{"kind":"Article","sha256":"da8cf080cdf34b215e134bbd71b2248a7360b7695db9c13ddd84b6c784500472","slug":"entanglement-4correlation","location":"/entanglement/4entanglement_4correlation.md","dependencies":[],"frontmatter":{"title":"Assessing the degree of entanglement of thermal Gaussian states with 2- and 4-body correlation functions","short_title":"Entanglement via g2 and g4","subtitle":"Witnessing entanglement with the second order correlation function","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/g2entanglement_crite-7ca9e5af5c75f5f4f197deaa8484dd54.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/g2entanglement_crite-7ca9e5af5c75f5f4f197deaa8484dd54.webp","exports":[{"format":"md","filename":"4entanglement_4correlation.md","url":"/~gondret/phd_manuscript/build/4entanglement_4corre-4bfd3ae339f4b3cfaa7c00c0c8d72da2.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":10,"column":1},"end":{"line":10,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"In this section, we explain how the second order correlation function can be used to witness entanglement and separability. We also show that if one has access to the fourth order correlation function, it is possible to assess the non-separability of the state. In other words, we show that the measurement of the populations and the second and fourth order correlation functions provides an entanglement ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"MxnseuxdGy"},{"type":"strong","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"criterion","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"inZxhzeOx0"}],"key":"SKeHMXsQ8B"},{"type":"text","value":" for thermal Gaussian states. Finally, we show that this measure gives access to the symplectic spectrum of the state and therefore quantifying entanglement, using the logarithmic negativity for example.","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"t2TAorP9Yi"}],"key":"jPZuqhpBsM"}],"data":{"part":"abstract"},"key":"xTpc3AQVOu"},{"type":"block","position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"children":[{"type":"comment","value":"In this section, I will use equivalently the index 1 for subsystem A and 2 for subsystem B.","position":{"start":{"line":15,"column":1},"end":{"line":15,"column":1}},"key":"zPpkCVrgjL"},{"type":"heading","depth":2,"position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"children":[{"type":"text","value":"What information can correlation functions say about the covariance matrix ?","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"BJCo7Fja4Y"}],"identifier":"what_info_cov_matrix","label":"what_info_cov_matrix","html_id":"what-info-cov-matrix","enumerator":"1","key":"OHiPGZLgx5"},{"type":"proof","kind":"theorem","label":"wick_theorem","identifier":"wick_theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Wick-Isserlis theorem","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"zx66jsmSGk"}],"key":"MycQKeaWwN"},{"type":"paragraph","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"children":[{"type":"text","value":"For a Gaussian state with zero mean, the expectation value of any operator reads ","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"key":"oRSfGvEHLH"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"children":[{"type":"cite","identifier":"wick_evaluation_1950","label":"wick_evaluation_1950","kind":"parenthetical","position":{"start":{"line":24,"column":83},"end":{"line":24,"column":104}},"children":[{"type":"text","value":"Wick, 1950","key":"P65PIXOlBZ"}],"enumerator":"1","key":"vWlTDf5mEt"},{"type":"cite","identifier":"isserlis_formula_1918","label":"isserlis_formula_1918","kind":"parenthetical","position":{"start":{"line":24,"column":105},"end":{"line":24,"column":127}},"children":[{"type":"text","value":"Isserlis, 1918","key":"wnD9GD3HTx"}],"enumerator":"2","key":"lzs9cRV5LM"}],"key":"ufQwZokjMD"}],"key":"mF5SwNO8GV"},{"type":"math","value":"\\braket{\\hat{b}_1\\hat{b}_2...\\hat{b}_N}=\\sum_\\text{binary contractions}\\braket{\\hat{b}_1\\hat{b}_\\alpha}\\braket{\\hat{b}_\\beta\\hat{b}_\\gamma}...\\braket{\\hat{b}_\\zeta\\hat{b}_\\omega}","tight":true,"html":"⟨b^1b^2...b^N⟩=∑binary contractions⟨b^1b^α⟩⟨b^βb^γ⟩...⟨b^ζb^ω⟩\\braket{\\hat{b}_1\\hat{b}_2...\\hat{b}_N}=\\sum_\\text{binary contractions}\\braket{\\hat{b}_1\\hat{b}_\\alpha}\\braket{\\hat{b}_\\beta\\hat{b}_\\gamma}...\\braket{\\hat{b}_\\zeta\\hat{b}_\\omega}⟨b^1​b^2​...b^N​⟩=binary contractions∑​⟨b^1​b^α​⟩⟨b^β​b^γ​⟩...⟨b^ζ​b^ω​⟩","enumerator":"1","key":"WlxqxjRhvV"},{"type":"paragraph","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"children":[{"type":"text","value":"and is zero if the number of operators is odd.","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"key":"DDrhKtE51u"}],"key":"cLeT372JoY"}],"html_id":"wick-theorem","key":"J2QS7sp942"},{"type":"paragraph","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"children":[{"type":"text","value":"The proof for this theorem and a nice discussion on its non applicability for the ","key":"Z62JrfQx7Y"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"he6qG6j6tx"}],"key":"ObXhO0qcBu"},{"type":"text","value":" condensed mode can be found in ","key":"Kn6WVoYnzW"},{"type":"cite","identifier":"castin_mecanique_2011","label":"castin_mecanique_2011","kind":"narrative","position":{"start":{"line":30,"column":118},"end":{"line":30,"column":140}},"children":[{"type":"text","value":"Castin (2011)","key":"Y0gUCTcHAj"}],"enumerator":"3","key":"PexWIAfIt1"},{"type":"text","value":". In particular, the Wick theorem under this form ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"iQl5JAXNKc"},{"type":"emphasis","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"children":[{"type":"text","value":"only applies to zero mean Gaussian states","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"Tnx4UuMB1Q"}],"key":"MUMtIhlZuI"},{"type":"text","value":" and therefore not for coherent states (displaced vacuum states). With a micro-channel plate, we measure the mode occupancy and any ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"MZhOhYhYrR"},{"type":"inlineMath","value":"n","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"html":"nnn","key":"e05qUBRUhn"},{"type":"text","value":" order correlation functions. We also measure the ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"ksL5qL1J9C"},{"type":"emphasis","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"children":[{"type":"text","value":"normal ordered field operators","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"HMkDLU6itf"}],"key":"Il8Yl7XCyn"},{"type":"footnoteReference","identifier":"foot_glauber","label":"foot_glauber","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"number":1,"enumerator":"1","key":"rJN3WNwd3t"},{"type":"text","value":" in the sense of ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"GSXl9miAvD"},{"type":"cite","identifier":"glauber_quantum_1963","label":"glauber_quantum_1963","kind":"narrative","position":{"start":{"line":30,"column":482},"end":{"line":30,"column":503}},"children":[{"type":"text","value":"Glauber (1963)","key":"DgkjJjeOO8"}],"enumerator":"4","key":"FYyDqODIAX"},{"type":"text","value":". Especially, we have access to the mode population and the second order correlation functions","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"QR6hQcrAQn"}],"key":"eVjjgKvSsC"},{"type":"math","identifier":"expression_correlation_functions","label":"expression_correlation_functions","value":"n_i := \\braket{\\hat{a}_i^\\dagger\\hat{a}_i} , \\quad \\quad \ng_{ij}^{(2)} :=\\frac{\\braket{\\hat{a}_i^\\dagger\\hat{a}_j^\\dagger\\hat{a}_i\\hat{a}_j}}{\\braket{\\hat{a}_i^\\dagger\\hat{a}_i}\\braket{\\hat{a}_j^\\dagger\\hat{a}_j}}.","tight":true,"html":"ni:=⟨a^i†a^i⟩,gij(2):=⟨a^i†a^j†a^ia^j⟩⟨a^i†a^i⟩⟨a^j†a^j⟩.n_i := \\braket{\\hat{a}_i^\\dagger\\hat{a}_i} , \\quad \\quad \ng_{ij}^{(2)} :=\\frac{\\braket{\\hat{a}_i^\\dagger\\hat{a}_j^\\dagger\\hat{a}_i\\hat{a}_j}}{\\braket{\\hat{a}_i^\\dagger\\hat{a}_i}\\braket{\\hat{a}_j^\\dagger\\hat{a}_j}}.ni​:=⟨a^i†​a^i​⟩,gij(2)​:=⟨a^i†​a^i​⟩⟨a^j†​a^j​⟩⟨a^i†​a^j†​a^i​a^j​⟩​.","enumerator":"2","html_id":"expression-correlation-functions","key":"vwqEzQg2F1"},{"type":"paragraph","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"children":[{"type":"text","value":"We now assume the state has zero mean, ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"f45AhWmQVp"},{"type":"emphasis","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"children":[{"type":"text","value":"i.e","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"jcCFEw9UXW"}],"key":"KWGddHCTrF"},{"type":"text","value":". we do not treat the ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"iI5nxbv6af"},{"type":"inlineMath","value":"k=0","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"k=0k=0k=0","key":"CIlCxCq1hz"},{"type":"text","value":" condensed mode. With that assumption, we introduce the following (complex) quantities","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"GYcTelepDh"}],"key":"U3WSSEHwLf"},{"type":"math","identifier":"definition_a_c_d","label":"definition_a_c_d","value":"\\alpha_i := \\braket{\\hat{a}_i^2} \\quad \\quad \nc := \\braket{\\hat{a}_1\\hat{a}_2} \\quad \\quad \nd := \\braket{\\hat{a}_1\\hat{a}_2^\\dagger}.","tight":true,"html":"αi:=⟨a^i2⟩c:=⟨a^1a^2⟩d:=⟨a^1a^2†⟩.\\alpha_i := \\braket{\\hat{a}_i^2} \\quad \\quad \nc := \\braket{\\hat{a}_1\\hat{a}_2} \\quad \\quad \nd := \\braket{\\hat{a}_1\\hat{a}_2^\\dagger}.αi​:=⟨a^i2​⟩c:=⟨a^1​a^2​⟩d:=⟨a^1​a^2†​⟩.","enumerator":"3","html_id":"definition-a-c-d","key":"tsuySmybzK"},{"type":"paragraph","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"children":[{"type":"text","value":"Those quantities fully determine the covariance matrix which reads","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"lm6xz46Gz3"}],"key":"hoFluDnXR4"},{"type":"math","identifier":"forme_covar_matrix","label":"forme_covar_matrix","value":"\\begin{split}\n & \\boldsymbol{A} = \\begin{pmatrix}\n2n_1+1+2\\mathcal{R}(\\alpha_1) & 2\\mathcal{I}(\\alpha_1) \\\\\n2\\mathcal{I}(\\alpha_1) &2n_1+1-2\\mathcal{R}(\\alpha_1) \\\\\n\\end{pmatrix}\\\\ \\\\\n\\boldsymbol{\\sigma}=\\begin{pmatrix}\\boldsymbol{A} & \\boldsymbol{C}\\\\\\boldsymbol{C}^\\intercal & \\boldsymbol{B}\\end{pmatrix} \\quad \\quad \\quad & \\boldsymbol{B} = \\begin{pmatrix}\n2n_2+1+2\\mathcal{R}(\\alpha_2) & 2\\mathcal{I}(\\alpha_2) \\\\\n 2\\mathcal{I}(\\alpha_2) &2n_1+1-2\\mathcal{R}(\\alpha_2) \\end{pmatrix}\\\\ \\\\\n & \\boldsymbol{C} = \\begin{pmatrix} \n2\\mathcal{R}(c+d) & 2\\mathcal{I}(c-d)\\\\\n2\\mathcal{I}(c+d) & 2\\mathcal{R}(-c+d)\n\\end{pmatrix},\n\\end{split}","tight":true,"html":"A=(2n1+1+2R(α1)2I(α1)2I(α1)2n1+1−2R(α1))σ=(ACC⊺B)B=(2n2+1+2R(α2)2I(α2)2I(α2)2n1+1−2R(α2))C=(2R(c+d)2I(c−d)2I(c+d)2R(−c+d)),\\begin{split}\n & \\boldsymbol{A} = \\begin{pmatrix}\n2n_1+1+2\\mathcal{R}(\\alpha_1) & 2\\mathcal{I}(\\alpha_1) \\\\\n2\\mathcal{I}(\\alpha_1) &2n_1+1-2\\mathcal{R}(\\alpha_1) \\\\\n\\end{pmatrix}\\\\ \\\\\n\\boldsymbol{\\sigma}=\\begin{pmatrix}\\boldsymbol{A} & \\boldsymbol{C}\\\\\\boldsymbol{C}^\\intercal & \\boldsymbol{B}\\end{pmatrix} \\quad \\quad \\quad & \\boldsymbol{B} = \\begin{pmatrix}\n2n_2+1+2\\mathcal{R}(\\alpha_2) & 2\\mathcal{I}(\\alpha_2) \\\\\n 2\\mathcal{I}(\\alpha_2) &2n_1+1-2\\mathcal{R}(\\alpha_2) \\end{pmatrix}\\\\ \\\\\n & \\boldsymbol{C} = \\begin{pmatrix} \n2\\mathcal{R}(c+d) & 2\\mathcal{I}(c-d)\\\\\n2\\mathcal{I}(c+d) & 2\\mathcal{R}(-c+d)\n\\end{pmatrix},\n\\end{split}σ=(AC⊺​CB​)​A=(2n1​+1+2R(α1​)2I(α1​)​2I(α1​)2n1​+1−2R(α1​)​)B=(2n2​+1+2R(α2​)2I(α2​)​2I(α2​)2n1​+1−2R(α2​)​)C=(2R(c+d)2I(c+d)​2I(c−d)2R(−c+d)​),​","enumerator":"4","html_id":"forme-covar-matrix","key":"y2eviMRS0H"},{"type":"paragraph","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"key":"myXrlKRNKv"},{"type":"inlineMath","value":"\\mathcal{I}(x)","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"html":"I(x)\\mathcal{I}(x)I(x)","key":"RsEt5DMWfO"},{"type":"text","value":" and ","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"key":"IyngwrBjd0"},{"type":"inlineMath","value":"\\mathcal{R}(x)","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"html":"R(x)\\mathcal{R}(x)R(x)","key":"GF9ByRfFJr"},{"type":"text","value":" refers to the imaginary and real parts of ","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"key":"WLTPYm1n7y"},{"type":"inlineMath","value":"x","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"html":"xxx","key":"YhYBNaw8nQ"},{"type":"text","value":". Using Wick expansion, one can show that the second order correlation functions is","position":{"start":{"line":60,"column":1},"end":{"line":60,"column":1}},"key":"QvDUU9dKFF"}],"key":"Bz2Texmr3I"},{"type":"math","identifier":"fct_g2_local","label":"fct_g2_local","value":"g_{ii}^{(2)} = 2 + \\frac{|\\alpha_i|^2}{|n_i|^2}.","tight":true,"html":"gii(2)=2+∣αi∣2∣ni∣2.g_{ii}^{(2)} = 2 + \\frac{|\\alpha_i|^2}{|n_i|^2}.gii(2)​=2+∣ni​∣2∣αi​∣2​.","enumerator":"5","html_id":"fct-g2-local","key":"x44SDgYiZN"},{"type":"paragraph","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"children":[{"type":"text","value":"In other words, if one finds that the second order correlation function is two, this implies that ","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"key":"Fo8NDOg3VX"},{"type":"inlineMath","value":"\\alpha_i","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"html":"αi\\alpha_iαi​","key":"DaN0rLIrft"},{"type":"text","value":" = 0. In the following, we will assume this is the case. This assumption is motivated by the fact that it is what we expect from the Hamiltonian seen in the first chapter. It is also observed experimentally: it was reported in different configurations (but still with the same physics at play) by other authors ","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"key":"nrLzXMD6mP"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"children":[{"type":"cite","identifier":"dall_ideal_2013","label":"dall_ideal_2013","kind":"parenthetical","position":{"start":{"line":65,"column":421},"end":{"line":65,"column":437}},"children":[{"type":"text","value":"Dall ","key":"Go76Y5Se0o"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"GPVuAOtrrF"}],"key":"pO25pkCCGq"},{"type":"text","value":", 2013","key":"fqzXk5Andv"}],"enumerator":"5","key":"rKbKV9MjZO"},{"type":"cite","identifier":"perrier_thermal_2019","label":"perrier_thermal_2019","kind":"parenthetical","position":{"start":{"line":65,"column":438},"end":{"line":65,"column":459}},"children":[{"type":"text","value":"Perrier ","key":"A1SWkJuYLa"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"gbY1dIvVRY"}],"key":"xS2urR5OwP"},{"type":"text","value":", 2019","key":"VuI3SisP82"}],"enumerator":"6","key":"JDPR9ws4Og"},{"type":"cite","identifier":"herce_full_2023","label":"herce_full_2023","kind":"parenthetical","position":{"start":{"line":65,"column":460},"end":{"line":65,"column":476}},"children":[{"type":"text","value":"Hercé ","key":"vNRrOEWmCP"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ZZsWWrUd8Y"}],"key":"iKmrr643PA"},{"type":"text","value":", 2023","key":"smKK1vccUu"}],"enumerator":"7","key":"D6wmWAHFNl"}],"key":"AZXVm7kqq0"},{"type":"text","value":". This assumption greatly simplifies the covariance matrix which reads","position":{"start":{"line":65,"column":1},"end":{"line":65,"column":1}},"key":"BmnTOmxr8f"}],"key":"K3KV77Y6QW"},{"type":"math","identifier":"covariance_matrix_part3","label":"covariance_matrix_part3","value":"\\begin{split}\n&\\boldsymbol{A} = (2n_1+1)\\mathbb{I}_2 \\quad \\quad \\boldsymbol{B} = (2n_2+1)\\mathbb{I}_2 \\\\ \\\\& \\boldsymbol{C}=\n\\begin{pmatrix} \n2\\mathcal{R}(c+d) & 2\\mathcal{I}(c-d)\\\\\n2\\mathcal{I}(c+d) & 2\\mathcal{R}(-c+d)\n\\end{pmatrix}.