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"R6w3G2KAKd"},{"type":"inlineMath","value":"r=0.5","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi><mo>=</mo><mn>0.5</mn></mrow><annotation encoding=\"application/x-tex\">r=0.5</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0.5</span></span></span></span>","key":"iAOsfrWIgE"},{"type":"text","value":"), rotated (","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"ZhLkLC7pvw"},{"type":"inlineMath","value":"-\\pi/5","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>−</mo><mi>π</mi><mi mathvariant=\"normal\">/</mi><mn>5</mn></mrow><annotation encoding=\"application/x-tex\">-\\pi/5</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">−</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">π</span><span class=\"mord\">/5</span></span></span></span>","key":"Fz8ohU4lVC"},{"type":"text","value":") and displaced (","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"O7kwQDgoqW"},{"type":"inlineMath","value":"\\bar{x}=-2","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>x</mi><mo>ˉ</mo></mover><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\bar{x}=-2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5678em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">−</span><span class=\"mord\">2</span></span></span></span>","key":"J5Jog5sYbi"},{"type":"text","value":") Gaussian state with purity ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"l5C0DJvFqK"},{"type":"inlineMath","value":"p=0.5<1","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0.5</mn><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">p=0.5&lt;1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6835em;vertical-align:-0.0391em;\"></span><span class=\"mord\">0.5</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">&lt;</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"HFCO8Br1Eg"},{"type":"text","value":". The mean photon population is also 2. ©The photon number distribution were computed using ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"ds0fsu51ov"},{"type":"link","url":"https://the-walrus.readthedocs.io/en/latest/","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"text","value":"the Walrus","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"laG6DlGXe1"}],"urlSource":"https://the-walrus.readthedocs.io/en/latest/","key":"VlwVWHRDIY"},{"type":"text","value":" library ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"PlNeNdkuji"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"cite","identifier":"gupt_walrus_2019","label":"gupt_walrus_2019","kind":"parenthetical","position":{"start":{"line":298,"column":594},"end":{"line":298,"column":611}},"children":[{"type":"text","value":"Gupt ","key":"U2ydV9mQ8v"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"W2VzvcQ0os"}],"key":"rYvEdfDqt8"},{"type":"text","value":", 2019","key":"Ax4Azhy1T4"}],"enumerator":"31","key":"qCvTCRKgga"}],"key":"lmy5gk2PS2"},{"type":"text","value":".","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"W8DQiw4avG"}],"key":"l2ZTWVuThn"}],"key":"MIQk5xtaYd"}],"enumerator":"3","html_id":"distribution-one-mode-gaussian-states","key":"zbj7YUFjN7"},{"type":"paragraph","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"strong","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"General one-mode Gaussian state:","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"GX9iTzAcLv"}],"key":"ASKbRVaCkf"},{"type":"text","value":" Any pure Gaussian state can be generated with squeezing, rotation and displacement operators acting on the vacuum, ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"KJgFldBpfx"},{"type":"emphasis","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"i.e","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"BuPc8WIS9D"}],"key":"fvjkoFbG4q"},{"type":"text","value":" one can characterize any one mode Gaussian state with four numbers ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"AxjaQAEe9v"},{"type":"inlineMath","value":"(\\bar{x},\\bar{p}, \\varphi, r)","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>x</mi><mo>ˉ</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>p</mi><mo>ˉ</mo></mover><mo separator=\"true\">,</mo><mi>φ</mi><mo separator=\"true\">,</mo><mi>r</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(\\bar{x},\\bar{p}, \\varphi, r)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">ˉ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">φ</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span></span></span></span>","key":"TDlgRYJqgE"},{"type":"text","value":". The most general form for a (non necessary pure) Gaussian state is therefore the following","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"QreRrOcrNJ"}],"key":"P2ANK18ylt"},{"type":"math","value":"\\boldsymbol{\\mu} = (\\bar{x}, \\bar{p}) \\quad \\quad \\boldsymbol{\\sigma} = \\frac{1}{p}\\boldsymbol{R}(\\varphi)  \\boldsymbol{S}(2r) \\boldsymbol{R^\\intercal}(\\varphi)","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>x</mi><mo>ˉ</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>p</mi><mo>ˉ</mo></mover><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mi mathvariant=\"bold-italic\">R</mi><mo stretchy=\"false\">(</mo><mi>φ</mi><mo stretchy=\"false\">)</mo><mi mathvariant=\"bold-italic\">S</mi><mo stretchy=\"false\">(</mo><mn>2</mn><mi>r</mi><mo stretchy=\"false\">)</mo><mi><msup><mi mathvariant=\"bold-italic\">R</mi><mo mathvariant=\"bold-italic\">⊺</mo></msup></mi><mo stretchy=\"false\">(</mo><mi>φ</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = (\\bar{x}, \\bar{p}) \\quad \\quad \\boldsymbol{\\sigma} = \\frac{1}{p}\\boldsymbol{R}(\\varphi)  \\boldsymbol{S}(2r) \\boldsymbol{R^\\intercal}(\\varphi)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">ˉ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2019em;vertical-align:-0.8804em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8804em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.00421em;\">R</span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">φ</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.00421em;\">R</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7144em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">φ</span><span class=\"mclose\">)</span></span></span></span></span>","enumerator":"28","key":"H85eGCyzjP"},{"type":"paragraph","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"HFMylaDYLu"},{"type":"inlineMath","value":"p","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi></mrow><annotation encoding=\"application/x-tex\">p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span></span></span></span>","key":"pdzN6Cp4yF"},{"type":"text","value":" is the purity of the single mode state ","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"Nt0TpocMKA"},{"type":"inlineMath","value":"p=1/\\sqrt{\\text{det}\\boldsymbol{\\sigma}}","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mi mathvariant=\"normal\">/</mi><msqrt><mrow><mtext>det</mtext><mi mathvariant=\"bold-italic\">σ</mi></mrow></msqrt></mrow><annotation encoding=\"application/x-tex\">p=1/\\sqrt{\\text{det}\\boldsymbol{\\sigma}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1822em;vertical-align:-0.25em;\"></span><span class=\"mord\">1/</span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9322em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord text\"><span class=\"mord\">det</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span></span></span><span style=\"top:-2.8922em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1078em;\"><span></span></span></span></span></span></span></span></span>","key":"kRw7QnWuiS"},{"type":"text","value":".","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"r6NbiP5nDN"}],"key":"Eo6ORir6mP"},{"type":"paragraph","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"children":[{"type":"strong","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"children":[{"type":"text","value":"Noise and loss channels:","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"XcsSaSvpK4"}],"key":"sg1VYgQ3jE"},{"type":"text","value":" the usual way to take into account the non-unit efficiency of detectors is by mixing the state on a beam-splitter with the environment. For pure loss channels, the environment is simply the vacuum. The action of a pure loss channel parametrized by ","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"NFK3bvna7C"},{"type":"text","value":"η","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"MML5CHhICA"},{"type":"text","value":" ","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"PLH1B90Hzf"},{"type":"emphasis","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"N8mrjLGjGw"}],"key":"nLHTpqXA9a"},{"type":"text","value":" the efficiency of the detector, is given by ","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"g5cbGbZmUL"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"children":[{"type":"cite","identifier":"barbosa_disentanglement_2011","label":"barbosa_disentanglement_2011","kind":"parenthetical","position":{"start":{"line":307,"column":337},"end":{"line":307,"column":366}},"children":[{"type":"text","value":"Barbosa ","key":"ubnvOYQQfW"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"LMGlmUjG3l"}],"key":"RcaHUNCJwk"},{"type":"text","value":", 2011","key":"lf7vaA0Yzd"}],"enumerator":"32","key":"OqARFEoyjY"}],"key":"RmTyMW9gTq"}],"key":"EnaInJ4U4m"},{"type":"math","identifier":"effect_finite_efficiency_equation","label":"effect_finite_efficiency_equation","value":"\\boldsymbol{\\mu}' = \\sqrt{\\eta}\\boldsymbol{\\mu}\\quad \\quad \\boldsymbol{\\sigma}' =\\eta\\boldsymbol{\\sigma} + (1-\\eta) \\mathbb{I}_2","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msup><mi mathvariant=\"bold-italic\">μ</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo>=</mo><msqrt><mi>η</mi></msqrt><mi mathvariant=\"bold-italic\">μ</mi><mspace width=\"1em\"/><mspace width=\"1em\"/><msup><mi mathvariant=\"bold-italic\">σ</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo>=</mo><mi>η</mi><mi mathvariant=\"bold-italic\">σ</mi><mo>+</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy=\"false\">)</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu}&#x27; = \\sqrt{\\eta}\\boldsymbol{\\mu}\\quad \\quad \\boldsymbol{\\sigma}&#x27; =\\eta\\boldsymbol{\\sigma} + (1-\\eta) \\mathbb{I}_2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8019em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.09em;vertical-align:-0.2881em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7519em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">η</span></span></span><span style=\"top:-2.7119em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2881em;\"><span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8019em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">η</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">η</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>","enumerator":"29","html_id":"effect-finite-efficiency-equation","key":"JfiOmq9pB9"},{"type":"paragraph","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"children":[{"type":"text","value":"Noisy channels can also be parametrized by a noise parameter, often noted ","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"key":"WvNbOykkjc"},{"type":"text","value":"Δ","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"key":"cUJ4bdFjPv"},{"type":"text","value":" that changes the covariance matrix according to ","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"key":"LvdERJCKBt"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"children":[{"type":"cite","identifier":"martin_comparing_2023","label":"martin_comparing_2023","kind":"parenthetical","position":{"start":{"line":312,"column":133},"end":{"line":312,"column":155}},"children":[{"type":"text","value":"Martin ","key":"hOSlnF4nEe"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"IkZ3d3BZlE"}],"key":"zE33tKIC6B"},{"type":"text","value":", 2023","key":"aTc79sfdrK"}],"enumerator":"25","key":"GsCNDev5Cq"}],"key":"BYlnBaGCcT"}],"key":"FUnS8pNgHj"},{"type":"math","value":"\\boldsymbol{\\sigma}' =\\boldsymbol{\\sigma} + \\Delta \\mathbb{I}_2","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msup><mi mathvariant=\"bold-italic\">σ</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo>=</mo><mi mathvariant=\"bold-italic\">σ</mi><mo>+</mo><mi mathvariant=\"normal\">Δ</mi><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma}&#x27; =\\boldsymbol{\\sigma} + \\Delta \\mathbb{I}_2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8019em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8019em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8389em;vertical-align:-0.15em;\"></span><span class=\"mord\">Δ</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>","enumerator":"30","key":"cRfc3XCbr4"},{"type":"heading","depth":2,"position":{"start":{"line":318,"column":1},"end":{"line":318,"column":1}},"children":[{"type":"text","value":"Two-mode Gaussian states and transformations","position":{"start":{"line":318,"column":1},"end":{"line":318,"column":1}},"key":"u1O5FFMKY2"}],"identifier":"section_bi_partite_gaussian_state_transofrmations","label":"section_bi_partite_gaussian_state_transofrmations","html_id":"section-bi-partite-gaussian-state-transofrmations","enumerator":"5","key":"GFHfXmJvUd"},{"type":"paragraph","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"children":[{"type":"strong","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"children":[{"type":"text","value":"Thermal state:","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"LM8aPdjTGZ"}],"key":"jxcgpHo7vV"},{"type":"text","value":" a two-mode thermal Gaussian state can be parametrized by two different thermal occupation ","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"R5FB1Scb9V"},{"type":"inlineMath","value":"\\bar{n}_1","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>1</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\bar{n}_1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7178em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"a45KJvTdlK"},{"type":"text","value":" and ","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"uRzLqc6kIT"},{"type":"inlineMath","value":"\\bar{n}_2","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>2</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\bar{n}_2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7178em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"rtlRpoaAH9"},{"type":"text","value":". Its covariance matrix is given by","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"ulyplGnbEo"}],"key":"dJxvsMhag7"},{"type":"math","value":"\\boldsymbol{\\mu} = (0,0,0,0) \\quad \\quad \\boldsymbol{\\sigma} = \\begin{pmatrix}(2\\bar{n}_1+1)\\mathbb{I}_2 & 0 \\\\ 0 & (2\\bar{n}_2+1)\\mathbb{I}_2 \\end{pmatrix}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">μ</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">σ</mi><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>2</mn></msub><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu} = (0,0,0,0) \\quad \\quad \\boldsymbol{\\sigma} = \\begin{pmatrix}(2\\bar{n}_1+1)\\mathbb{I}_2 &amp; 0 \\\\ 0 &amp; (2\\bar{n}_2+1)\\mathbb{I}_2 \\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6389em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"31","key":"Z7nuC5bycY"},{"type":"paragraph","position":{"start":{"line":323,"column":1},"end":{"line":323,"column":1}},"children":[{"type":"text","value":"and the purity of this state is ","position":{"start":{"line":323,"column":1},"end":{"line":323,"column":1}},"key":"zrt7G5Q2Xi"},{"type":"inlineMath","value":"p^{-1}=(2\\bar{n}_1+1)(2\\bar{n}_2+1)","position":{"start":{"line":323,"column":1},"end":{"line":323,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>p</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy=\"false\">(</mo><mn>2</mn><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mn>2</mn><msub><mover accent=\"true\"><mi>n</mi><mo>ˉ</mo></mover><mn>2</mn></msub><mo>+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">p^{-1}=(2\\bar{n}_1+1)(2\\bar{n}_2+1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5678em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">n</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">ˉ</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span>","key":"Iga8odvdWr"},{"type":"text","value":".","position":{"start":{"line":323,"column":1},"end":{"line":323,"column":1}},"key":"y6pGhfthwv"}],"key":"Ph0woudL9C"},{"type":"paragraph","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"children":[{"type":"strong","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"children":[{"type":"text","value":"Beam-Splitter:","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"key":"whhYkn8LEh"}],"key":"MDDrAqH0Rt"},{"type":"text","value":" is a transformation that mixes the two modes and is quite useful to model an interferometer with Gaussian states. It writes ","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"key":"VprT7HbqfB"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"children":[{"type":"cite","identifier":"brady_symplectic_2022","label":"brady_symplectic_2022","kind":"parenthetical","position":{"start":{"line":327,"column":145},"end":{"line":327,"column":167}},"children":[{"type":"text","value":"Brady ","key":"uhTztiGdna"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"HE6cJibOTG"}],"key":"Yi7H4yfc4x"},{"type":"text","value":", 2022","key":"S6YkGsSuys"}],"enumerator":"24","key":"Ebzwkt77vr"}],"key":"jkgvRCZHR1"}],"key":"LxV7v9GFJV"},{"type":"math","identifier":"cov_mat_thermal_state","label":"cov_mat_thermal_state","value":"\\boldsymbol{S}_{BS}(\\theta) = \\begin{pmatrix} \\cos \\theta & 0 & \\sin \\theta & 0\\\\\n0 &\\cos \\theta & 0 & \\sin \\theta\\\\\n-\\sin \\theta & 0 & \\cos \\theta & 0 \\\\\n0 & -\\sin \\theta &0 & \\cos \\theta\n\\end{pmatrix}","tight":"before","html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi mathvariant=\"bold-italic\">S</mi><mrow><mi>B</mi><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>θ</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{S}_{BS}(\\theta) = \\begin{pmatrix} \\cos \\theta &amp; 0 &amp; \\sin \\theta &amp; 0\\\\\n0 &amp;\\cos \\theta &amp; 0 &amp; \\sin \\theta\\\\\n-\\sin \\theta &amp; 0 &amp; \\cos \\theta &amp; 0 \\\\\n0 &amp; -\\sin \\theta &amp;0 &amp; \\cos \\theta\n\\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05764em;\">BS</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:4.8em;vertical-align:-2.15em;\"></span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.65em;\"><span class=\"pstrut\" style=\"height:6.8em;\"></span><span style=\"width:0.875em;height:4.800em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"0.875em\" height=\"4.800em\" viewBox=\"0 0 875 4800\"><path d=\"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,1284c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-1292c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-1.