\n\\end{split}","tight":true,"html":"A=(2n1+1)I2B=(2n2+1)I2C=(2R(c+d)2I(c−d)2I(c+d)2R(−c+d)).\\begin{split}\n&\\boldsymbol{A} = (2n_1+1)\\mathbb{I}_2 \\quad \\quad \\boldsymbol{B} = (2n_2+1)\\mathbb{I}_2 \\\\ \\\\& \\boldsymbol{C}=\n\\begin{pmatrix} \n2\\mathcal{R}(c+d) & 2\\mathcal{I}(c-d)\\\\\n2\\mathcal{I}(c+d) & 2\\mathcal{R}(-c+d)\n\\end{pmatrix}.\n\\end{split}​A=(2n1​+1)I2​B=(2n2​+1)I2​C=(2R(c+d)2I(c+d)​2I(c−d)2R(−c+d)​).​","enumerator":"6","html_id":"covariance-matrix-part3","key":"h5mJVogCtT"},{"type":"paragraph","position":{"start":{"line":76,"column":1},"end":{"line":76,"column":1}},"children":[{"type":"text","value":"We can now consider the second-order cross correlation function","position":{"start":{"line":76,"column":1},"end":{"line":76,"column":1}},"key":"tlKIvHX8Z5"}],"key":"JgC36gDlQb"},{"type":"math","identifier":"g2_expression_math","label":"g2_expression_math","value":"g^{(2)}_{12} = \\frac{n_1n_2 + |c|^2 + |d|^2}{n_1n_2}.","tight":true,"html":"g12(2)=n1n2+∣c∣2+∣d∣2n1n2.g^{(2)}_{12} = \\frac{n_1n_2 + |c|^2 + |d|^2}{n_1n_2}.g12(2)​=n1​n2​n1​n2​+∣c∣2+∣d∣2​.","enumerator":"7","html_id":"g2-expression-math","key":"ymaPPxklko"},{"type":"paragraph","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"If we ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"i0nswUJDf3"},{"type":"emphasis","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"assume","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"eFheutjhi9"}],"key":"WkWYfnhO9A"},{"type":"text","value":" ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"f35w1bqoVB"},{"type":"inlineMath","value":"d=\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}=0","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"d=⟨a^1a^2†⟩=0d=\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}=0d=⟨a^1​a^2†​⟩=0","key":"MOrA2u4DC5"},{"type":"text","value":", ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"njj2HBWgHn"},{"type":"emphasis","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"i.e","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"nl0HSCUZaO"}],"key":"KVm4pxYWGY"},{"type":"text","value":" that we directly measure the pure correlation term ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"HuqXZ3nSWL"},{"type":"inlineMath","value":"|c|","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"∣c∣|c|∣c∣","key":"ksMByycR49"},{"type":"text","value":", an observation of ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"EicnwT48Af"},{"type":"inlineMath","value":"g^{(2)}_{12}>2","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"g12(2)>2g^{(2)}_{12}>2g12(2)​>2","key":"OkpnhWvL1G"},{"type":"text","value":" implies entanglement. The ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"D6tkVtji7o"},{"type":"inlineMath","value":"d=0","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"d=0d=0d=0","key":"HOcG5GxfI0"},{"type":"text","value":" assumption implies we exactly measure ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"rnJjNCS9XK"},{"type":"inlineMath","value":"\\Delta ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"Δ\\Delta Δ","key":"gfFybyEl5l"},{"type":"text","value":": this is the ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"ZKsaorosiB"},{"type":"crossReference","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"Campo-Parentani","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"eoT8DtDpML"}],"identifier":"busch","label":"busch","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"busch","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"HBDU9PKso8"},{"type":"text","value":" criterion, defined in ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"KJZZQPUhnl"},{"type":"crossReference","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"m2I06QsQZ6"}],"identifier":"other_entanglement_witness","label":"other_entanglement_witness","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"other-entanglement-witness","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"WxkQHE0L2z"},{"type":"text","value":" ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"vNU7MfTfyo"},{"type":"crossReference","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"4","key":"dAvoOb3KuC"}],"identifier":"other_entanglement_witness","label":"other_entanglement_witness","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"other-entanglement-witness","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"grpJU6WDv3"},{"type":"text","value":". In ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"AtdMLk16XT"},{"type":"crossReference","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"L4XBClFJGS"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","key":"IyMvNWHTVd"},{"type":"text","value":" ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"tpeL9AUneF"},{"type":"crossReference","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"children":[{"type":"text","value":"2","key":"ipLLj207W0"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","key":"UaMPb4WTK6"},{"type":"text","value":", we relax the ","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"hRyvkBzWlR"},{"type":"inlineMath","value":"d=0","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"html":"d=0d=0d=0","key":"fTpPWLWsDj"},{"type":"text","value":" assumption and show that that the second order correlation function still witnesses entanglement.","position":{"start":{"line":81,"column":1},"end":{"line":81,"column":1}},"key":"mGqb3XnPoS"}],"key":"cgVYqQkWTT"},{"type":"heading","depth":2,"position":{"start":{"line":88,"column":1},"end":{"line":88,"column":1}},"children":[{"type":"text","value":"The second order correlation function to probe non-separability","position":{"start":{"line":88,"column":1},"end":{"line":88,"column":1}},"key":"au4XH7Vcty"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","html_id":"g2-fnc-sect","enumerator":"2","key":"CQI13dmWDA"},{"type":"paragraph","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"children":[{"type":"text","value":"As introduced in the ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"DgEmkQ2MT7"},{"type":"crossReference","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"children":[{"type":"text","value":"first section","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"hxfvbqpGmi"}],"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","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"bgrbyhC9Rm"},{"type":"text","value":", because ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"NvJ401xFIk"},{"type":"inlineMath","value":"\\hat{\\rho}","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"html":"ρ^\\hat{\\rho}ρ^​","key":"bkIT7qzTBK"},{"type":"text","value":" is positive definite, the Cauchy-Schwarz inequality ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"ruR2aGxf7p"},{"type":"crossReference","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"children":[{"type":"text","value":"(","key":"fkL96gqrLX"},{"type":"text","value":"2","key":"gHrgXIJr4l"},{"type":"text","value":")","key":"drnDvbxXF1"}],"identifier":"cauchy_schwarz_inequality_always","label":"cauchy_schwarz_inequality_always","kind":"equation","template":"(%s)","enumerator":"2","resolved":true,"html_id":"cauchy-schwarz-inequality-always","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"wYcMZ322HT"},{"type":"text","value":" implies that ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"z5A7PpxlfV"},{"type":"inlineMath","value":"\\forall \\hat{A},\\hat{B}, \\, \\, |\\braket{\\hat{A}^\\dagger \\hat{B}}|^2 \\leq \\braket{\\hat{A}^\\dagger\\hat{A}}\\braket{\\hat{B}^\\dagger\\hat{B}}","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"html":"∀A^,B^,  ∣⟨A^†B^⟩∣2≤⟨A^†A^⟩⟨B^†B^⟩\\forall \\hat{A},\\hat{B}, \\, \\, |\\braket{\\hat{A}^\\dagger \\hat{B}}|^2 \\leq \\braket{\\hat{A}^\\dagger\\hat{A}}\\braket{\\hat{B}^\\dagger\\hat{B}}∀A^,B^,∣⟨A^†B^⟩∣2≤⟨A^†A^⟩⟨B^†B^⟩","key":"WmBxz3Ky5y"},{"type":"text","value":". We can use this inequality with ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"IUlH7BfmlI"},{"type":"inlineMath","value":"\\hat{a}_{1}","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"html":"a^1\\hat{a}_{1}a^1​","key":"kbOAVqjUnS"},{"type":"text","value":" and ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"VKqjbDvNZ8"},{"type":"inlineMath","value":"\\hat{a}_2","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"html":"a^2\\hat{a}_2a^2​","key":"SpFkGUZF0f"},{"type":"text","value":" to show that ","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"key":"ZhJMDFE4PK"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":90,"column":1},"end":{"line":90,"column":1}},"children":[{"type":"cite","identifier":"robertson_notes_2021","label":"robertson_notes_2021","kind":"parenthetical","position":{"start":{"line":90,"column":410},"end":{"line":90,"column":431}},"children":[{"type":"text","value":"Robertson, 2021","key":"NytRlHXoVD"}],"enumerator":"8","key":"IslUWfhQcR"}],"key":"kny9wfXgPx"}],"key":"bUpRBsNa3P"},{"type":"math","identifier":"bounds_for_a1a2","label":"bounds_for_a1a2","value":"\\begin{split}\n|d|^2 = |\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}|^2&\\leq n_1n_2\\\\\n|c|^2 = |\\braket{\\hat{a}_1\\hat{a}_2}|^2&\\leq n_1n_2 +n_1\\\\\n& \\leq n_1n_2 +n_2.\n\\end{split}","tight":true,"html":"∣d∣2=∣⟨a^1a^2†⟩∣2≤n1n2∣c∣2=∣⟨a^1a^2⟩∣2≤n1n2+n1≤n1n2+n2.\\begin{split}\n|d|^2 = |\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}|^2&\\leq n_1n_2\\\\\n|c|^2 = |\\braket{\\hat{a}_1\\hat{a}_2}|^2&\\leq n_1n_2 +n_1\\\\\n& \\leq n_1n_2 +n_2.\n\\end{split}∣d∣2=∣⟨a^1​a^2†​⟩∣2∣c∣2=∣⟨a^1​a^2​⟩∣2​≤n1​n2​≤n1​n2​+n1​≤n1​n2​+n2​.​","enumerator":"8","html_id":"bounds-for-a1a2","key":"ygWuhF5fjm"},{"type":"paragraph","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"children":[{"type":"text","value":"We can express the modulus of ","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"key":"s4Bw1fXP9g"},{"type":"inlineMath","value":"c","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"html":"ccc","key":"SmKP5o0Fjo"},{"type":"text","value":" as a function of the second order correlation function ","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"key":"t4htc6j0Mn"},{"type":"crossReference","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"children":[{"type":"text","value":"(","key":"LDWmoJAltl"},{"type":"text","value":"7","key":"GOsgRysHaM"},{"type":"text","value":")","key":"PoXk6q2eap"}],"identifier":"g2_expression_math","label":"g2_expression_math","kind":"equation","template":"(%s)","enumerator":"7","resolved":true,"html_id":"g2-expression-math","key":"wCEDPbO38A"}],"key":"XJdap5hnIQ"},{"type":"math","value":"|c|^2 = (g^{(2)}_{12} - 1)n_1n_2 - |d|^2","tight":true,"html":"∣c∣2=(g12(2)−1)n1n2−∣d∣2|c|^2 = (g^{(2)}_{12} - 1)n_1n_2 - |d|^2∣c∣2=(g12(2)​−1)n1​n2​−∣d∣2","enumerator":"9","key":"FfFXeKLetL"},{"type":"paragraph","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"children":[{"type":"text","value":"From the Cauchy-Schwarz inequality ","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"key":"BElBAxsuRC"},{"type":"crossReference","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"children":[{"type":"text","value":"(","key":"az16laY1Zf"},{"type":"text","value":"8","key":"w8dMPHl1lB"},{"type":"text","value":")","key":"P9Ob0XmKhn"}],"identifier":"bounds_for_a1a2","label":"bounds_for_a1a2","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"bounds-for-a1a2","key":"cEL0Qq4mrK"},{"type":"text","value":", we have ","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"key":"E1cnFxndLX"},{"type":"inlineMath","value":"|d|^2∣d∣2<n1n2|d|^2<n_1n_2∣d∣2<n1​n2​","key":"Z2RdMcCggM"},{"type":"text","value":" which means that","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"key":"M56WNLcZyC"}],"key":"vgLr8MrGAe"},{"type":"math","value":"|c|^2 \\geq (g^{(2)}_{12} - 2)n_1n_2","tight":true,"html":"∣c∣2≥(g12(2)−2)n1n2|c|^2 \\geq (g^{(2)}_{12} - 2)n_1n_2∣c∣2≥(g12(2)​−2)n1​n2​","enumerator":"10","key":"G1rC8DCG1K"},{"type":"paragraph","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"children":[{"type":"text","value":"If ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"ir1raPiGhF"},{"type":"inlineMath","value":"g^{(2)}>3","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"g(2)>3g^{(2)}>3g(2)>3","key":"ZTS3TtctNH"},{"type":"text","value":", this means that ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"X037n0ocPO"},{"type":"inlineMath","value":"|c|^2>n_1n_2","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"∣c∣2>n1n2|c|^2>n_1n_2∣c∣2>n1​n2​","key":"jYPSWaN5PY"},{"type":"text","value":" and