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-1.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-1.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.81em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sin</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-1.21em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cos</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">θ</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.65em;\"><span style=\"top:-4.65em;\"><span class=\"pstrut\" style=\"height:6.8em;\"></span><span style=\"width:0.875em;height:4.800em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"0.875em\" height=\"4.800em\" viewBox=\"0 0 875 4800\"><path d=\"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,1209\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-1344c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.15em;\"><span></span></span></span></span></span></span></span></span></span></span></span>","enumerator":"32","html_id":"cov-mat-thermal-state","key":"e7PEbGlzdQ"},{"type":"paragraph","position":{"start":{"line":339,"column":1},"end":{"line":339,"column":1}},"children":[{"type":"strong","position":{"start":{"line":339,"column":1},"end":{"line":339,"column":1}},"children":[{"type":"text","value":"Noises and pure losses:","position":{"start":{"line":339,"column":1},"end":{"line":339,"column":1}},"key":"lv9BTN6e3W"}],"key":"JYetHXwIFl"},{"type":"text","value":" each subsystem might undergo different pure loss channel. In this case, the covariance matrix is transformed as ","position":{"start":{"line":339,"column":1},"end":{"line":339,"column":1}},"key":"n7AAWPRP0W"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":339,"column":1},"end":{"line":339,"column":1}},"children":[{"type":"cite","identifier":"barbosa_disentanglement_2011","label":"barbosa_disentanglement_2011","kind":"parenthetical","position":{"start":{"line":339,"column":142},"end":{"line":339,"column":171}},"children":[{"type":"text","value":"Barbosa ","key":"x1FQ4iwdBh"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"WTvv50SCuc"}],"key":"FTSofziniQ"},{"type":"text","value":", 2011","key":"vXsZHBbyqE"}],"enumerator":"32","key":"hvm21kQuog"}],"key":"zhD13WWJ3r"}],"key":"rXsRM55se5"},{"type":"math","value":"\\boldsymbol{\\mu}' =  \\boldsymbol{L\\mu} \\quad \\quad \\boldsymbol{\\sigma}' = \\boldsymbol{L}(\\boldsymbol{\\sigma} − \\mathbb{I}_4 )\\boldsymbol{L} + \\mathbb{I}_4,\\quad \\quad \\boldsymbol{L} = \\begin{pmatrix}\\sqrt{\\eta}_1\\mathbb{I}_2 & 0 \\\\ 0 & \\sqrt{\\eta}_2\\mathbb{I}_2 \\end{pmatrix}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msup><mi mathvariant=\"bold-italic\">μ</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo>=</mo><mi><mrow><mi mathvariant=\"bold-italic\">L</mi><mi mathvariant=\"bold-italic\">μ</mi></mrow></mi><mspace width=\"1em\"/><mspace width=\"1em\"/><msup><mi mathvariant=\"bold-italic\">σ</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo>=</mo><mi mathvariant=\"bold-italic\">L</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-italic\">σ</mi><mo>−</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>4</mn></msub><mo stretchy=\"false\">)</mo><mi mathvariant=\"bold-italic\">L</mi><mo>+</mo><msub><mi mathvariant=\"double-struck\">I</mi><mn>4</mn></msub><mo separator=\"true\">,</mo><mspace width=\"1em\"/><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">L</mi><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><msqrt><mi>η</mi></msqrt><mn>1</mn></msub><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><msqrt><mi>η</mi></msqrt><mn>2</mn></msub><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\mu}&#x27; =  \\boldsymbol{L\\mu} \\quad \\quad \\boldsymbol{\\sigma}&#x27; = \\boldsymbol{L}(\\boldsymbol{\\sigma} − \\mathbb{I}_4 )\\boldsymbol{L} + \\mathbb{I}_4,\\quad \\quad \\boldsymbol{L} = \\begin{pmatrix}\\sqrt{\\eta}_1\\mathbb{I}_2 &amp; 0 \\\\ 0 &amp; \\sqrt{\\eta}_2\\mathbb{I}_2 \\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">μ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8019em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">Lμ</span></span></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8019em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">L</span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">L</span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">L</span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4533em;vertical-align:-0.9766em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4766em;\"><span style=\"top:-3.6366em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7031em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">η</span></span></span><span style=\"top:-2.6631em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3369em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.0645em;\"><span style=\"top:-2.3134em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3866em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9766em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4766em;\"><span style=\"top:-3.6366em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">0</span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7031em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">η</span></span></span><span style=\"top:-2.6631em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3369em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.0645em;\"><span style=\"top:-2.3134em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3866em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9766em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"33","key":"GxP6TCiPZR"},{"type":"paragraph","position":{"start":{"line":343,"column":1},"end":{"line":343,"column":1}},"children":[{"type":"text","value":"The same could apply for noisy channels, see ","position":{"start":{"line":343,"column":1},"end":{"line":343,"column":1}},"key":"DzoR2XlGj8"},{"type":"cite","identifier":"serafini_entanglement_2004","label":"serafini_entanglement_2004","kind":"narrative","position":{"start":{"line":343,"column":46},"end":{"line":343,"column":73}},"children":[{"type":"text","value":"Serafini ","key":"Cwee4yUybu"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"IbIXsu6KVK"}],"key":"dRRumYuumA"},{"type":"text","value":" (2004)","key":"EZk7ZzQTkR"}],"enumerator":"22","key":"wHfSFjiAy2"},{"type":"text","value":" for which one can define the noise on channel 1 and the noise on channel 2, particularly relevant in the frame of quantum information to take into account the transmission channel.","position":{"start":{"line":343,"column":1},"end":{"line":343,"column":1}},"key":"s5xfJmkWT1"}],"key":"wxXgwDvM6m"},{"type":"paragraph","position":{"start":{"line":346,"column":1},"end":{"line":346,"column":1}},"children":[{"type":"strong","position":{"start":{"line":346,"column":1},"end":{"line":346,"column":1}},"children":[{"type":"text","value":"Two-mode squeezed state:","position":{"start":{"line":346,"column":1},"end":{"line":346,"column":1}},"key":"ljz5bnCAko"}],"key":"emyNPPh8zR"},{"type":"text","value":" a two-mode squeezer is defined by a squeezing parameter ","position":{"start":{"line":346,"column":1},"end":{"line":346,"column":1}},"key":"MNlkzwZYpc"},{"type":"inlineMath","value":"r","position":{"start":{"line":346,"column":1},"end":{"line":346,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi></mrow><annotation encoding=\"application/x-tex\">r</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span></span></span>","key":"iavzk0kBxp"}],"key":"zm46xuuz9w"},{"type":"math","identifier":"two_mode_squeezer_operator","label":"two_mode_squeezer_operator","value":"\\hat{S}_2(r) = \\exp[r(\\hat{a}\\hat{b} - \\hat{a}^\\dagger\\hat{b}^\\dagger)/2] \\quad \\leftrightarrow \\quad \\boldsymbol{S}(r) = \\begin{pmatrix}\\cosh r \\mathbb{I}_2 & \\sinh r \\boldsymbol{\\sigma}_z\\\\ \\sinh r \\boldsymbol{\\sigma}_z & \\cosh r \\mathbb{I}_2  \\end{pmatrix}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mover accent=\"true\"><mi>S</mi><mo>^</mo></mover><mn>2</mn></msub><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy=\"false\">[</mo><mi>r</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mover accent=\"true\"><mi>b</mi><mo>^</mo></mover><mo>−</mo><msup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mo>†</mo></msup><msup><mover accent=\"true\"><mi>b</mi><mo>^</mo></mover><mo>†</mo></msup><mo stretchy=\"false\">)</mo><mi mathvariant=\"normal\">/</mi><mn>2</mn><mo stretchy=\"false\">]</mo><mspace width=\"1em\"/><mo>↔</mo><mspace width=\"1em\"/><mi mathvariant=\"bold-italic\">S</mi><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cosh</mi><mo>⁡</mo><mi>r</mi><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sinh</mi><mo>⁡</mo><mi>r</mi><msub><mi mathvariant=\"bold-italic\">σ</mi><mi>z</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sinh</mi><mo>⁡</mo><mi>r</mi><msub><mi mathvariant=\"bold-italic\">σ</mi><mi>z</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cosh</mi><mo>⁡</mo><mi>r</mi><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">\\hat{S}_2(r) = \\exp[r(\\hat{a}\\hat{b} - \\hat{a}^\\dagger\\hat{b}^\\dagger)/2] \\quad \\leftrightarrow \\quad \\boldsymbol{S}(r) = \\begin{pmatrix}\\cosh r \\mathbb{I}_2 &amp; \\sinh r \\boldsymbol{\\sigma}_z\\\\ \\sinh r \\boldsymbol{\\sigma}_z &amp; \\cosh r \\mathbb{I}_2  \\end{pmatrix}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">S</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2079em;vertical-align:-0.25em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">[</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9579em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">b</span></span><span style=\"top:-3.2634em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2079em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9579em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">b</span></span><span style=\"top:-3.2634em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8991em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">/2</span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">↔</span><span class=\"mspace\" style=\"margin-right:1em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cosh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sinh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sinh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cosh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span></span></span></span></span>","enumerator":"34","html_id":"two-mode-squeezer-operator","key":"vONG0GVYkw"},{"type":"paragraph","position":{"start":{"line":351,"column":1},"end":{"line":351,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":351,"column":1},"end":{"line":351,"column":1}},"key":"h1cLJqSwvN"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma_z}","position":{"start":{"line":351,"column":1},"end":{"line":351,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi><msub><mi mathvariant=\"bold-italic\">σ</mi><mi mathvariant=\"bold-italic\">z</mi></msub></mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{\\sigma_z}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5944em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1611em;\"><span style=\"top:-2.55em;margin-left:-0.037em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord boldsymbol mtight\" style=\"margin-right:0.04213em;\">z</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span></span>","key":"wathbS9IdF"},{"type":"text","value":" is the Pauli matrix diag(1,-1). When the two-mode squeezing operator acts on the vacuum, one obtains a two-mode squeezed vacuum state","position":{"start":{"line":351,"column":1},"end":{"line":351,"column":1}},"key":"ZfQr2nucH3"}],"key":"Db996RfKaC"},{"type":"math","value":"\\ket{\\text{TMSv}}(r) = \\sqrt{1 - \\tanh^2r}\\sum_i (\\tanh r)^i\\ket{i,i}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mtext>TMSv</mtext><mo stretchy=\"false\">⟩</mo></mpadded><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mi>tanh</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>r</mi></mrow></msqrt><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy=\"false\">(</mo><mi>tanh</mi><mo>⁡</mo><mi>r</mi><msup><mo stretchy=\"false\">)</mo><mi>i</mi></msup><mpadded><mi mathvariant=\"normal\">∣</mi><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>i</mi></mrow><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{\\text{TMSv}}(r) = \\sqrt{1 - \\tanh^2r}\\sum_i (\\tanh r)^i\\ket{i,i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord\">TMSv</span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4191em;vertical-align:-1.2777em;\"></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.1414em;\"><span class=\"svg-align\" style=\"top:-3.2em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mop\"><span class=\"mop\">tanh</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8984em;\"><span style=\"top:-3.1473em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span></span></span><span style=\"top:-3.1014em;\"><span class=\"pstrut\" style=\"height:3.2em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.0986em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8723em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2777em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mop\">tanh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8747em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span></span><span class=\"mclose\">⟩</span></span></span></span></span></span>","enumerator":"35","key":"I1GuELdNyc"},{"type":"paragraph","position":{"start":{"line":355,"column":1},"end":{"line":355,"column":1}},"children":[{"type":"text","value":"whose covariance matrix reads","position":{"start":{"line":355,"column":1},"end":{"line":355,"column":1}},"key":"UamcKFL1lH"}],"key":"dmhYFRrhvl"},{"type":"math","value":" \\boldsymbol{\\sigma}_{\\text{TMSv}}(r) = \\begin{pmatrix}\\cosh 2r \\mathbb{I}_2 & \\sinh 2r \\boldsymbol{\\sigma}_z\\\\ \\sinh 2r \\boldsymbol{\\sigma}_z & \\cosh 2r \\mathbb{I}_2  \\end{pmatrix}.","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi mathvariant=\"bold-italic\">σ</mi><mtext>TMSv</mtext></msub><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo fence=\"true\">(</mo><mtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cosh</mi><mo>⁡</mo><mn>2</mn><mi>r</mi><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sinh</mi><mo>⁡</mo><mn>2</mn><mi>r</mi><msub><mi mathvariant=\"bold-italic\">σ</mi><mi>z</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>sinh</mi><mo>⁡</mo><mn>2</mn><mi>r</mi><msub><mi mathvariant=\"bold-italic\">σ</mi><mi>z</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>cosh</mi><mo>⁡</mo><mn>2</mn><mi>r</mi><msub><mi mathvariant=\"double-struck\">I</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence=\"true\">)</mo></mrow><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\"> \\boldsymbol{\\sigma}_{\\text{TMSv}}(r) = \\begin{pmatrix}\\cosh 2r \\mathbb{I}_2 &amp; \\sinh 2r \\boldsymbol{\\sigma}_z\\\\ \\sinh 2r \\boldsymbol{\\sigma}_z &amp; \\cosh 2r \\mathbb{I}_2  \\end{pmatrix}.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">TMSv</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(</span></span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cosh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sinh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"arraycolsep\" style=\"width:0.5em;\"></span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.45em;\"><span style=\"top:-3.61em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">sinh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span style=\"top:-2.41em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop\">cosh</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord\"><span class=\"mord mathbb\">I</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.95em;\"><span></span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"36","key":"gzGX6DQAQ8"},{"type":"paragraph","position":{"start":{"line":359,"column":1},"end":{"line":359,"column":1}},"children":[{"type":"text","value":"The variance of the quadrature difference or sum vanishes in the limit ","position":{"start":{"line":359,"column":1},"end":{"line":359,"column":1}},"key":"Af4IZjAedo"},{"type":"inlineMath","value":"r\\rightarrow\\infty","position":{"start":{"line":359,"column":1},"end":{"line":359,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi><mo>→</mo><mi mathvariant=\"normal\">∞</mi></mrow><annotation encoding=\"application/x-tex\">r\\rightarrow\\infty</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord\">∞</span></span></span></span>","key":"NvTm6spQXm"},{"type":"text","value":" as","position":{"start":{"line":359,"column":1},"end":{"line":359,"column":1}},"key":"WsMCENyuYh"}],"key":"sBQ5hxTZ0A"},{"type":"math","value":"V(\\hat{x}_a-\\hat{x}_b) = V(\\hat{p}_a+ \\hat{p}_b) \\propto e^{-2r} .","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>V</mi><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mi>a</mi></msub><mo>−</mo><msub><mover accent=\"true\"><mi>x</mi><mo>^</mo></mover><mi>b</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>V</mi><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mi>a</mi></msub><mo>+</mo><msub><mover accent=\"true\"><mi>p</mi><mo>^</mo></mover><mi>b</mi></msub><mo stretchy=\"false\">)</mo><mo>∝</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>2</mn><mi>r</mi></mrow></msup><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">V(\\hat{x}_a-\\hat{x}_b) = V(\\hat{p}_a+ \\hat{p}_b) \\propto e^{-2r} .</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">a</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">x</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.2222em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">b</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">a</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">p</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.1667em;\"><span class=\"mord\">^</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1944em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">b</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∝</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8641em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">2</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span></span></span></span></span></span></span></span></span><span class=\"mord\">.</span></span></span></span></span>","enumerator":"37","key":"R3hHQAFpov"},{"type":"paragraph","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"children":[{"type":"text","value":"This state is often referred as an EPR state, for Einstein-Podolski-Rosen because it exhibits perfect correlations between subsystem ","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"key":"hG6X3WtEmf"},{"type":"inlineMath","value":"a","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>a</mi></mrow><annotation encoding=\"application/x-tex\">a</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">a</span></span></span></span>","key":"nJGWjldlcJ"},{"type":"text","value":" and ","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"key":"uYigYOBnJa"},{"type":"inlineMath","value":"b","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>","key":"uc9CUGzfEz"},{"type":"text","value":". In experiments however, one never deals with zero temperature systems and the temperature of the environment must be taken into account. A thermal squeezed state is therefore a thermal state ","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"key":"Nq7ri5qIrw"},{"type":"crossReference","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"children":[{"type":"text","value":"(","key":"xSJS71QPba"},{"type":"text","value":"32","key":"M1EXYxonJR"},{"type":"text","value":")","key":"wNHp1q3u9U"}],"identifier":"cov_mat_thermal_state","label":"cov_mat_thermal_state","kind":"equation","template":"(%s)","enumerator":"32","resolved":true,"html_id":"cov-mat-thermal-state","key":"XNK71JJlMJ"},{"type":"text","value":" that has been squeezed ","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"key":"KZlYEfPPJ7"},{"type":"crossReference","position":{"start":{"line":363,"column":1},"end":{"line":363,"column":1}},"children":[{"type":"text","value":"(","key":"pksMyQ7Ys7"},{"type":"text","value":"34","key":"wfhXehjQKO"},{"type":"text","value":")","key":"REWPuuIGzg"}],"identifier":"two_mode_squeezer_operator","label":"two_mode_squeezer_operator","kind":"equation","template":"(%s)","enumerator":"34","resolved":true,"html_id":"two-mode-squeezer-operator","key":"jKV5FFB7SG"}],"key":"GMhcxjc5SC"},{"type":"comment","value":":  $\\hat{p}_a=-\\hat{p}_b$ and $\\hat{q}_a=\\hat{q}_b$. The EPR state is widely used among Gaussian states and has maximally entangled quadratures for a fix average photon number [@weedbrook_gaussian_2012].","position":{"start":{"line":364,"column":1},"end":{"line":364,"column":1}},"key":"ZikgPFh5xV"},{"type":"math","identifier":"tmsth_expression","label":"tmsth_expression","value":" \\boldsymbol{\\sigma}_{\\text{TMSth}} = \\boldsymbol{S\\sigma}_{\\text{th}}\\boldsymbol{S}^\\intercal=(1+2n_e) \\boldsymbol{\\sigma}_{\\text{TMSv}}(r)","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><msub><mi mathvariant=\"bold-italic\">σ</mi><mtext>TMSth</mtext></msub><mo>=</mo><msub><mi><mrow><mi mathvariant=\"bold-italic\">S</mi><mi mathvariant=\"bold-italic\">σ</mi></mrow></mi><mtext>th</mtext></msub><msup><mi mathvariant=\"bold-italic\">S</mi><mo>⊺</mo></msup><mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><mn>2</mn><msub><mi>n</mi><mi>e</mi></msub><mo stretchy=\"false\">)</mo><msub><mi mathvariant=\"bold-italic\">σ</mi><mtext>TMSv</mtext></msub><mo stretchy=\"false\">(</mo><mi>r</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\"> \\boldsymbol{\\sigma}_{\\text{TMSth}} = \\boldsymbol{S\\sigma}_{\\text{th}}\\boldsymbol{S}^\\intercal=(1+2n_e) \\boldsymbol{\\sigma}_{\\text{TMSv}}(r)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5944em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">TMSth</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8904em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">Sσ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">th</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.05382em;\">S</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7404em;\"><span style=\"top:-3.139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">n</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03704em;\">σ</span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">TMSv</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span></span></span></span></span>","enumerator":"38","html_id":"tmsth-expression","key":"h3yHkMVjY4"},{"type":"paragraph","position":{"start":{"line":369,"column":1},"end":{"line":369,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":369,"column":1},"end":{"line":369,"column":1}},"key":"NZEeM9XNsv"},{"type":"inlineMath","value":"n_e","position":{"start":{"line":369,"column":1},"end":{"line":369,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>n</mi><mi>e</mi></msub></mrow><annotation encoding=\"application/x-tex\">n_e</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">n</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"aZEv9gjNBk"},{"type":"text","value":" is the temperature environment that we take equal for the two subsystems for simplicity.","position":{"start":{"line":369,"column":1},"end":{"line":369,"column":1}},"key":"sMzNWR94vB"}],"key":"pv5fibbPdP"},{"type":"comment","value":" In the next section, we will study entanglement criterion that ables one to study the strength of the correlation/non-classicality between  ","key":"KcQg8NzBN3"},{"type":"comment","value":"  \\text{diag}(\\sqrt{\\eta}_1,\\sqrt{\\eta}_1, \\sqrt{\\eta}_2, \\sqrt{\\eta}_2) ","key":"Z8jOTpUkHy"},{"type":"heading","depth":2,"position":{"start":{"line":375,"column":1},"end":{"line":375,"column":1}},"children":[{"type":"text","value":"Joint probability distribution","position":{"start":{"line":375,"column":1},"end":{"line":375,"column":1}},"key":"XLelTu85ZC"}],"identifier":"joint_proba_distrib_section","label":"joint_proba_distrib_section","html_id":"joint-proba-distrib-section","enumerator":"6","key":"n3AfVZrZay"},{"type":"paragraph","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"children":[{"type":"text","value":"The covariance matrix defines the state entirely. In principle, it is possible to obtain the probability distribution of the particle number ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"KU1q6PJ3sR"},{"type":"inlineMath","value":"\\mathcal{P}(n,m)","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"script\">P</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo separator=\"true\">,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\mathcal{P}(n,m)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathcal\" style=\"margin-right:0.08222em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">n</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mclose\">)</span></span></span></span>","key":"n5hr5HfdGU"},{"type":"text","value":", namely the probability of having ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"TlMzPkmij3"},{"type":"inlineMath","value":"n","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>n</mi></mrow><annotation encoding=\"application/x-tex\">n</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">n</span></span></span></span>","key":"m6NDdJzBvo"},{"type":"text","value":" particles in mode A and ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"DBToFxIqVP"},{"type":"inlineMath","value":"m","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>m</mi></mrow><annotation encoding=\"application/x-tex\">m</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">m</span></span></span></span>","key":"JKEsiGAXF8"},{"type":"text","value":" particles in mode B. In practice, projecting a Gaussian state onto the Fock space is not trivial. ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"gfArzc4Uat"},{"type":"emphasis","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"children":[{"type":"text","value":"A priori","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"BrW8GG8nwY"}],"key":"pj4Dq8c5PU"},{"type":"text","value":", this distribution can be computed from the Weyl transform of the Fock basis projectors ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"NM3mmNUoDY"},{"type":"inlineMath","value":"\\ket{i}\\bra{i}","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><mi>i</mi><mo stretchy=\"false\">⟩</mo></mpadded><mpadded><mo stretchy=\"false\">⟨</mo><mi>i</mi><mi mathvariant=\"normal\">∣</mi></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{i}\\bra{i}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span></span><span class=\"mord\">∣</span></span></span></span></span>","key":"sUEoagZySI"},{"type":"text","value":", as given in equation ","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"vjIDvAl3mx"},{"type":"crossReference","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"children":[{"type":"text","value":"(","key":"cdrvxP5jfI"},{"type":"text","value":"8","key":"v4C8t0vUSg"},{"type":"text","value":")","key":"AZSlo2TuaP"}],"identifier":"weyl_fock","label":"weyl_fock","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"weyl-fock","key":"EmVmYt2u5o"},{"type":"text","value":". For our system, it reads","position":{"start":{"line":376,"column":1},"end":{"line":376,"column":1}},"key":"EV3wzH9yBD"}],"key":"ljvsdEG7Sf"},{"type":"math","value":"\\mathcal{P}(n,m) \\propto \\int \\text{d}\\boldsymbol{r} \\, \\,  W_n(r_1, r_2)W_m(r_3, r_4)e^{-\\sum_{i,j}(r_i-\\mu_i)\\sigma_{ij}^{-1}(r_j-\\mu_j)}","tight":true,"html":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"script\">P</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo separator=\"true\">,</mo><mi>m</mi><mo stretchy=\"false\">)</mo><mo>∝</mo><mo>∫</mo><mtext>d</mtext><mi mathvariant=\"bold-italic\">r</mi><mtext> </mtext><mtext> </mtext><msub><mi>W</mi><mi>n</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>r</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>r</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo><msub><mi>W</mi><mi>m</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>r</mi><mn>3</mn></msub><mo separator=\"true\">,</mo><msub><mi>r</mi><mn>4</mn></msub><mo stretchy=\"false\">)</mo><msup><mi>e</mi><mrow><mo>−</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>j</mi></mrow></munder><mo stretchy=\"false\">(</mo><msub><mi>r</mi><mi>i</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>i</mi></msub><mo stretchy=\"false\">)</mo><msubsup><mi>σ</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>r</mi><mi>j</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>j</mi></msub><mo stretchy=\"false\">)</mo></mrow></msup></mrow><annotation encoding=\"application/x-tex\">\\mathcal{P}(n,m) \\propto \\int \\text{d}\\boldsymbol{r} \\, \\,  W_n(r_1, r_2)W_m(r_3, r_4)e^{-\\sum_{i,j}(r_i-\\mu_i)\\sigma_{ij}^{-1}(r_j-\\mu_j)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathcal\" style=\"margin-right:0.08222em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">n</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∝</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.2222em;vertical-align:-0.8622em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord text\"><span class=\"mord\">d</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.03194em;\">r</span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathnormal\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0539em;\"><span style=\"top:-3.13em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mspace mtight\" style=\"margin-right:0.1952em;\"></span><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:0em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1496em;\"><span style=\"top:-2.1786em;margin-left:0em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4603em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3281em;\"><span style=\"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3281em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mclose mtight\">)</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913em;\"><span style=\"top:-2.2044em;margin-left:-0.0359em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">ij</span></span></span></span><span style=\"top:-2.931em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4345em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3281em;\"><span style=\"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2819em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3281em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2819em;\"><span></span></span></span></span></span></span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span></span></span></span></span>","enumerator":"39","key":"TUipSaYZAL"},{"type":"paragraph","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"children":[{"type":"text","value":"which is neither analytical nor easy to evaluate. In fact, the analytical expression for the joint photon-number distribution was first derived for a displaced but pure two-mode squeezed state by ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"S7NyosBtCh"},{"type":"cite","identifier":"caves_photons_1991","label":"caves_photons_1991","kind":"narrative","position":{"start":{"line":380,"column":197},"end":{"line":380,"column":216}},"children":[{"type":"text","value":"Caves ","key":"TsczoI4gqs"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"uljPIiMEa2"}],"key":"A7m6p8xEtN"},{"type":"text","value":" (1991)","key":"RQsUJ7fMqZ"}],"enumerator":"33","key":"NLASZn0c3Y"},{"type":"text","value":". For a mixed state, ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"DpZHpMCxJc"},{"type":"cite","identifier":"dodonov_photon_1994","label":"dodonov_photon_1994","kind":"narrative","position":{"start":{"line":380,"column":237},"end":{"line":380,"column":257}},"children":[{"type":"text","value":"Dodonov ","key":"G3QNAnWlkm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"rcgotoapwV"}],"key":"kXI3MZELkv"},{"type":"text","value":" (1994)","key":"MjOrbUHAYW"}],"enumerator":"34","key":"TNW2KQt31f"},{"type":"text","value":" started by deriving the photon distribution for a single-mode state. Finally, ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"KVHs62M787"},{"type":"cite","identifier":"dodonov_multidimensional_1994","label":"dodonov_multidimensional_1994","kind":"narrative","position":{"start":{"line":380,"column":336},"end":{"line":380,"column":366}},"children":[{"type":"text","value":"Dodonov ","key":"Z1gaaqznxE"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"whzN96WwNZ"}],"key":"XCSBmo9D7f"},{"type":"text","value":" (1994)","key":"KMf1X6P2bw"}],"enumerator":"35","key":"s7R8d7qMaX"},{"type":"text","value":" extended this result to a general N-mode Gaussian mixed state of light. The latter formula involves “diagonal multidimensional Hermite polynomials” ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"f8TiYZ9InU"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"children":[{"type":"cite","identifier":"berkowitz_calculation_1970","label":"berkowitz_calculation_1970","kind":"parenthetical","position":{"start":{"line":380,"column":516},"end":{"line":380,"column":543}},"children":[{"type":"text","value":"Berkowitz & Garner, 1970","key":"SEMuftpaQ6"}],"enumerator":"36","key":"nl4kQYiQ9M"},{"type":"cite","identifier":"kok_multi_dimensional_2001","label":"kok_multi_dimensional_2001","kind":"parenthetical","position":{"start":{"line":380,"column":544},"end":{"line":380,"column":572}},"children":[{"type":"text","value":"Kok & Braunstein, 2001","key":"YlV1bLlCc4"}],"enumerator":"37","key":"WAeqwOc6IS"}],"key":"EfR2lHNSFY"},{"type":"text","value":". Fortunately, this has been implemented in a Python package named ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"KaKxXdyCrQ"},{"type":"link","url":"https://the-walrus.readthedocs.io/en/latest/gbs.html","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"children":[{"type":"text","value":"The Walrus","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"A15OcPPPl1"}],"urlSource":"https://the-walrus.readthedocs.io/en/latest/gbs.html","key":"R5bAPvIgC8"},{"type":"text","value":" by ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"nOw8hfnE2h"},{"type":"cite","identifier":"gupt_walrus_2019","label":"gupt_walrus_2019","kind":"narrative","position":{"start":{"line":380,"column":710},"end":{"line":380,"column":727}},"children":[{"type":"text","value":"Gupt ","key":"ovyCX6IYAq"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"qL6RFe7nHK"}],"key":"GYW5ab2Wl6"},{"type":"text","value":" (2019)","key":"uEtjRB8ZdP"}],"enumerator":"31","key":"kLM6Rh3iMk"},{"type":"text","value":". Also, note the existence of ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"ylfb6m9dVs"},{"type":"link","url":"https://github.com/IgorBrandao42/Quantum-Gaussian-Information-Toolbox","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"children":[{"type":"text","value":"Qugit","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"jysz6XFF8K"}],"urlSource":"https://github.com/IgorBrandao42/Quantum-Gaussian-Information-Toolbox","error":true,"key":"SBWAKZ8tcY"},{"type":"text","value":", a Python package to simulate the evolution of Gaussian states developed by ","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"wHfNSUvsxd"},{"type":"cite","identifier":"brandao_qugit_2022","label":"brandao_qugit_2022","kind":"narrative","position":{"start":{"line":380,"column":912},"end":{"line":380,"column":931}},"children":[{"type":"text","value":"Brandão ","key":"juLULJjZGa"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"pQMBv0ME2G"}],"key":"ZHOA0qqdTS"},{"type":"text","value":" (2022)","key":"LeXhF421Wy"}],"enumerator":"38","key":"yiBfEz1obC"},{"type":"text","value":", but it does not currently implement the routine to obtain the joint particle distribution.","position":{"start":{"line":380,"column":1},"end":{"line":380,"column":1}},"key":"lriAlE3C3v"}],"key":"cyYQ1CzIRU"},{"type":"container","kind":"figure","identifier":"distribution_two_mode_gaussian_states","label":"distribution_two_mode_gaussian_states","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/bi-distribution-14d818a006446acd6801ed132b4f5539.png","alt":"Joint distribution function for Gaussian two-mode","width":"100%","align":"center","key":"VkarQBD36o","urlSource":"images/bi-distribution.png","urlOptimized":"/~gondret/phd_manuscript/build/bi-distribution-14d818a006446acd6801ed132b4f5539.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"distribution_two_mode_gaussian_states","identifier":"distribution_two_mode_gaussian_states","html_id":"distribution-two-mode-gaussian-states","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"tnuPjTxF58"},{"type":"text","value":"4","key":"sIx0UdyP57"},{"type":"text","value":":","key":"vqNih9IioH"}],"template":"Figure %s:","key":"qXqmk91xcC"},{"type":"text","value":"Joint probability distributions of different two-mode Gaussian states. A) Two-mode squeezed vacuum state with squeezing parameter ","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"iGjGH7iDzF"},{"type":"inlineMath","value":"r=1.3","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1.3</mn></mrow><annotation encoding=\"application/x-tex\">r=1.3</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1.3</span></span></span></span>","key":"VVCJMM61fC"},{"type":"text","value":" and detected with a 100% efficiency detector. A pure TMSv has non zero elements on the diagonal and 0 elsewhere. B) Two-mode squeezed vacuum state detected with a detector that has 50% efficiency: the distribution exhibits non-diagonal elements. C) Two-mode squeezed (","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"AgXnJVctCP"},{"type":"inlineMath","value":"r=1","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">r=1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">1</span></span></span></span>","key":"djNHSzFCUd"},{"type":"text","value":") thermal (","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"SLQjdPwGK8"},{"type":"inlineMath","value":"n^{in}_{th} = 0.4","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msubsup><mi>n</mi><mrow><mi>t</mi><mi>h</mi></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msubsup><mo>=</mo><mn>0.4</mn></mrow><annotation encoding=\"application/x-tex\">n^{in}_{th} = 0.4</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1078em;vertical-align:-0.2831em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">n</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8247em;\"><span style=\"top:-2.4169em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t</span><span class=\"mord mathnormal mtight\">h</span></span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">in</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2831em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0.4</span></span></span></span>","key":"u4QpSSIEKN"},{"type":"text","value":") state. The initial thermal population broadens the diagonal distribution. D) Two-mode thermal state (","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"LRrVO1sLKF"},{"type":"inlineMath","value":"n_{th} = 2.8","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>n</mi><mrow><mi>t</mi><mi>h</mi></mrow></msub><mo>=</mo><mn>2.8</mn></mrow><annotation encoding=\"application/x-tex\">n_{th} = 2.8</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">n</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t</span><span class=\"mord mathnormal mtight\">h</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">2.8</span></span></span></span>","key":"RD8jNRiETe"},{"type":"text","value":") with no correlations. This state is completely symmetric and exhibits no correlations at all. ©Distributions obtained using ","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"YjXajrIjWC"},{"type":"link","url":"https://the-walrus.readthedocs.io/en/latest/","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"children":[{"type":"text","value":"the Walrus","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"piwPSdS1sX"}],"urlSource":"https://the-walrus.readthedocs.io/en/latest/","key":"etvVZkn3SB"},{"type":"text","value":" library ","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"eP7uXgQOyH"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"children":[{"type":"cite","identifier":"gupt_walrus_2019","label":"gupt_walrus_2019","kind":"parenthetical","position":{"start":{"line":389,"column":753},"end":{"line":389,"column":770}},"children":[{"type":"text","value":"Gupt ","key":"MSdEA3nhX5"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"o2ZYDWflj2"}],"key":"QIyifY2sGN"},{"type":"text","value":", 2019","key":"w9ZD3CBjES"}],"enumerator":"31","key":"lamezugwps"}],"key":"rvOeTJpcom"},{"type":"text","value":".","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"ht0khZDoeQ"}],"key":"nt9ZEHnMzY"}],"key":"oEtvdFBWmD"}],"enumerator":"4","html_id":"distribution-two-mode-gaussian-states","key":"vmzESggN5t"},{"type":"comment","value":" \n```{math}\n\\text{Tr}(\\hat{\\rho}\\ket{j}\\bra{j}) = \\int \\text{d}x\\text{d}p \\frac{(-1)^j}{\\pi}e^{-\\boldsymbol{r}^\\intercal \\boldsymbol{r}}\\mathcal{L}_j(2\\boldsymbol{r}^\\intercal \\boldsymbol{r})\\frac{e^{ -(\\boldsymbol{r-\\mu})^\\intercal\\boldsymbol{\\sigma}^{-1}(\\boldsymbol{r-\\mu}) }}{\\pi^N \\sqrt{\\text{Det}[\\boldsymbol{\\sigma}]}}\n```\n ","key":"jLnCCECoKK"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Summary","position":{"start":{"line":402,"column":1},"end":{"line":402,"column":1}},"key":"Z0dGz5j4KG"}],"key":"fZDkR4HgmL"},{"type":"paragraph","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"children":[{"type":"text","value":"This section introduced Gaussian states, that can be fully characterized by their first and second moment, the covariance matrix ","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"key":"LzFLaHAoUe"},{"type":"crossReference","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"children":[{"type":"text","value":"(","key":"eSyaJoIY7y"},{"type":"text","value":"9","key":"Y3JMoojfjg"},{"type":"text","value":")","key":"T9Th1CEv1f"}],"identifier":"covariance_matrix_def","label":"covariance_matrix_def","kind":"equation","template":"(%s)","enumerator":"9","resolved":true,"html_id":"covariance-matrix-def","key":"LE8Y7k8RBo"},{"type":"text","value":". Any covariance matrix that represents a quantum state must satisfy a ","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"key":"WYJf3h6vXg"},{"type":"emphasis","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"key":"VudXj88TLf"}],"key":"MQWfaHpEU3"},{"type":"text","value":" condition, the Schrödinger-Robertson inequality ","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"key":"b3fCCVzFeV"},{"type":"crossReference","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"children":[{"type":"text","value":"(","key":"slLrrCviUq"},{"type":"text","value":"14","key":"tl8x38q21U"},{"type":"text","value":")","key":"mHKOZJBJtu"}],"identifier":"shrodingerrobertsonequation","label":"ShrodingerRobertsonEquation","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"shrodingerrobertsonequation","key":"vQ7iKG0nVn"},{"type":"text","value":".  