therefore that the state is entangled. As a conclusion, ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"V7fYh5oHgK"},{"type":"inlineMath","value":"g^{(2)}> 3","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"g(2)>3g^{(2)}> 3g(2)>3","key":"wtwnxf1Zb0"},{"type":"text","value":" implies entanglement. This bound is however really constraining: for an ideal two-mode squeezed vacuum state, the second order correlation function is ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"H6GIPcvEbu"},{"type":"inlineMath","value":"2+1/n","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"2+1/n2+1/n2+1/n","key":"mmsozYjORo"},{"type":"text","value":" where ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"uq45sYbVX8"},{"type":"inlineMath","value":"n","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"nnn","key":"W39eJ5Ekl3"},{"type":"text","value":" is the population of one mode. Therefore, this ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"PqwKh7zS34"},{"type":"inlineMath","value":"g^{(2)}>3","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"g(2)>3g^{(2)}>3g(2)>3","key":"mm1CgcTyrP"},{"type":"text","value":" bound allows us to witness entanglement only for low population ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"fx5RripuXU"},{"type":"emphasis","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"KnsFxcJRZi"}],"key":"ubIXTXs8GA"},{"type":"text","value":" for ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"Hh9LgkziR0"},{"type":"inlineMath","value":"n<1","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"html":"n<1n<1n<1","key":"bw5oKvB06Y"},{"type":"text","value":".","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"L5Zm9rpu3x"}],"key":"LuWSt1Ykzc"},{"type":"paragraph","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"children":[{"type":"text","value":"The defect of this entanglement witness is due to the fact that the increase we took for ","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"Ad9uurevns"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"html":"∣d∣|d|∣d∣","key":"OUjG9rpiWU"},{"type":"text","value":" is the highest possible: ","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"ZJp4338GRP"},{"type":"inlineMath","value":"n_1n_2","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"html":"n1n2n_1n_2n1​n2​","key":"RO1QeOolL1"},{"type":"text","value":". We also assumed that a non-zero value of ","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"xaoBjGJqQm"},{"type":"inlineMath","value":"d","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"html":"ddd","key":"clWRZ3LOSM"},{"type":"text","value":" cannot participate in the non-separability. We now find a more constraining bound on ","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"GCzaeL4V53"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"html":"∣d∣|d|∣d∣","key":"WNlQPLadEO"},{"type":"text","value":" to derive the following “","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"L6C0IDk86t"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"a5Q69QyYSt"},{"type":"text","value":" entanglement witness”.","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"mZIAejdwq4"}],"key":"KkIFvzXrKG"},{"type":"proof","kind":"theorem","label":"g2_criterion_theorem","identifier":"g2_criterion_theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"g2 entanglement witness","position":{"start":{"line":113,"column":1},"end":{"line":113,"column":1}},"key":"nZArQgZETT"}],"key":"aPHd7kfsek"},{"type":"paragraph","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"children":[{"type":"text","value":"Assuming that the state is Gaussian with zero mean (","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"key":"NDHsbLGaDn"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i}=0","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"html":"⟨a^i⟩=0\\braket{\\hat{a}_i}=0⟨a^i​⟩=0","key":"Q6E4kUyF9s"},{"type":"text","value":") and that ","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"key":"cEY8MizkJu"},{"type":"inlineMath","value":"|\\hat{a}_1^2|=|\\hat{a}_2^2|=0","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"html":"∣a^12∣=∣a^22∣=0|\\hat{a}_1^2|=|\\hat{a}_2^2|=0∣a^12​∣=∣a^22​∣=0","key":"TzXjuHvw0j"},{"type":"footnoteReference","identifier":"footnote_measure_not_assume","label":"footnote_measure_not_assume","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"number":2,"enumerator":"2","key":"nEpllzDjeh"},{"type":"text","value":" the two-mode state 1-2 is entangled if ","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"key":"oOrZs5LjAe"},{"type":"inlineMath","value":"g_{12}^{(2)}","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"html":"g12(2)g_{12}^{(2)}g12(2)​","key":"DkBtL1E8cg"},{"type":"text","value":" is slightly above 2. In particular,","position":{"start":{"line":116,"column":1},"end":{"line":116,"column":1}},"key":"AAvvj1ndDt"}],"key":"OYVvIKnVa6"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":117,"column":1},"end":{"line":118,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"key":"ZUqD03j7Yc"},{"type":"inlineMath","value":"n_1n_2 \\geq 1/2","position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"html":"n1n2≥1/2n_1n_2 \\geq 1/2n1​n2​≥1/2","key":"lCYAkrWdVq"},{"type":"text","value":", then ","position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"key":"wDYjIEIGnl"},{"type":"inlineMath","value":"g_{12}^{(2)}>2","position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"html":"g12(2)>2g_{12}^{(2)}>2g12(2)​>2","key":"O8AC8sgjgK"},{"type":"text","value":" is a sufficient condition,","position":{"start":{"line":117,"column":1},"end":{"line":117,"column":1}},"key":"eFQ0GLiz2B"}],"key":"xYE5CW3QVU"},{"type":"listItem","spread":true,"position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"ypjot4mpF8"},{"type":"inlineMath","value":"n_1n_2<1/2","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"n1n2<1/2n_1n_2<1/2n1​n2​<1/2","key":"g53vtZTCy2"},{"type":"text","value":", the threshold for ","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"IvV1QuwGIr"},{"type":"inlineMath","value":"g_{12}^{(2)}","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"html":"g12(2)g_{12}^{(2)}g12(2)​","key":"eHwm5pRVg3"},{"type":"text","value":" is shifted and the sufficient condition for non-separability reads","position":{"start":{"line":118,"column":1},"end":{"line":118,"column":1}},"key":"baPC73Ysiw"}],"key":"iAsPOlLtWt"}],"key":"LNWIHGtlIp"},{"type":"math","identifier":"fg2","label":"Fg2","value":"g_{12}^{(2)}>2 +\\frac{1/2 -n_1n_2}{2n_1n_2 + n_1 + n_2 +1/2}\\, \\, .","tight":"before","html":"g12(2)>2+1/2−n1n22n1n2+n1+n2+1/2  .g_{12}^{(2)}>2 +\\frac{1/2 -n_1n_2}{2n_1n_2 + n_1 + n_2 +1/2}\\, \\, .g12(2)​>2+2n1​n2​+n1​+n2​+1/21/2−n1​n2​​.","enumerator":"11","html_id":"fg2","key":"kR8AarQtBQ"}],"html_id":"g2-criterion-theorem","key":"TfQmbh6ozq"},{"type":"proof","kind":"proof","label":"proof_nonseparability_criterion","identifier":"proof_nonseparability_criterion","class":"dropdown","enumerated":false,"children":[{"type":"paragraph","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"children":[{"type":"text","value":"As we assumed the state is Gaussian, the g","key":"nL3CFQndzL"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"PUOeDFR6rn"}],"key":"J5tTlmEWdf"},{"type":"text","value":" criterion assesses the non-separability of the state. It is entangled if and only if Simon’s quantity ","key":"BUDOJfuROA"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"VahPGYWyso"},{"type":"text","value":" defined in equation ","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"key":"zV7W9D5m2l"},{"type":"crossReference","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"children":[{"type":"text","value":"(","key":"nsI1bBnF2x"},{"type":"text","value":"10","key":"uBXdjGarf9"},{"type":"text","value":")","key":"TrauMQStll"}],"identifier":"pminus_definition","label":"pminus_definition","kind":"equation","template":"(%s)","enumerator":"10","resolved":true,"html_id":"pminus-definition","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"o88jXIxN9I"},{"type":"text","value":" is negative ","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"key":"xeLxc6Fjbv"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"children":[{"type":"cite","identifier":"simon_peres_horodecki_2000","label":"simon_peres_horodecki_2000","kind":"parenthetical","position":{"start":{"line":130,"column":221},"end":{"line":130,"column":248}},"children":[{"type":"text","value":"Simon, 2000","key":"XjE71lgLEf"}],"enumerator":"9","key":"HgLCF9ls9b"}],"key":"ZcxuvDwKIq"},{"type":"text","value":". In our case, this quantity reads","position":{"start":{"line":130,"column":1},"end":{"line":130,"column":1}},"key":"r0Irci63sr"}],"key":"XrgWHnZGHD"},{"type":"math","identifier":"pminus_demonstration","label":"pminus_demonstration","value":"\\begin{split}\n\\mathcal{P}_- = 16 \\biggl[ & (1 + n_1)(1 + n_2)(n_1n_2 - |d|^2 - |c|^2) \\\\\n & +\\left( \\frac{1}{2}- n_1n_2\\right) \\left(|d|^2 + |c|^2\\right) \\\\\n &+ (|c|^2 - |d|^2)^2 - \\frac{1}{2}\\big| |c|^2 - |d|^2\\big| \\biggr].\n\\end{split}","tight":true,"html":"P−=16[(1+n1)(1+n2)(n1n2−∣d∣2−∣c∣2)+(12−n1n2)(∣d∣2+∣c∣2)+(∣c∣2−∣d∣2)2−12∣∣c∣2−∣d∣2∣].\\begin{split}\n\\mathcal{P}_- = 16 \\biggl[ & (1 + n_1)(1 + n_2)(n_1n_2 - |d|^2 - |c|^2) \\\\\n & +\\left( \\frac{1}{2}- n_1n_2\\right) \\left(|d|^2 + |c|^2\\right) \\\\\n &+ (|c|^2 - |d|^2)^2 - \\frac{1}{2}\\big| |c|^2 - |d|^2\\big| \\biggr].\n\\end{split}P−​=16[​(1+n1​)(1+n2​)(n1​n2​−∣d∣2−∣c∣2)+(21​−n1​n2​)(∣d∣2+∣c∣2)+(∣c∣2−∣d∣2)2−21​​∣c∣2−∣d∣2​].​","enumerator":"12","html_id":"pminus-demonstration","key":"fzv2g3T4KP"},{"type":"paragraph","position":{"start":{"line":139,"column":1},"end":{"line":140,"column":1}},"children":[{"type":"text","value":"In this expression, we grouped the terms involving the sum ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"aqQa0q6u9g"},{"type":"inlineMath","value":"|d|^2+|c|^2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"∣d∣2+∣c∣2|d|^2+|c|^2∣d∣2+∣c∣2","key":"uHFfVVGoZY"},{"type":"text","value":": they are known through the value of ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"Ly00Tiyork"},{"type":"inlineMath","value":"g^{(2)}_{12} = 1+(|c|^2+|d|^2)/n_1n_2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"g12(2)=1+(∣c∣2+∣d∣2)/n1n2g^{(2)}_{12} = 1+(|c|^2+|d|^2)/n_1n_2g12(2)​=1+(∣c∣2+∣d∣2)/n1​n2​","key":"OfmrFe2bFl"},{"type":"text","value":". The ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"UIu6V9bVRX"},{"type":"emphasis","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"children":[{"type":"text","value":"a priori","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"pu1sMlIQ0h"}],"key":"wVuSXi7lY9"},{"type":"text","value":" unknown quantity is the difference ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"Hs031pT75O"},{"type":"inlineMath","value":"|c|^2-|d|^2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"∣c∣2−∣d∣2|c|^2-|d|^2∣c∣2−∣d∣2","key":"Cq7VsRBGJD"},{"type":"text","value":". This proof consists of showing that this value is bounded and that it cannot be arbitrarily large: this quantity cannot change too much the sign of ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"kgqVaqCjou"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"Fk0sagvBUW"},{"type":"text","value":".","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"gtUJ6iaUyF"},{"type":"break","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"KMpXRPTd1O"},{"type":"text","value":"As stated in equation ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"fSL6GPtuXP"},{"type":"crossReference","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"children":[{"type":"text","value":"(","key":"pOoHUT8TB9"},{"type":"text","value":"8","key":"QR92KlTjVL"},{"type":"text","value":")","key":"zm3w7mXxth"}],"identifier":"bounds_for_a1a2","label":"bounds_for_a1a2","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"bounds-for-a1a2","key":"pMfcvZoyuy"},{"type":"text","value":", the value of ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"zAtkGhmMTI"},{"type":"inlineMath","value":"|c|^2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"∣c∣2|c|^2∣c∣2","key":"W8tcg4oakD"},{"type":"text","value":" and ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"w9dok7IOUK"},{"type":"inlineMath","value":"|d|^2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"∣d∣2|d|^2∣d∣2","key":"ASN9SassG5"},{"type":"text","value":" are bounded by ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"OAc9TxeWiu"},{"type":"inlineMath","value":"n_1n_2+n_1","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"n1n2+n1n_1n_2+n_1n1​n2​+n1​","key":"mKrCjus2oO"},{"type":"text","value":" and ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"kZ5c121uTk"},{"type":"inlineMath","value":"n_1n_2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"n1n2n_1n_2n1​n2​","key":"YghHtCo6aT"},{"type":"text","value":". Here we consider that ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"OFnbGlb0Cl"},{"type":"inlineMath","value":"n_1\\leq n_2","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"n1≤n2n_1\\leq n_2n1​≤n2​","key":"ipYQuAYFpe"},{"type":"text","value":" without loss of generality, not to make the discussion more cumbersome. We can therefore define ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"ljQTekOb1P"},{"type":"text","value":"δ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"PByJJ34Tgf"},{"type":"text","value":" such that","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"DJ5pq6gZ1n"}],"key":"kE5qWdAtuH"},{"type":"math","identifier":"definition_delta_pminus","label":"definition_delta_pminus","value":"\\big| |c|^2 - |d|^2\\big| := \\delta n_1n_2 ,\\quad \\quad \\delta \\in \\big[0, 1+\\frac{1}{n_2}\\big].","tight":true,"html":"∣∣c∣2−∣d∣2∣:=δn1n2,δ∈[0,1+1n2].\\big| |c|^2 - |d|^2\\big| := \\delta n_1n_2 ,\\quad \\quad \\delta \\in \\big[0, 1+\\frac{1}{n_2}\\big].