It requires its (symplectic) eigenvalues to be reater than 1. This inequality is fundamental in the next sections, to assess the non-separability of Gaussian states.","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"key":"CZHUwu1RPM"}],"key":"vPCm2GuaMp"}],"key":"MazG5ZZXO0"},{"type":"footnoteDefinition","identifier":"positive-semi-definite","label":"positive-semi-definite","position":{"start":{"line":403,"column":1},"end":{"line":403,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"text","value":"A positive-definite matrix ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"ioywcpBfl4"},{"type":"inlineMath","value":"\\boldsymbol{M}","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">M</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{M}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.11424em;\">M</span></span></span></span></span></span>","key":"xeCjdcEP15"},{"type":"text","value":" is such that for any non-zero vector ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"PrjRL3hhnA"},{"type":"inlineMath","value":"x","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>","key":"rVhkNZYWST"},{"type":"text","value":", the real quantity ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"cMzZ3zrvRB"},{"type":"inlineMath","value":"\\boldsymbol{x}^\\intercal \\boldsymbol{M}\\boldsymbol{x}","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi mathvariant=\"bold-italic\">x</mi><mo>⊺</mo></msup><mi mathvariant=\"bold-italic\">M</mi><mi mathvariant=\"bold-italic\">x</mi></mrow><annotation encoding=\"application/x-tex\">\\boldsymbol{x}^\\intercal \\boldsymbol{M}\\boldsymbol{x}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6861em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">x</span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6644em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin amsrm mtight\">⊺</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right:0.11424em;\">M</span></span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">x</span></span></span></span></span></span>","key":"C4Fkn9Hp7W"},{"type":"text","value":" is strictly positive. A positive semi-definite matrix requires just positivity of this quantity (it allows zero value).","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"c24pxFARrJ"}],"key":"FwVp2xlbrR"}],"number":1,"enumerator":"1","key":"hVB6tGhGd3"},{"type":"footnoteDefinition","identifier":"note_wigner_helium","label":"note_wigner_helium","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":85,"column":1},"end":{"line":85,"column":1}},"children":[{"type":"text","value":"Note also the measurement of the Wigner function of an atomic wave after a double-slit by ","position":{"start":{"line":85,"column":1},"end":{"line":85,"column":1}},"key":"dU8j2HNOp2"},{"type":"cite","identifier":"kurtsiefer_measurement_1997","label":"kurtsiefer_measurement_1997","kind":"narrative","position":{"start":{"line":85,"column":91},"end":{"line":85,"column":119}},"children":[{"type":"text","value":"Kurtsiefer 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","position":{"start":{"line":82,"column":1},"end":{"line":82,"column":1}},"key":"S93bLwBA0B"},{"type":"cite","identifier":"lvovsky_production_2020","label":"lvovsky_production_2020","kind":"narrative","position":{"start":{"line":82,"column":96},"end":{"line":82,"column":120}},"children":[{"type":"text","value":"Lvovsky ","key":"A0EQouIB9d"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"jX9DXckMfS"}],"key":"Lhny5D9yss"},{"type":"text","value":" (2020)","key":"N4Sq6cw5HQ"}],"enumerator":"40","key":"MQnagFqh0l"},{"type":"text","value":".","position":{"start":{"line":82,"column":1},"end":{"line":82,"column":1}},"key":"uFYAxpMevG"}],"key":"poKHka198J"}],"number":3,"enumerator":"3","key":"h30JcCVGto"},{"type":"footnoteDefinition","identifier":"footnote_bcs","label":"footnote_bcs","position":{"start":{"line":82,"column":1},"end":{"line":82,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"children":[{"type":"text","value":"BCS stands for Bardeen–Cooper–Schrieffer, the three scientists that devlopped the “Microscopic Theory of Superconductivity” ","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"key":"iZPVdq3XMF"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"children":[{"type":"cite","identifier":"bardeen_theory_1957","label":"bardeen_theory_1957","kind":"parenthetical","position":{"start":{"line":140,"column":126},"end":{"line":140,"column":146}},"children":[{"type":"text","value":"Bardeen ","key":"ZlOlZfR4Mx"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"UNcc3gu3t8"}],"key":"CAHXxuVCvw"},{"type":"text","value":", 1957","key":"gVnTe0VaBr"}],"enumerator":"41","key":"xTuVBBB10p"}],"key":"uUzNqWQ1il"},{"type":"text","value":".","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"key":"XwkQ0cmuP7"}],"key":"G5FdpR2YWp"}],"number":4,"enumerator":"4","key":"wjyczF1hc7"}],"key":"VKDIs3Lm1r"}],"key":"ozRX8dyJeH"},"references":{"cite":{"order":["horn_cauchy_schwarz_1990","robertson_notes_2021","wigner_quantum_1932","leonhardt_essential_2010","vogel_determination_1989","smithey_measurement_1993","lvovsky_quantum_2001","ourjoumtsev_quantum_2006","cooper_experimental_2013","ourjoumtsev_generating_2006","leibfried_experimental_1996","ourjoumtsev_etude_2007","weyl_theory_1950","case_wigner_2008","arvind_real_1995","serafini_quantum_2017","brady_gaussian_2023","weedbrook_gaussian_2012","adesso_continuous_2014","braunstein_quantum_2005","brask_gaussian_2022","serafini_entanglement_2004","pirandola_correlation_2009","brady_symplectic_2022","martin_comparing_2023","simon_gaussian_wigner_1987","simon_quantum_noise_1994","williamson_algebraic_1936","serafini_symplectic_2004","yuen_two_photon_1976","gupt_walrus_2019","barbosa_disentanglement_2011","caves_photons_1991","dodonov_photon_1994","dodonov_multidimensional_1994","berkowitz_calculation_1970","kok_multi_dimensional_2001","brandao_qugit_2022","kurtsiefer_measurement_1997","lvovsky_production_2020","bardeen_theory_1957"],"data":{"horn_cauchy_schwarz_1990":{"label":"horn_cauchy_schwarz_1990","enumerator":"1","doi":"10.1016/0024-3795(90)90256-C","html":"Horn, R. A., & Mathias, R. (1990). Cauchy-Schwarz inequalities associated with positive semidefinite matrices. <i>Linear Algebra and Its Applications</i>, <i>142</i>, 63–82. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1016/0024-3795(90)90256-C\">10.1016/0024-3795(90)90256-C</a>","url":"https://doi.org/10.1016/0024-3795(90)90256-C"},"robertson_notes_2021":{"label":"robertson_notes_2021","enumerator":"2","html":"Robertson, S. (2021). <i>Notes on COSQUA: Separability, g2, all that stuff</i>. Private communication."},"wigner_quantum_1932":{"label":"wigner_quantum_1932","enumerator":"3","doi":"10.1103/PhysRev.40.749","html":"Wigner, E. (1932). On the Quantum Correction For Thermodynamic Equilibrium. <i>Physical Review</i>, <i>40</i>(5), 749–759. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1103/PhysRev.40.749\">10.1103/PhysRev.40.749</a>","url":"https://doi.org/10.1103/PhysRev.40.749"},"leonhardt_essential_2010":{"label":"leonhardt_essential_2010","enumerator":"4","html":"Leonhardt, U. (2010). <i>Essential Quantum Optics: from quantum measurements to black holes</i>. Cambridge University Press. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://www.cambridge.org/fr/universitypress/subjects/physics/optics-optoelectronics-and-photonics/essential-quantum-optics-quantum-measurements-black-holes?format=HB&isbn=9780521869782\">https://www.cambridge.org/fr/universitypress/subjects/physics/optics-optoelectronics-and-photonics/essential-quantum-optics-quantum-measurements-black-holes?format=HB&isbn=9780521869782</a>","url":"https://www.cambridge.org/fr/universitypress/subjects/physics/optics-optoelectronics-and-photonics/essential-quantum-optics-quantum-measurements-black-holes?format=HB&isbn=9780521869782"},"vogel_determination_1989":{"label":"vogel_determination_1989","enumerator":"5","doi":"10.1103/PhysRevA.40.2847","html":"Vogel, K., & Risken, H. (1989). Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. <i>Physical Review A</i>, <i>40</i>(5), 2847–2849. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1103/PhysRevA.40.2847\">10.1103/PhysRevA.40.2847</a>","url":"https://doi.org/10.1103/PhysRevA.40.2847"},"smithey_measurement_1993":{"label":"smithey_measurement_1993","enumerator":"6","doi":"10.1103/PhysRevLett.70.1244","html":"Smithey, D. T., Beck, M., Raymer, M. G., & Faridani, A. (1993). 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Quantum State Reconstruction of the Single-Photon Fock State. <i>Physical Review Letters</i>, <i>87</i>(5), 050402. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1103/PhysRevLett.87.050402\">10.1103/PhysRevLett.87.050402</a>","url":"https://doi.org/10.1103/PhysRevLett.87.050402"},"ourjoumtsev_quantum_2006":{"label":"ourjoumtsev_quantum_2006","enumerator":"8","doi":"10.1103/PhysRevLett.96.213601","html":"Ourjoumtsev, A., Tualle-Brouri, R., & Grangier, P. (2006). Quantum Homodyne Tomography of a Two-Photon Fock State. <i>Physical Review Letters</i>, <i>96</i>(21), 213601. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1103/PhysRevLett.96.213601\">10.1103/PhysRevLett.96.213601</a>","url":"https://doi.org/10.1103/PhysRevLett.96.213601"},"cooper_experimental_2013":{"label":"cooper_experimental_2013","enumerator":"9","doi":"10.1364/OE.21.005309","html":"Cooper, M., Wright, L. J., Söller, C., & Smith, B. J. (2013). 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