​∣c∣2−∣d∣2​:=δn1​n2​,δ∈[0,1+n2​1​].","enumerator":"13","html_id":"definition-delta-pminus","key":"IvzokrpUnI"},{"type":"paragraph","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"children":[{"type":"text","value":"With that notation, the entanglement criterion ","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"key":"BBC6Ss8g3e"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"FJAfiZNSWN"},{"type":"text","value":" reads","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"key":"A7MWBInonN"}],"key":"o7lO366Zzs"},{"type":"math","identifier":"pminus_calculation_1","label":"Pminus_calculation_1","value":"\\begin{split}\n\\mathcal{P}_- = 16 n_1n_2\\biggl[ & (1 + n_1)(1 + n_2)(2 -g^{(2)}_{12} ) \\\\\n & +\\left( \\frac{1}{2}- n_1n_2\\right) \\left(g^{(2)}_{12} -1\\right) \\\\\n &+ \\delta\\left(n_1n_2\\delta - \\frac{1}{2}\\right) \\biggr].\n\\end{split}","tight":true,"html":"P−=16n1n2[(1+n1)(1+n2)(2−g12(2))+(12−n1n2)(g12(2)−1)+δ(n1n2δ−12)].\\begin{split}\n\\mathcal{P}_- = 16 n_1n_2\\biggl[ & (1 + n_1)(1 + n_2)(2 -g^{(2)}_{12} ) \\\\\n & +\\left( \\frac{1}{2}- n_1n_2\\right) \\left(g^{(2)}_{12} -1\\right) \\\\\n &+ \\delta\\left(n_1n_2\\delta - \\frac{1}{2}\\right) \\biggr].\n\\end{split}P−​=16n1​n2​[​(1+n1​)(1+n2​)(2−g12(2)​)+(21​−n1​n2​)(g12(2)​−1)+δ(n1​n2​δ−21​)].​","enumerator":"14","html_id":"pminus-calculation-1","key":"dkVpY5Uk9r"},{"type":"paragraph","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"children":[{"type":"text","value":"Assume the sum of the first line is negative: the state is entangled if the last term ","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"key":"SCkfbiYsuC"},{"type":"inlineMath","value":"\\delta (n_1n_2\\delta -1/2)","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"html":"δ(n1n2δ−1/2)\\delta (n_1n_2\\delta -1/2)δ(n1​n2​δ−1/2)","key":"CahvUNBVje"},{"type":"text","value":" is not too large. The question we are interested in here is: what is the highest value of ","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"key":"vYesAYJeHd"},{"type":"inlineMath","value":"\\delta (n_1n_2\\delta -1/2)","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"html":"δ(n1n2δ−1/2)\\delta (n_1n_2\\delta -1/2)δ(n1​n2​δ−1/2)","key":"A8SqmBKdlI"},{"type":"text","value":" so that ","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"key":"wND5ClWOY9"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"aDPauTfMxH"},{"type":"text","value":" is negative without it and positive when taking it into account. We look for an upper bound for ","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"key":"wUJoJLP1KB"},{"type":"inlineMath","value":"\\delta (n_1n_2\\delta -1/2)","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"html":"δ(n1n2δ−1/2)\\delta (n_1n_2\\delta -1/2)δ(n1​n2​δ−1/2)","key":"UnwjmdxsYh"},{"type":"text","value":".","position":{"start":{"line":154,"column":1},"end":{"line":154,"column":1}},"key":"MWCjKGQRCv"}],"key":"Af7X3bK4IH"},{"type":"paragraph","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"We first prove that we can take ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"dKJw6ibGHW"},{"type":"inlineMath","value":"\\delta \\leq 1","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"δ≤1\\delta \\leq 1δ≤1","key":"Iqvc8qTI6J"},{"type":"text","value":" by absurdity. Let’s assume a state is such that ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"SYZdOqFllL"},{"type":"inlineMath","value":"\\mathcal{P}_-\\geq 0","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"P−≥0\\mathcal{P}_-\\geq 0P−​≥0","key":"YLLZZVAysZ"},{"type":"text","value":" and ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"UIhC2PgoZh"},{"type":"inlineMath","value":"\\delta >1","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"δ>1\\delta >1δ>1","key":"a1eFtgIbqP"},{"type":"text","value":". The latter condition implies from ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"d348I5ESiO"},{"type":"crossReference","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"(","key":"LG0aL96Wan"},{"type":"text","value":"13","key":"ldqctIHU5K"},{"type":"text","value":")","key":"qEhDoqdJ34"}],"identifier":"definition_delta_pminus","label":"definition_delta_pminus","kind":"equation","template":"(%s)","enumerator":"13","resolved":true,"html_id":"definition-delta-pminus","key":"CFQBGQqM9Q"},{"type":"text","value":" that either ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"b0fKOmz3Vi"},{"type":"inlineMath","value":"|c|^2 > n_1n_2","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"∣c∣2>n1n2|c|^2 > n_1n_2∣c∣2>n1​n2​","key":"rROZsIweEA"},{"type":"text","value":" or ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"gWfVij7LNa"},{"type":"inlineMath","value":"|d|^2> n_1n_2","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"∣d∣2>n1n2|d|^2> n_1n_2∣d∣2>n1​n2​","key":"cPGQDLQGaG"},{"type":"text","value":". This second possibility is impossible because of the Cauchy-Schwarz inequality ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"X6GGF68Hin"},{"type":"crossReference","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"(","key":"QJQZuOC4N0"},{"type":"text","value":"8","key":"YsSAcKvxMx"},{"type":"text","value":")","key":"gxzaNfBGak"}],"identifier":"bounds_for_a1a2","label":"bounds_for_a1a2","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"bounds-for-a1a2","key":"iyWeHn6GVe"},{"type":"text","value":". We therefore must have that ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"RQeuGFS45O"},{"type":"inlineMath","value":"|c|^2>n_1n_2","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"∣c∣2>n1n2|c|^2>n_1n_2∣c∣2>n1​n2​","key":"mOPJavAY76"},{"type":"text","value":". This last inequality implies that the state is entangled as it is the ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"DAa0hl57C5"},{"type":"crossReference","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"children":[{"type":"text","value":"Hillery-Zubairy","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"bFeT8y9PM7"}],"identifier":"hillery_zubairy_witness","label":"hillery_zubairy_witness","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"hillery-zubairy-witness","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"ti6BYwIubT"},{"type":"text","value":" witness. Inequality ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"nfW7esvSlI"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"s1NVXfN4iJ"},{"type":"text","value":" is not only sufficient but necessary for entanglement: it must therefore be negative. We conclude that it is not possible to have a state for which ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"mEfhcCw4Vi"},{"type":"inlineMath","value":"\\delta >1","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"δ>1\\delta >1δ>1","key":"JTEP94Ab6E"},{"type":"text","value":" and ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"eWW7FpnKyI"},{"type":"inlineMath","value":"\\mathcal{P}_-\\geq 0","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"P−≥0\\mathcal{P}_-\\geq 0P−​≥0","key":"jrWcb9sgG0"},{"type":"text","value":" and we can restrict our analysis to ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"C0POY2vVfc"},{"type":"inlineMath","value":"\\delta \\in \\big[0, 1\\big].","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"δ∈[0,1].\\delta \\in \\big[0, 1\\big].δ∈[0,1].","key":"DTs9XkA7JW"}],"key":"rjaDSJiIoV"},{"type":"paragraph","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"children":[{"type":"text","value":"The maximum of the quantity ","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"key":"gSMRUCUl82"},{"type":"inlineMath","value":"\\delta (n_1n_2\\delta -1/2)","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"html":"δ(n1n2δ−1/2)\\delta (n_1n_2\\delta -1/2)δ(n1​n2​δ−1/2)","key":"DeNpmqMJC9"},{"type":"text","value":" depends on the population. We are therefore left to distinguish two cases, depending on the population ","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"key":"YnEp2DVNSR"},{"type":"inlineMath","value":"n_1n_2","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"html":"n1n2n_1n_2n1​n2​","key":"gl51ETG1ms"},{"type":"text","value":".","position":{"start":{"line":159,"column":1},"end":{"line":159,"column":1}},"key":"WQtH6x0GYx"}],"key":"hjrHMFDmgT"},{"type":"comment","value":"et us reformulate what we are looking for: we want to find the highest value of $\\delta$ that can take $\\mathcal{P}_1$ from negative to positive. Suppose $\\delta>1$ and $\\mathcal{P}_-$ positive. If $\\delta 1$, we have $|c|^2$ or $|d|^2$ that are greater than $n_1n_2$. $|d|^2$ cannot be greater than $n_1n_2$ therefore we must have $|c|^2>n_1n_2$. The latter condition implies that the bi-partite state is entangled as it is the [Hillery-Zubairy](#hillery_zubairy_witness) entanglement witness. If the state is entangled, it implies $\\mathcal{P}_-$ to be negative as it is an entanglement criterion and not a witness. We therefore conclude that we can never have a separable state with $\\delta>1$ and","position":{"start":{"line":161,"column":1},"end":{"line":161,"column":1}},"key":"K1kcAhQKKh"},{"type":"paragraph","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"If ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"kZ3DxBWfGV"},{"type":"inlineMath","value":"n_1n_2<1/2","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"html":"n1n2<1/2n_1n_2<1/2n1​n2​<1/2","key":"xYf8kbJPyU"},{"type":"text","value":", the last parenthesis of equation ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"WtipbzAoP9"},{"type":"crossReference","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"(","key":"CBUaYse9gd"},{"type":"text","value":"14","key":"sHql3m9jbF"},{"type":"text","value":")","key":"HGeM4xGrNb"}],"identifier":"pminus_calculation_1","label":"Pminus_calculation_1","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"pminus-calculation-1","key":"Mk6A59dIwY"},{"type":"text","value":" is always negative thus an upper bound for ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"B2bL3EoHfF"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"B9TzTkOop0"},{"type":"text","value":" is obtained for ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"vlAhBJlzgc"},{"type":"inlineMath","value":"\\delta=0","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"html":"δ=0\\delta=0δ=0","key":"D7DbeaEzVL"},{"type":"text","value":". Negativity of the first two lines ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"D1yFm6lMQR"},{"type":"emphasis","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"implies","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"lmbRlgM1G7"}],"key":"IIVudmu87o"},{"type":"text","value":" negativity of ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"Ko5sALnWvo"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"SO5AG5e08Q"},{"type":"text","value":" and therefore non-separability. A bit of algebra leads then to the critical value in equation ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"ih1ITMKuWF"},{"type":"crossReference","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"(","key":"iOlsAJ1p0I"},{"type":"text","value":"11","key":"pMMD4EO1Vo"},{"type":"text","value":")","key":"ehUnwkTERO"}],"identifier":"fg2","label":"Fg2","kind":"equation","template":"(%s)","enumerator":"11","resolved":true,"html_id":"fg2","key":"toO1lQ4rQG"},{"type":"text","value":" for the second order correlation function ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"hTNs2aFWF3"},{"type":"inlineMath","value":"g^{(2)}_{12}","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"html":"g12(2)g^{(2)}_{12}g12(2)​","key":"y59DD9lVKB"},{"type":"text","value":" to ensure non-separability.","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"DUkVz2m8QV"}],"key":"LqHO9gprZR"},{"type":"paragraph","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"children":[{"type":"text","value":"If ","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"key":"RU9xwXoPme"},{"type":"inlineMath","value":"n_1n_2>1/2","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"html":"n1n2>1/2n_1n_2>1/2n1​n2​>1/2","key":"eNSaeu0mLV"},{"type":"text","value":", this last term might increase ","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"key":"rn7kk8jQ1K"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"qKlv9hVhH8"},{"type":"text","value":" and we must consider the case where it could be maximal, that is when ","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"key":"idSksKgcFV"},{"type":"inlineMath","value":"\\delta=1","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"html":"δ=1\\delta=1δ=1","key":"EYZzoSI14C"},{"type":"text","value":". When doing so, the two last lines simplify and factorize as","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"key":"KpQOCPnO0L"}],"key":"LsDN2gwYzS"},{"type":"math","value":"\\begin{split}\n\\mathcal{P}_- \\leq 16 n_1n_2\\biggl[ & (1 + n_1)(1 + n_2)\\left(2 -g^{(2)}_{12} \\right) \\\\\n & +\\left( n_1n_2- \\frac{1}{2}\\right) \\left(2-g^{(2)}_{12} \\right) \\biggr].\n\\end{split}","tight":true,"html":"P−≤16n1n2[(1+n1)(1+n2)(2−g12(2))+(n1n2−12)(2−g12(2))].\\begin{split}\n\\mathcal{P}_- \\leq 16 n_1n_2\\biggl[ & (1 + n_1)(1 + n_2)\\left(2 -g^{(2)}_{12} \\right) \\\\\n & +\\left( n_1n_2- \\frac{1}{2}\\right) \\left(2-g^{(2)}_{12} \\right) \\biggr].\n\\end{split}P−​≤16n1​n2​[​(1+n1​)(1+n2​)(2−g12(2)​)+(n1​n2​−21​)(2−g12(2)​)].​","enumerator":"15","key":"W84J3zDC9F"},{"type":"paragraph","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"children":[{"type":"text","value":"We see that we have a positive term that is multiplied by ","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"key":"N470ol38Px"},{"type":"inlineMath","value":"2-g^{(2)}_{12}","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"html":"2−g12(2)2-g^{(2)}_{12}2−g12(2)​","key":"Nrn7HagdYg"},{"type":"text","value":". We conclude that in this case, ","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"key":"aWtWrqALPg"},{"type":"inlineMath","value":"g^{(2)}_{12}>2","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"html":"g12(2)>2g^{(2)}_{12}>2g12(2)​>2","key":"wSjWxOJbRU"},{"type":"text","value":" implies non-separability.","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"key":"qiTDQg50eI"}],"key":"EwjhJ1KURY"}],"html_id":"proof-nonseparability-criterion","key":"qfech4fJVN"},{"type":"paragraph","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"children":[{"type":"text","value":"In ","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"key":"DZYTiMlcqk"},{"type":"crossReference","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"children":[{"type":"text","value":"Figure ","key":"tL3ZzgmHVJ"},{"type":"text","value":"1","key":"xRNRlW0jOO"}],"identifier":"g2_criterion_witness","label":"g2_criterion_witness","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"g2-criterion-witness","key":"Xtg9dGxS8i"},{"type":"text","value":", we represented the threshold value for ","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"key":"vVIq9SbNmW"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"QqHxJgAbeK"},{"type":"text","value":" given by equation ","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"key":"lem8Kw7c91"},{"type":"crossReference","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"children":[{"type":"text","value":"(","key":"P4cGb7cpSD"},{"type":"text","value":"11","key":"zyG6xExz2l"},{"type":"text","value":")","key":"vhfI41BIbY"}],"identifier":"fg2","label":"Fg2","kind":"equation","template":"(%s)","enumerator":"11","resolved":true,"html_id":"fg2","key":"wh3dVOuiy5"},{"type":"text","value":". This threshold reaches 3 in the limit of vanishing population: we recover the threshold discussed in the introduction.","position":{"start":{"line":176,"column":1},"end":{"line":176,"column":1}},"key":"wObY6rLxNt"}],"key":"alTLgvi3o4"},{"type":"admonition","kind":"note","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Consequence on the Cauchy-Schwarz inequality","position":{"start":{"line":179,"column":1},"end":{"line":179,"column":1}},"key":"nUIwRR0wlb"}],"key":"DjPz0TSjgC"},{"type":"paragraph","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"children":[{"type":"text","value":"The shift of the ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"odbZfrdRfM"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"Vfx69yt7QB"},{"type":"text","value":" non-separability bound shifts as well the Cauchy-Schwarz ratio ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"hHjQbbNJeu"},{"type":"inlineMath","value":"\\mathcal{C_S}","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"html":"CS\\mathcal{C_S}CS​","key":"ITXg4ehMy0"},{"type":"text","value":" bound. For a population larger than 0.7, the observation of ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"WNIpieZlq1"},{"type":"inlineMath","value":"\\mathcal{C_S}\\geq 1","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"html":"CS≥1\\mathcal{C_S}\\geq 1CS​≥1","key":"C1Za0yUZe6"},{"type":"text","value":" does not need to assume anymore the vanishing coherence of the ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"W6NAWxYQ1C"},{"type":"inlineMath","value":"\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"html":"⟨a^1a^2†⟩\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}⟨a^1​a^2†​⟩","key":"IEmPlQ4Ulr"},{"type":"text","value":" term. For a lower population, the bound of the non-separability threshold is just increased as half of equation ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"cr69AhoAh3"},{"type":"crossReference","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"children":[{"type":"text","value":"(","key":"dW6HXhO4xC"},{"type":"text","value":"11","key":"tukpRfDnWJ"},{"type":"text","value":")","key":"whCQNb5nEJ"}],"identifier":"fg2","label":"Fg2","kind":"equation","template":"(%s)","enumerator":"11","resolved":true,"html_id":"fg2","key":"W6VST32zH9"},{"type":"text","value":" ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"kQMaL4JWBP"},{"type":"emphasis","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"MbcsFhH4WN"}],"key":"PGU3ZtZck6"},{"type":"text","value":" the bound for ","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"k7pU4fCTHG"},{"type":"inlineMath","value":"\\mathcal{C_S}","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"html":"CS\\mathcal{C_S}CS​","key":"c6S6OxjE5o"},{"type":"text","value":" shifts from 1 to 1.5.","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"Gre0Nr1JtV"}],"key":"G5ys8C3nZO"}],"key":"cQnG0unpOO"},{"type":"heading","depth":2,"position":{"start":{"line":183,"column":1},"end":{"line":183,"column":1}},"children":[{"type":"text","value":"The second order correlation function to probe separability","position":{"start":{"line":183,"column":1},"end":{"line":183,"column":1}},"key":"a1R1b8HMnR"}],"identifier":"the-second-order-correlation-function-to-probe-separability","label":"The second order correlation function to probe separability","html_id":"the-second-order-correlation-function-to-probe-separability","implicit":true,"enumerator":"3","key":"SFVzc65ZaG"},{"type":"paragraph","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"children":[{"type":"text","value":"We can also take advantage that ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"iePRHYxL78"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"gMHDazCXXV"},{"type":"text","value":" is an entanglement criterion to derive a bound for ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"KUin8ddQ8I"},{"type":"inlineMath","value":"g^{(2)}_{12}","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"html":"g12(2)g^{(2)}_{12}g12(2)​","key":"GVCOA0ePgH"},{"type":"text","value":" to ensure separability of the state. The derivation will exactly follow the proof of the ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"COMubWeQBU"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"ab6144c0fq"},{"type":"text","value":" non-separability witness, but this time in finding decrease for ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"D3Iune44KO"},{"type":"text","value":"δ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"krh1u5SK5q"},{"type":"text","value":" so that ","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"kgWjyEDTMk"},{"type":"inlineMath","value":"\\mathcal{P}_s","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"html":"Ps\\mathcal{P}_sPs​","key":"Tosi3iFUXf"},{"type":"text","value":" is positive.","position":{"start":{"line":185,"column":1},"end":{"line":185,"column":1}},"key":"NiiHQDBtCM"}],"key":"JoifRKSiaS"},{"type":"proof","kind":"theorem","label":"g2_classical_theorem","identifier":"g2_classical_theorem","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"g2 maximal bound for separability","position":{"start":{"line":186,"column":1},"end":{"line":186,"column":1}},"key":"NYOVYLi5rP"}],"key":"IjdWIvSveW"},{"type":"paragraph","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"children":[{"type":"text","value":"Assuming that the state is Gaussian and that ","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"MPCEIhShxj"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i}=\\braket{\\hat{a}_i^2}=0","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"html":"⟨a^i⟩=⟨a^i2⟩=0\\braket{\\hat{a}_i}=\\braket{\\hat{a}_i^2}=0⟨a^i​⟩=⟨a^i2​⟩=0","key":"zPw97ppxBP"},{"type":"text","value":", the two-mode state 1-2 is separable if ","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"VaQVdwx14e"},{"type":"inlineMath","value":"g_{12}^{(2)}","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"html":"g12(2)g_{12}^{(2)}g12(2)​","key":"RuhOx0AXwz"},{"type":"text","value":" is slightly below 2. Especially,","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"vcvEK4uXpa"}],"key":"vHDXhu0r9L"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":190,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"fhtr4lia8a"},{"type":"inlineMath","value":"n_1n_2 \\leq 1/4","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"n1n2≤1/4n_1n_2 \\leq 1/4n1​n2​≤1/4","key":"YHBwgu4OIn"},{"type":"text","value":", then ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"LtWyPJmWJh"},{"type":"inlineMath","value":"g_{12}^{(2)}<2","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"g12(2)<2g_{12}^{(2)}<2g12(2)​<2","key":"t2NjsRQyfl"},{"type":"text","value":" is a sufficient condition,","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"C9jpDmpR9a"}],"key":"TA5ybJaVSA"},{"type":"listItem","spread":true,"position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"VWXDd3Oh69"},{"type":"inlineMath","value":"n_1n_2\\geq 1/4","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"html":"n1n2≥1/4n_1n_2\\geq 1/4n1​n2​≥1/4","key":"L3mt1eG5g6"},{"type":"text","value":", the threshold for ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"DuEtt4PGkE"},{"type":"inlineMath","value":"g_{12}^{(2)}","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"html":"g12(2)g_{12}^{(2)}g12(2)​","key":"LWJkoy0l37"},{"type":"text","value":" is shifted and the sufficient condition for separability reads","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"KK2feoMvkj"}],"key":"GtJah8Ogqm"}],"key":"HJjbj9r3WB"},{"type":"math","identifier":"classical_witness_equation","label":"classical_witness_equation","value":"g_{12}^{(2)}<2 -\\frac{(1-4n_1n_2)^2}{8n_1n_2(1+2n_1)(1+2n_2)}\\, \\, .","tight":true,"html":"g12(2)<2−(1−4n1n2)28n1n2(1+2n1)(1+2n2)  .g_{12}^{(2)}<2 -\\frac{(1-4n_1n_2)^2}{8n_1n_2(1+2n_1)(1+2n_2)}\\, \\, .g12(2)​<2−8n1​n2​(1+2n1​)(1+2n2​)(1−4n1​n2​)2​.","enumerator":"16","html_id":"classical-witness-equation","key":"fsQuQ5SXQn"},{"type":"paragraph","position":{"start":{"line":196,"column":1},"end":{"line":196,"column":1}},"children":[{"type":"text","value":"which asymptotically reaches 1.5.","position":{"start":{"line":196,"column":1},"end":{"line":196,"column":1}},"key":"VAjXCM3H2s"}],"key":"SxvqyZGKQv"}],"html_id":"g2-classical-theorem","key":"yOEP2sJlGj"},{"type":"proof","kind":"proof","class":"dropdown","enumerated":false,"children":[{"type":"paragraph","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"children":[{"type":"text","value":"The proof follows the same recipe as the one for the ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"fbXNKp0DXe"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"CpnswqeLzN"},{"type":"text","value":" entanglement witness. The difference is that we now look for a lower bound for ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"JLFMIoPnzF"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"BgMbkC3tA2"},{"type":"text","value":". The minimum of the ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"bBh3wzyKrn"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"JNErt3OVoX"},{"type":"text","value":" polynomial is reached for ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"yvezRlSAzQ"},{"type":"inlineMath","value":"\\delta = 1/4n_1n_2","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"δ=1/4n1n2\\delta = 1/4n_1n_2δ=1/4n1​n2​","key":"pG3oLuUvuV"},{"type":"text","value":". The value for the separability lower bound is bounded by 2, which is the value when ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"EDosFH9GQH"},{"type":"inlineMath","value":"d=0","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"html":"d=0d=0d=0","key":"MyHZ41VxZn"},{"type":"text","value":". As a result, ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"UUbhvArzAW"},{"type":"text","value":"δ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"koABGjvVWW"},{"type":"text","value":" is bounded by ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"qPpVjlOwir"},{"type":"text","value":"1","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"ZpkhAgIpkP"},{"type":"text","value":". We are therefore left with the following values for ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"vs63crNZO3"},{"type":"text","value":"δ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"fWWNLFJDA1"},{"type":"text","value":":","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"tXq0rhzm6e"}],"key":"tiGSUIoH3a"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":204,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"YM1VwVM1BO"},{"type":"inlineMath","value":"n_1n_2 \\leq 1/4","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"html":"n1n2≤1/4n_1n_2 \\leq 1/4n1​n2​≤1/4","key":"wkE01xxXa2"},{"type":"text","value":", the lower bound for the g","key":"Jg1sd2GtFE"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"A9Fpv4itgc"}],"key":"wFeuka62Ge"},{"type":"text","value":" criterion is obtained by replacing ","key":"d0GYoUYqmd"},{"type":"text","value":"δ","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"Vxly3dFXsN"},{"type":"text","value":" by 1 in equation ","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"U5auFz5XnK"},{"type":"crossReference","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"children":[{"type":"text","value":"(","key":"eEBQ94jx2g"},{"type":"text","value":"14","key":"TdH5TmEs7C"},{"type":"text","value":")","key":"UJhrZPSNCZ"}],"identifier":"pminus_calculation_1","label":"Pminus_calculation_1","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"pminus-calculation-1","key":"bG6CBJaRhf"},{"type":"text","value":" which leads immediately to the critical value of 2 for ","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"N7nbiQ7og1"},{"type":"inlineMath","value":"g^{(2)}_{12}","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"html":"g12(2)g^{(2)}_{12}g12(2)​","key":"hGP40Eddtw"},{"type":"text","value":".","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"yPpzEhHu0Q"}],"key":"I6en07Rc66"},{"type":"listItem","spread":true,"position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"text","value":"if ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"GkA2Tok6go"},{"type":"inlineMath","value":"n_1n_2\\geq1/4","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"html":"n1n2≥1/4n_1n_2\\geq1/4n1​n2​≥1/4","key":"IRGDcFUunc"},{"type":"text","value":", the minimum is reached for ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"yjPNKO5xji"},{"type":"inlineMath","value":"\\delta = 1/4n_1n_2","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"html":"δ=1/4n1n2\\delta = 1/4n_1n_2δ=1/4n1​n2​","key":"DsvXVWeZi5"},{"type":"text","value":". Inserting this value in equation ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"jr8Vibm98V"},{"type":"crossReference","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"text","value":"(","key":"k7UkamEbvw"},{"type":"text","value":"14","key":"aWibL2FEVO"},{"type":"text","value":")","key":"C4aTJPBC5x"}],"identifier":"pminus_calculation_1","label":"Pminus_calculation_1","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"pminus-calculation-1","key":"GN8gvBCBPy"},{"type":"text","value":" leads after some algebra to the critical value for the second order correlation function given by equation ","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"OI1ngUFeKD"},{"type":"crossReference","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"children":[{"type":"text","value":"(","key":"qm14Sk1joo"},{"type":"text","value":"16","key":"akM9L1KKNW"},{"type":"text","value":")","key":"v2YsDCvP6i"}],"identifier":"classical_witness_equation","label":"classical_witness_equation","kind":"equation","template":"(%s)","enumerator":"16","resolved":true,"html_id":"classical-witness-equation","key":"HJrWEE1YDV"},{"type":"text","value":".","position":{"start":{"line":205,"column":1},"end":{"line":205,"column":1}},"key":"R0DtkjzshQ"}],"key":"BrFZCZAv6e"}],"key":"x84VBSnWAW"}],"key":"g6JzK05oBf"},{"type":"container","kind":"figure","identifier":"g2_criterion_witness","label":"g2_criterion_witness","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/g2entanglement_crite-7ca9e5af5c75f5f4f197deaa8484dd54.png","alt":"Second-order correlation function to probe non-separability and separability","width":"60%","align":"center","key":"EP6Tyeg5X2","urlSource":"images/g2entanglement_criterion.png","urlOptimized":"/~gondret/phd_manuscript/build/g2entanglement_crite-7ca9e5af5c75f5f4f197deaa8484dd54.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"g2_criterion_witness","identifier":"g2_criterion_witness","html_id":"g2-criterion-witness","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"TJsEyin9QJ"},{"type":"text","value":"1","key":"IBJptKVXSp"},{"type":"text","value":":","key":"MgBHhig0QF"}],"template":"Figure %s:","key":"qOLgryy6Tu"},{"type":"text","value":"Critical value for the second order correlation function to assess non-separability and separability of the bipartite state 1-2. The maroon dashed curve corresponds to the separability witness: any state lying below this curve is separable. The solid green curve and the area above represents a region where states are entangled. In between these two curves, it is not possible to assess the separability of the state using only the population in each modes and the second order correlation function. The un-physical limit corresponds to a value of ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"saJxoOEqqp"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"L6aC5CMrEL"},{"type":"text","value":" greater than the one of a two-mode squeezed state. It is ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"nea5dFQkJX"},{"type":"inlineMath","value":"2+1/n","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"html":"2+1/n2+1/n2+1/n","key":"YdIVpgoyYb"},{"type":"text","value":" where ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"JuiY6r8yBS"},{"type":"inlineMath","value":"n","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"html":"nnn","key":"TnYsyhMKr4"},{"type":"text","value":" is the mode population.","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"V0LMtzRlzE"}],"key":"yUeaWlk92y"}],"key":"F6SmhDJQAG"}],"enumerator":"1","html_id":"g2-criterion-witness","key":"ZFNYofcgDk"},{"type":"heading","depth":2,"position":{"start":{"line":217,"column":1},"end":{"line":217,"column":1}},"children":[{"type":"text","value":"Graphical resolution of the ","position":{"start":{"line":217,"column":1},"end":{"line":217,"column":1}},"key":"XwTdoBaOOF"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":217,"column":1},"end":{"line":217,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"m7tdTKlmIJ"},{"type":"text","value":" witness","position":{"start":{"line":217,"column":1},"end":{"line":217,"column":1}},"key":"IOePntvjB8"}],"identifier":"graphical-resolution-of-the-g-2-witness","label":"Graphical resolution of the g^{(2)} witness","html_id":"graphical-resolution-of-the-g-2-witness","implicit":true,"enumerator":"4","key":"EpSmmyTnVX"},{"type":"container","kind":"figure","identifier":"g2_entanglement_heatmap","label":"g2_entanglement_heatmap","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/g2_entanglement_heat-d8a1d6989b543aee6b56087e4eefc95e.png","alt":"Second-order correlation function to probe non-separability and separability","width":"100%","align":"center","key":"tZhkNF7klm","urlSource":"images/g2_entanglement_heatmap.png","urlOptimized":"/~gondret/phd_manuscript/build/g2_entanglement_heat-d8a1d6989b543aee6b56087e4eefc95e.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"g2_entanglement_heatmap","identifier":"g2_entanglement_heatmap","html_id":"g2-entanglement-heatmap","enumerator":"2","children":[{"type":"text","value":"Figure ","key":"vbDa6NpiIB"},{"type":"text","value":"2","key":"nlqt8YBUk5"},{"type":"text","value":":","key":"I3RyLZShkI"}],"template":"Figure %s:","key":"Hw4dMIWAV8"},{"type":"text","value":"Colormap of ","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"w5MZkAFYLL"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"K9bHqcQGFP"},{"type":"text","value":" for Gaussian states in the ","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"PI7LIWXb3J"},{"type":"inlineMath","value":"(c,d)","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"html":"(c,d)(c,d)(c,d)","key":"nIGtVri3b1"},{"type":"text","value":" plane for different mean population (see title). The grey region represents unphysical Gaussian states. Black solid curve is the g","key":"B7JGIVctaq"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"dbqLyKksQu"}],"key":"m3qjyEkAaR"},{"type":"text","value":" criterion, brown dashed line the ","key":"e0SA46GQAC"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"ZBk1bxzWBm"},{"type":"text","value":" ","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"Sep2NDAMf8"},{"type":"crossReference","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"children":[{"type":"text","value":"separable witness","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"AR8Pntv4UM"}],"identifier":"g2_classical_theorem","label":"g2_classical_theorem","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"g2-classical-theorem","key":"i6KacRGzHG"},{"type":"text","value":" threshold and the blue dashed-dotted curve is the ","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"Dcf0meb1Kv"},{"type":"crossReference","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"children":[{"type":"text","value":"non-separability witness","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"RBtkhPo1PY"}],"identifier":"g2_criterion_theorem","label":"g2_criterion_theorem","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"g2-criterion-theorem","key":"mbacYRkFAy"},{"type":"text","value":" threshold.","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"emmQp8m0i0"}],"key":"ql1OhYhKLT"}],"key":"nbNj4GrJvR"}],"enumerator":"2","html_id":"g2-entanglement-heatmap","key":"xfXBtWu1tK"},{"type":"paragraph","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"children":[{"type":"crossReference","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"children":[{"type":"text","value":"Figure ","key":"gstvlkitk2"},{"type":"text","value":"2","key":"jxFXJMSWZ5"}],"identifier":"g2_entanglement_heatmap","label":"g2_entanglement_heatmap","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"g2-entanglement-heatmap","key":"AFthqpV7dW"},{"type":"text","value":" provides a graphical illustration of the separable and non-separable bounds. We represent the second order correlation function on the ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"Be0jBNz1Nu"},{"type":"inlineMath","value":"(c,d)","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"(c,d)(c,d)(c,d)","key":"e9bDSicGnd"},{"type":"text","value":" map at fixed population. The color encodes the value of ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"faBbKKHffI"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"TwLP76HNrT"},{"type":"text","value":" and the solid black curve is the g","key":"WE6yDdaDi7"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"SwKfKpYdBr"}],"key":"pEs7sqcor6"},{"type":"text","value":" criterion. On the left of the figure lie separable states and on the right entangled states. The grey region represents unphysical states. The form of the ","key":"MVkiTEpshm"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"Tb9P78PbUV"},{"type":"text","value":" curves (solid dashed green and dashed brown) drawn gives a glimpse of the behavior of iso-","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"bVtzkXmet3"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"NwbVvKH0ur"},{"type":"text","value":" curves on this plan. We can interpret the different regimes (small and high populations) in terms of convexity of the g","key":"iVcL9VmMrs"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"aUrTWyCrIY"}],"key":"OXoDoZcteC"},{"type":"text","value":" curve in the ","key":"gV90Uv6LRP"},{"type":"inlineMath","value":"(|c|, |d|)","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"(∣c∣,∣d∣)(|c|, |d|)(∣c∣,∣d∣)","key":"zw72JTyksH"},{"type":"text","value":" plane, and compare it to the convexity of the iso-","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"rNcdjhdfiY"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"GUTynvLKJ6"},{"type":"text","value":" curves. When the population is low (top panel), the iso-","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"SqaabXXTCi"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"lVXEQoZzHP"},{"type":"text","value":" curves are more convex than the g","key":"ZEl2N2MyjZ"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"w0vWM0XCCJ"}],"key":"GjLEGi6DjT"},{"type":"text","value":" curve. As a result, the threshold value of ","key":"Tqo4G5E1W9"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"FDavHrsunC"},{"type":"text","value":" to certify entanglement is given by the value of ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"dJEi3Ibz1m"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"aAPBehehsa"},{"type":"text","value":" at the top corner, for which ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"Aofw0xtLrI"},{"type":"inlineMath","value":"d\\neq 0","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"d≠0d\\neq 0d=0","key":"H6CZl534i1"},{"type":"text","value":". For higher population - bottom right panel C - we observe that the g","key":"p5KVbrgpaJ"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"MlY8J6meUv"}],"key":"b1oY4YqIQ6"},{"type":"text","value":" curve is more convex than the iso-","key":"xByxjsTLcX"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"zkddV59oBa"},{"type":"text","value":". It is therefore the value at the bottom of the graph that fixes the minimum value for ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"ag3A96oTQx"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"rDLCfsJ0u8"},{"type":"text","value":". In this case, entanglement is certified by ","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"SM7V2AItVN"},{"type":"inlineMath","value":"g^{(2)}>2","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"html":"g(2)>2g^{(2)}>2g(2)>2","key":"XBFZTo8IFu"},{"type":"text","value":".","position":{"start":{"line":228,"column":1},"end":{"line":228,"column":1}},"key":"PEhya7SFte"}],"key":"PKehU5Wcmw"},{"type":"comment","value":"This means that gPPT curve is always under the iso-curve $g^{(2)}=2$, which assess entanglement for this population. We see for example that the entanglement threshold of the first panel touches the gPPT curve at the top which makes ot the smaller value of $g^{(2)}$ to ensure entanglement.","position":{"start":{"line":230,"column":1},"end":{"line":230,"column":1}},"key":"pnWnPjVSXJ"},{"type":"heading","depth":2,"position":{"start":{"line":234,"column":1},"end":{"line":234,"column":1}},"children":[{"type":"text","value":"Second and fourth order correlation function as an entanglement criterion","position":{"start":{"line":234,"column":1},"end":{"line":234,"column":1}},"key":"HUDdeILA8C"}],"identifier":"fourth_order_corr_func","label":"fourth_order_corr_func","html_id":"fourth-order-corr-func","enumerator":"5","key":"wWjyDNLGHp"},{"type":"paragraph","position":{"start":{"line":235,"column":1},"end":{"line":235,"column":1}},"children":[{"type":"text","value":"It was proposed by ","position":{"start":{"line":235,"column":1},"end":{"line":235,"column":1}},"key":"aOHpr42EFC"},{"type":"cite","identifier":"clement_fourth_2022","label":"clement_fourth_2022","kind":"narrative","position":{"start":{"line":235,"column":20},"end":{"line":235,"column":40}},"children":[{"type":"text","value":"Clément (2022)","key":"xM4p80lr9F"}],"enumerator":"10","key":"zZJG6TWysn"},{"type":"text","value":" to use the fourth order correlation function to retrieve more information and assess the (non)-separability of the state. The cross four-body normalized correlation function is given by","position":{"start":{"line":235,"column":1},"end":{"line":235,"column":1}},"key":"KzRjmbopJp"}],"key":"KkLgrbXWVg"},{"type":"math","identifier":"fourth_order_correlation_expression","label":"fourth_order_correlation_expression","value":"\\begin{split}\ng_{12}^{(4)} &\\equiv \\frac{\\braket{:(\\hat{a}_1^\\dagger\\hat{a}_1)^2(\\hat{a}_2^\\dagger\\hat{a}_2)^2:}}{ \\braket{\\hat{a}_1^\\dagger\\hat{a}_1}^2\\braket{\\hat{a}_2^\\dagger\\hat{a}_2}^2}\\\\\n&= 4\\left[1 +\\frac{\\left(|d|^2 + |c|^2\\right)^2}{n_1^2 n_2^2}+4\\frac{|c|^2+|d|^2}{n_1n_2}+2\\frac{|d|^2|c|^2}{n_1^2n_2^2} \\right].\n\\end{split}","tight":true,"html":"g12(4)≡⟨:(a^1†a^1)2(a^2†a^2)2:⟩⟨a^1†a^1⟩2⟨a^2†a^2⟩2=4[1+(∣d∣2+∣c∣2)2n12n22+4∣c∣2+∣d∣2n1n2+2∣d∣2∣c∣2n12n22].\\begin{split}\ng_{12}^{(4)} &\\equiv \\frac{\\braket{:(\\hat{a}_1^\\dagger\\hat{a}_1)^2(\\hat{a}_2^\\dagger\\hat{a}_2)^2:}}{ \\braket{\\hat{a}_1^\\dagger\\hat{a}_1}^2\\braket{\\hat{a}_2^\\dagger\\hat{a}_2}^2}\\\\\n&= 4\\left[1 +\\frac{\\left(|d|^2 + |c|^2\\right)^2}{n_1^2 n_2^2}+4\\frac{|c|^2+|d|^2}{n_1n_2}+2\\frac{|d|^2|c|^2}{n_1^2n_2^2} \\right].\n\\end{split}g12(4)​​≡⟨a^1†​a^1​⟩2⟨a^2†​a^2​⟩2⟨:(a^1†​a^1​)2(a^2†​a^2​)2:⟩​=4[1+n12​n22​(∣d∣2+∣c∣2)2​+4n1​n2​∣c∣2+∣d∣2​+2n12​n22​∣d∣2∣c∣2​].​","enumerator":"17","html_id":"fourth-order-correlation-expression","key":"jjFt4LnHHK"},{"type":"paragraph","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"children":[{"type":"text","value":"Surprisingly, it involves non-only the sum of the square modulus of ","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"key":"PtJZV62JGs"},{"type":"inlineMath","value":"c","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"html":"ccc","key":"OkpO2mPkS3"},{"type":"text","value":" and ","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"key":"XPk9jOSWSs"},{"type":"inlineMath","value":"d","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"html":"ddd","key":"YixsJH6xEx"},{"type":"text","value":" but also their product, which might allow one to access both values. Having information about the individual values for ","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"key":"mS5RCPxZN6"},{"type":"inlineMath","value":"|c|^2","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"html":"∣c∣2|c|^2∣c∣2","key":"fsQUGgEMIu"},{"type":"text","value":" and ","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"key":"dJ0TmypRmn"},{"type":"inlineMath","value":"|d|^2","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"html":"∣d∣2|d|^2∣d∣2","key":"Ab7un3GTNY"},{"type":"text","value":" would therefore allow us to completely characterize the separability of the state. While powerful, this method might not be relevant for photonic experiments: measuring the fourth order correlation function requires the use of two beam-splitters combined with four single photon avalanche photo-diodes that are quite expansive. In this case, a complete tomography of the state seems more appropriate. However, this might be relevant for atom counting experiments in which one can often count precisely several particles per mode while measuring the full tomography of the state is less common.","position":{"start":{"line":243,"column":1},"end":{"line":243,"column":1}},"key":"g2o9TfeB48"}],"key":"ReXNWZM4GN"},{"type":"proof","kind":"definition","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"The Bona fide function","position":{"start":{"line":246,"column":1},"end":{"line":246,"column":1}},"key":"VEJTI8RQOq"}],"key":"fgKvh9uWpg"},{"type":"paragraph","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"children":[{"type":"text","value":"As we saw in the first section, any Gaussian state must respect the ","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"VTCp9yyhbk"},{"type":"emphasis","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"lFKMfxtC6K"}],"key":"C22wQPkGQa"},{"type":"text","value":" condition ","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"icYV0d7CuV"},{"type":"crossReference","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"children":[{"type":"text","value":"(","key":"igvhqkl1sj"},{"type":"text","value":"7","key":"gJ0QWydGpD"},{"type":"text","value":")","key":"Fjvq5TCE4m"}],"identifier":"schrodingerrobertsonsimonequation","label":"SchrodingerRobertsonSimonEquation","kind":"equation","template":"(%s)","enumerator":"7","resolved":true,"html_id":"schrodingerrobertsonsimonequation","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"tr5VkkYASr"},{"type":"text","value":". In our particular case where a state is characterized by its mean population (","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"ljIMO01qSv"},{"type":"inlineMath","value":"n_1","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"html":"n1n_1n1​","key":"B264eMIvSp"},{"type":"text","value":", ","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"cGQxBcODmt"},{"type":"inlineMath","value":"n_2","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"html":"n2n_2n2​","key":"jQNX0Laq3b"},{"type":"text","value":") and its correlation ","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"aUGpc5QAeE"},{"type":"inlineMath","value":"(c,d)","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"html":"(c,d)(c,d)(c,d)","key":"bekIMl37Js"},{"type":"text","value":". It is therefore practical to define the following function","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"EDSLHnayMf"}],"key":"cyuJ5GXtyB"},{"type":"math","identifier":"pplus_def_bonafide","label":"pplus_def_bonafide","value":"\\begin{split}\n\\mathcal{P}_+(n_1, n_2, c, d) = & (1 + n_1)(1 + n_2)(n_1n_2 - |c|^2 - |d|^2) \\\\\n &+\\left( \\frac{1}{2}- n_1n_2\\right) \\left(|d|^2 + |c|^2\\right) \\\\\n&+ (|c|^2 - |d|^2)^2 -\\frac{1}{2}\\big( |d|^2 - |c|^2\\big).\n\\end{split}","tight":true,"html":"P+(n1,n2,c,d)=(1+n1)(1+n2)(n1n2−∣c∣2−∣d∣2)+(12−n1n2)(∣d∣2+∣c∣2)+(∣c∣2−∣d∣2)2−12(∣d∣2−∣c∣2).\\begin{split}\n\\mathcal{P}_+(n_1, n_2, c, d) = & (1 + n_1)(1 + n_2)(n_1n_2 - |c|^2 - |d|^2) \\\\\n &+\\left( \\frac{1}{2}- n_1n_2\\right) \\left(|d|^2 + |c|^2\\right) \\\\\n&+ (|c|^2 - |d|^2)^2 -\\frac{1}{2}\\big( |d|^2 - |c|^2\\big).\n\\end{split}P+​(n1​,n2​,c,d)=​(1+n1​)(1+n2​)(n1​n2​−∣c∣2−∣d∣2)+(21​−n1​n2​)(∣d∣2+∣c∣2)+(∣c∣2−∣d∣2)2−21​(∣d∣2−∣c∣2).​","enumerator":"18","html_id":"pplus-def-bonafide","key":"sEfMwyp6jV"},{"type":"paragraph","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"children":[{"type":"text","value":"Any Gaussian state whose covariance matrix is of the form ","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"key":"PZOzMbJGHx"},{"type":"crossReference","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"children":[{"type":"text","value":"(","key":"EaMcf8WZex"},{"type":"text","value":"4","key":"xJJgoyHhO7"},{"type":"text","value":")","key":"mDia9lMuAg"}],"identifier":"forme_covar_matrix","label":"forme_covar_matrix","kind":"equation","template":"(%s)","enumerator":"4","resolved":true,"html_id":"forme-covar-matrix","key":"TDHKQy5BE0"},{"type":"text","value":" must satisfy ","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"key":"oqpd2PqfXw"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, c, d) \\geq 0","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"html":"P+(n1,n2,c,d)≥0\\mathcal{P}_+(n_1, n_2, c, d) \\geq 0P+​(n1​,n2​,c,d)≥0","key":"m5qN499kBM"},{"type":"text","value":".","position":{"start":{"line":258,"column":1},"end":{"line":258,"column":1}},"key":"eZdBhd2FEw"}],"key":"RU1smBw79J"}],"key":"msfuo9VNQ7"},{"type":"paragraph","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"children":[{"type":"text","value":"A couple of remarks that will help in understanding the ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"IZOQH70bWQ"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"g(2)/g(4)g^{(2)}/g^{(4)}g(2)/g(4)","key":"jHAeXW5VG8"},{"type":"text","value":" criterion. First, note the difference between ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"FlILVWJb72"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"ypipyVTsSW"},{"type":"text","value":" defined in eq. ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"MP5Y9QuKX0"},{"type":"crossReference","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"children":[{"type":"text","value":"(","key":"JuZSJBZwZa"},{"type":"text","value":"12","key":"QRmfb1TKxS"},{"type":"text","value":")","key":"y2ijMeohD8"}],"identifier":"pminus_demonstration","label":"pminus_demonstration","kind":"equation","template":"(%s)","enumerator":"12","resolved":true,"html_id":"pminus-demonstration","key":"xAjvj9iArT"},{"type":"text","value":" and ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"sf3yeVkhuL"},{"type":"inlineMath","value":"\\mathcal{P}_+","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P+\\mathcal{P}_+P+​","key":"rJARjVzMlK"},{"type":"text","value":" in ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"UVGyC70I7X"},{"type":"crossReference","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"children":[{"type":"text","value":"(","key":"E1ZSidzGlF"},{"type":"text","value":"18","key":"tbUCka3c5i"},{"type":"text","value":")","key":"qXoaxjXwGY"}],"identifier":"pplus_def_bonafide","label":"pplus_def_bonafide","kind":"equation","template":"(%s)","enumerator":"18","resolved":true,"html_id":"pplus-def-bonafide","key":"fyDZGyf3dN"},{"type":"text","value":": the absolute value in the third line disappears. This means that an entangled quantum state will exhibit ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"QHqj2xxPTP"},{"type":"inlineMath","value":"\\mathcal{P}_+ \\geq 0","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P+≥0\\mathcal{P}_+ \\geq 0P+​≥0","key":"JNSpZL71wO"},{"type":"text","value":" while having ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"BMizq7XgtI"},{"type":"inlineMath","value":"\\mathcal{P}_- < 0","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P−<0\\mathcal{P}_- < 0P−​<0","key":"AVHBdT8O2i"},{"type":"text","value":". Second, note that ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"eiwdvNyMOP"},{"type":"inlineMath","value":"\\mathcal{P}_-\\leq \\mathcal{P}_+","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P−≤P+\\mathcal{P}_-\\leq \\mathcal{P}_+P−​≤P+​","key":"dFJeRvNBWy"},{"type":"text","value":". As a result, if a state is entangled, it implies that ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"HsaUgD2Dgq"},{"type":"inlineMath","value":"|c|>|d|","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"∣c∣>∣d∣|c|>|d|∣c∣>∣d∣","key":"UFePE3GXw6"},{"type":"text","value":". The three conditions ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"b8asJROYQi"},{"type":"inlineMath","value":"\\mathcal{P}_- < 0","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P−<0\\mathcal{P}_- < 0P−​<0","key":"MRkuqHrwlw"},{"type":"text","value":", ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"CvLKqqvRPL"},{"type":"inlineMath","value":"\\, \\, |c|<|d|","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"  ∣c∣<∣d∣\\, \\, |c|<|d|∣c∣<∣d∣","key":"dwlTq5ftMs"},{"type":"text","value":" and ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"MYKz9aj4ZL"},{"type":"inlineMath","value":"\\mathcal{P}_+ \\geq 0","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"P+≥0\\mathcal{P}_+ \\geq 0P+​≥0","key":"JcRMZwy0DE"},{"type":"text","value":" are incompatible: as discussed by ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"zBQxF8RDp4"},{"type":"cite","identifier":"simon_peres_horodecki_2000","label":"simon_peres_horodecki_2000","kind":"narrative","position":{"start":{"line":261,"column":626},"end":{"line":261,"column":653}},"children":[{"type":"text","value":"Simon (2000)","key":"rS2v6JADjJ"}],"enumerator":"9","key":"M0z8wGuWos"},{"type":"text","value":", a entangled quantum state must have ","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"hKyQQpznJb"},{"type":"inlineMath","value":"\\text{det}C<0","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"html":"detC<0\\text{det}C<0detC<0","key":"OZFTdjmhk4"},{"type":"text","value":".","position":{"start":{"line":261,"column":1},"end":{"line":261,"column":1}},"key":"mKgce3ItG7"}],"key":"CDuN2pecht"},{"type":"paragraph","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"children":[{"type":"text","value":"We now prove the following theorem.","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"key":"ckRzbY2lfh"}],"key":"F1CzvTmXFY"},{"type":"proof","kind":"theorem","label":"g2g4criterion","identifier":"g2g4criterion","enumerated":false,"children":[{"type":"admonitionTitle","children":[{"type":"text","value":"g2 and g4 criterion","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"DqfUFfmKN4"}],"key":"ayPkLW6erq"},{"type":"paragraph","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"children":[{"type":"text","value":"Assuming that the state is Gaussian and that ","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"bnCxJvWVjX"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i}=\\braket{\\hat{a}_i^2}=0","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"html":"⟨a^i⟩=⟨a^i2⟩=0\\braket{\\hat{a}_i}=\\braket{\\hat{a}_i^2}=0⟨a^i​⟩=⟨a^i2​⟩=0","key":"REWlWmPKRE"},{"type":"text","value":", the measure of the population of each mode as well as the second and fourth order correlation functions ","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"ETkxqhB679"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"fw99FozCFM"},{"type":"text","value":" and ","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"Mmc1cXmyPx"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"ADzFhF8Xl2"},{"type":"text","value":" provide an entanglement criterion. We define","position":{"start":{"line":268,"column":1},"end":{"line":268,"column":1}},"key":"AoBEEqYPaT"}],"key":"V40tK21mVB"},{"type":"math","value":"\\beta_\\pm^2 = \\frac{n_1n_2}{2}\\left[ g_{12}^{(2)} - 1\\pm \\sqrt{2+8\\left(g_{12}^{(2)}-1\\right)+3\\left(g_{12}^{(2)}-1\\right)^2 - \\frac{g_{12}^{(4)}}{2}}\\right].","tight":true,"html":"β±2=n1n22[g12(2)−1±2+8(g12(2)−1)+3(g12(2)−1)2−g12(4)2].\\beta_\\pm^2 = \\frac{n_1n_2}{2}\\left[ g_{12}^{(2)} - 1\\pm \\sqrt{2+8\\left(g_{12}^{(2)}-1\\right)+3\\left(g_{12}^{(2)}-1\\right)^2 - \\frac{g_{12}^{(4)}}{2}}\\right].β±2​=2n1​n2​​​g12(2)​−1±2+8(g12(2)​−1)+3(g12(2)​−1)2−2g12(4)​​​​.","enumerator":"19","key":"nJuu4wzySm"},{"type":"paragraph","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"children":[{"type":"text","value":"The state is entangled if and only if ","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"key":"Qdd9yCbWFd"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2,\\beta_-, \\beta_+)<0","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"html":"P+(n1,n2,β−,β+)<0\\mathcal{P}_+(n_1, n_2,\\beta_-, \\beta_+)<0P+​(n1​,n2​,β−​,β+​)<0","key":"shkPQsMhVz"},{"type":"text","value":". In particular, if the state is non-separable, it is then characterized by","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"key":"VKdCqPuX8u"}],"key":"L4OvjNp8Ip"},{"type":"math","value":"|\\braket{\\hat{a}_1\\hat{a}_2}|=\\beta_+ \\quad \\quad |\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}| = \\beta_-.","tight":"before","html":"∣⟨a^1a^2⟩∣=β+∣⟨a^1a^2†⟩∣=β−.|\\braket{\\hat{a}_1\\hat{a}_2}|=\\beta_+ \\quad \\quad |\\braket{\\hat{a}_1\\hat{a}_2^\\dagger}| = \\beta_-.∣⟨a^1​a^2​⟩∣=β+​∣⟨a^1​a^2†​⟩∣=β−​.","enumerator":"20","key":"DcTO8dmRc8"}],"html_id":"g2g4criterion","key":"d630EdPdVY"},{"type":"proof","kind":"proof","class":"dropdown","enumerated":false,"children":[{"type":"paragraph","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"children":[{"type":"text","value":"From the ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"p4JZ5zkMR9"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"IQqBsP7W9E"},{"type":"text","value":" expression in equation ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"iZnaDwqvop"},{"type":"crossReference","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"children":[{"type":"text","value":"(","key":"A3FejHSyTJ"},{"type":"text","value":"17","key":"QJ4pIEKNIz"},{"type":"text","value":")","key":"q81EIWwzjS"}],"identifier":"fourth_order_correlation_expression","label":"fourth_order_correlation_expression","kind":"equation","template":"(%s)","enumerator":"17","resolved":true,"html_id":"fourth-order-correlation-expression","key":"slyEDIEcIu"},{"type":"text","value":", the fourth order correlation function involves not only the sum ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"Ruw12oAOoE"},{"type":"inlineMath","value":"|d|^2+|c|^2","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"∣d∣2+∣c∣2|d|^2+|c|^2∣d∣2+∣c∣2","key":"MlujfOcDMa"},{"type":"text","value":" but also the product ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"E1mZPFXasD"},{"type":"inlineMath","value":"|d|^2|c|^2","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"∣d∣2∣c∣2|d|^2|c|^2∣d∣2∣c∣2","key":"FuaGKFT77W"},{"type":"text","value":". We can therefore obtain the value of ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"QZScu8jHHc"},{"type":"inlineMath","value":"|c|","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"∣c∣|c|∣c∣","key":"w7J8wOg5nM"},{"type":"text","value":" and ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"QFfzYJUUpl"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"∣d∣|d|∣d∣","key":"AmiQ7Daz0O"},{"type":"text","value":", using ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"ipiUubp96Y"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"XQuABlayGZ"},{"type":"text","value":" and ","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"NLy8RW76hP"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"RyxJisQlUN"},{"type":"text","value":" through the system","position":{"start":{"line":283,"column":1},"end":{"line":283,"column":1}},"key":"BchtV4FQRs"}],"key":"M2x9r5bKrL"},{"type":"math","identifier":"g2g4_systeme_deux_equations","label":"g2g4_systeme_deux_equations","value":"\\begin{split}\n|c|^2|d|^2 =&\\, \\, \\frac{n_1^2n_2^2}{2}\\left[\\frac{g_{12}^{(4)}}{4} - \\left(g_{12}^{(2)}-1\\right)^2 - 1 - 4\\left(g_{12}^{(2)}-1\\right) \\right]\\\\\n|c|^2+|d|^2 =&\\, \\, n_1n_2\\left(g_{12}^{(2)}-1\\right) .\n\\end{split}","tight":true,"html":"∣c∣2∣d∣2=  n12n222[g12(4)4−(g12(2)−1)2−1−4(g12(2)−1)]∣c∣2+∣d∣2=  n1n2(g12(2)−1).\\begin{split}\n|c|^2|d|^2 =&\\, \\, \\frac{n_1^2n_2^2}{2}\\left[\\frac{g_{12}^{(4)}}{4} - \\left(g_{12}^{(2)}-1\\right)^2 - 1 - 4\\left(g_{12}^{(2)}-1\\right) \\right]\\\\\n|c|^2+|d|^2 =&\\, \\, n_1n_2\\left(g_{12}^{(2)}-1\\right) .\n\\end{split}∣c∣2∣d∣2=∣c∣2+∣d∣2=​2n12​n22​​[4g12(4)​​−(g12(2)​−1)2−1−4(g12(2)​−1)]n1​n2​(g12(2)​−1).​","enumerator":"21","html_id":"g2g4-systeme-deux-equations","key":"DFZR4mcWkf"},{"type":"paragraph","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"children":[{"type":"text","value":"This system has ","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"key":"YI2aStxdMJ"},{"type":"emphasis","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"children":[{"type":"text","value":"a priori","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"key":"nOHvHH7BvL"}],"key":"CNt4RjDQQs"},{"type":"text","value":" two indiscernible solutions for ","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"key":"PjCd0Ua0nN"},{"type":"inlineMath","value":"|c|","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"html":"∣c∣|c|∣c∣","key":"EFmz2dGq7D"},{"type":"text","value":" and ","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"key":"oP2UEH5vGj"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"html":"∣d∣|d|∣d∣","key":"Sb27ODcJlK"},{"type":"text","value":", and we introduce the following quantity","position":{"start":{"line":291,"column":1},"end":{"line":291,"column":1}},"key":"IAkJqaoDBK"}],"key":"uLXQUFqiqb"},{"type":"math","identifier":"cd_values_g2g4","label":"cd_values_g2g4","value":"\\beta_\\pm^2 = \\frac{n_1n_2}{2}\\left[ g_{12}^{(2)} - 1\\pm \\sqrt{2+8\\left(g_{12}^{(2)}-1\\right)+3\\left(g_{12}^{(2)}-1\\right)^2 - \\frac{g_{12}^{(4)}}{2}}\\right].","tight":true,"html":"β±2=n1n22[g12(2)−1±2+8(g12(2)−1)+3(g12(2)−1)2−g12(4)2].\\beta_\\pm^2 = \\frac{n_1n_2}{2}\\left[ g_{12}^{(2)} - 1\\pm \\sqrt{2+8\\left(g_{12}^{(2)}-1\\right)+3\\left(g_{12}^{(2)}-1\\right)^2 - \\frac{g_{12}^{(4)}}{2}}\\right].β±2​=2n1​n2​​​g12(2)​−1±2+8(g12(2)​−1)+3(g12(2)​−1)2−2g12(4)​​​​.","enumerator":"22","html_id":"cd-values-g2g4","key":"WEIaG9ubrm"},{"type":"paragraph","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"children":[{"type":"text","value":"There are two possible candidates for the value of ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"RF9mrcomL3"},{"type":"inlineMath","value":"(c,d)","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"(c,d)(c,d)(c,d)","key":"Fi6Q1w6LNK"},{"type":"text","value":" that are ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"lVyUzikeBj"},{"type":"inlineMath","value":"(\\beta_+, \\beta_-)","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"(β+,β−)(\\beta_+, \\beta_-)(β+​,β−​)","key":"T9yzQccnTC"},{"type":"text","value":" and ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"ZRZFsucnVT"},{"type":"inlineMath","value":"(\\beta_-, \\beta_+)","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"(β−,β+)(\\beta_-, \\beta_+)(β−​,β+​)","key":"qWuRJfBJTI"},{"type":"text","value":". The outcome of our measurement is therefore either the (family of) quantum states characterized by ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"HAu2GqbTG2"},{"type":"inlineMath","value":"(n_1,n_2, \\beta_+, \\beta_-)","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"(n1,n2,β+,β−)(n_1,n_2, \\beta_+, \\beta_-)(n1​,n2​,β+​,β−​)","key":"xy67L6nsf7"},{"type":"text","value":" or the one characterized by ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"jRC2j3y9Er"},{"type":"inlineMath","value":"(n_1,n_2, \\beta_-, \\beta_+)","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"(n1,n2,β−,β+)(n_1,n_2, \\beta_-, \\beta_+)(n1​,n2​,β−​,β+​)","key":"JT0PVZLaLK"},{"type":"text","value":". Here, the term that come first in the parenthesis refer to the value of ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"zCeVyhGQgm"},{"type":"inlineMath","value":"c","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ccc","key":"sqk3RSmV9q"},{"type":"text","value":" and the one that comes after to the value of ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"tyjpkFARMi"},{"type":"inlineMath","value":"d","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ddd","key":"LYiROIIFBK"},{"type":"text","value":". I added “family” in parentheses because our measurement does not completely characterize the state: we do not measure the phase of ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"f7Sp9ePytG"},{"type":"inlineMath","value":"c","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ccc","key":"GgITBKV7gk"},{"type":"text","value":" and ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"e73iv8x2Rq"},{"type":"inlineMath","value":"d","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ddd","key":"Gw6hlyHeuL"},{"type":"text","value":". However, for thermal states, the phase does not appear in the ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"Be0oklsCGt"},{"type":"emphasis","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"Bo0Ig43CfA"}],"key":"Mw9mtIL6KT"},{"type":"text","value":" condition and in the measure of the entanglement","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"de6x0mteIW"},{"type":"footnoteReference","identifier":"note_phase_thermal_states","label":"note_phase_thermal_states","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"number":3,"enumerator":"3","key":"cUX5FFWRiC"},{"type":"text","value":". This is why I will drop the absolute value of ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"JWm9JGwrYg"},{"type":"inlineMath","value":"c","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ccc","key":"r1UmCHPv9h"},{"type":"text","value":" and ","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"TdTZHuL1a7"},{"type":"inlineMath","value":"d","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"html":"ddd","key":"IElta5mEjE"},{"type":"text","value":" to lighten the notations.","position":{"start":{"line":296,"column":1},"end":{"line":296,"column":1}},"key":"X37bRRxjMs"}],"key":"EENTQCIfUq"},{"type":"paragraph","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"Before accepting one candidate, we must check that it is physically acceptable, ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"vzZZzcl4hM"},{"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":"lJWbW95mF3"}],"key":"g8Zf7Jflcd"},{"type":"text","value":" that it satisfies the ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"fvTGTS6ODW"},{"type":"emphasis","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"Kmqbq5D1K5"}],"key":"I4CBk9CUUf"},{"type":"text","value":" condition ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"xJrSKrwIlC"},{"type":"crossReference","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"(","key":"se2igODXeD"},{"type":"text","value":"7","key":"LQtj8N5um9"},{"type":"text","value":")","key":"WNqyuLJOzX"}],"identifier":"schrodingerrobertsonsimonequation","label":"SchrodingerRobertsonSimonEquation","kind":"equation","template":"(%s)","enumerator":"7","resolved":true,"html_id":"schrodingerrobertsonsimonequation","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"qVRmGNbOku"},{"type":"text","value":". In our case, it writes using the quantity ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"GGmJaxJdmM"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, c, d)","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"html":"P+(n1,n2,c,d)\\mathcal{P}_+(n_1, n_2, c, d)P+​(n1​,n2​,c,d)","key":"rOKCOUEsq2"},{"type":"text","value":" we defined above, in equation ","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"AwrsUHcPMr"},{"type":"crossReference","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"children":[{"type":"text","value":"(","key":"zSQKdPuww3"},{"type":"text","value":"18","key":"ZaL40NqjU3"},{"type":"text","value":")","key":"QSwPHsesjx"}],"identifier":"pplus_def_bonafide","label":"pplus_def_bonafide","kind":"equation","template":"(%s)","enumerator":"18","resolved":true,"html_id":"pplus-def-bonafide","key":"fggRYhcXoa"},{"type":"text","value":". Last but not least, we remark that each solution is the partial transpose of the other:","position":{"start":{"line":300,"column":1},"end":{"line":300,"column":1}},"key":"H8hTZQy74c"}],"key":"RR7qnUc7Xv"},{"type":"math","value":"\\hat{\\rho}_{(n_1,n_2, \\beta_+, \\beta_-)}^{\\intercal_B} = \\hat{\\rho}_{(n_1,n_2, \\beta_-, \\beta_+)}.","tight":true,"html":"ρ^(n1,n2,β+,β−)⊺B=ρ^(n1,n2,β−,β+).\\hat{\\rho}_{(n_1,n_2, \\beta_+, \\beta_-)}^{\\intercal_B} = \\hat{\\rho}_{(n_1,n_2, \\beta_-, \\beta_+)}.ρ^​(n1​,n2​,β+​,β−​)⊺B​​=ρ^​(n1​,n2​,β−​,β+​)​.","enumerator":"23","key":"PGezmgYl4p"},{"type":"paragraph","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"children":[{"type":"text","value":"To clarify, let’s distinguish between the different scenarios. Because ","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"T4r0Gc0IAd"},{"type":"inlineMath","value":"\\beta_+>\\beta_-","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"html":"β+>β−\\beta_+>\\beta_-β+​>β−​","key":"b9WETnLbhc"},{"type":"text","value":", we have that ","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"jqNBZZ7vfQ"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, \\beta_+, \\beta_-)\\geq \\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"html":"P+(n1,n2,β+,β−)≥P+(n1,n2,β−,β+)\\mathcal{P}_+(n_1, n_2, \\beta_+, \\beta_-)\\geq \\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)P+​(n1​,n2​,β+​,β−​)≥P+​(n1​,n2​,β−​,β+​)","key":"kEL5OvT5zv"},{"type":"text","value":", which makes only three cases to distinguish.","position":{"start":{"line":304,"column":1},"end":{"line":304,"column":1}},"key":"DRONNLc5eb"}],"key":"OOzmpnIVOE"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":305,"column":1},"end":{"line":308,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"children":[{"type":"text","value":"If ","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"G0YHAOmGdN"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)\\geq 0","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"html":"P+(n1,n2,β−,β+)≥0\\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)\\geq 0P+​(n1​,n2​,β−​,β+​)≥0","key":"ecTvsntP57"},{"type":"text","value":". We cannot distinguish between the two states: they both respect the ","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"iloi7mlblS"},{"type":"emphasis","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"k86WPcHeAz"}],"key":"dwfbgLkncE"},{"type":"text","value":" condition. However, we know the state is separable because the partial transpose of this state does respect the ","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"PIy5gl17XQ"},{"type":"emphasis","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"N2GMj7nXdY"}],"key":"VmboAUdoQI"},{"type":"text","value":" condition.","position":{"start":{"line":305,"column":1},"end":{"line":305,"column":1}},"key":"eKhFDtyppv"}],"key":"lhL3I6SHAl"},{"type":"listItem","spread":true,"position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"children":[{"type":"text","value":"If ","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"key":"UjMDLWg2dk"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, \\beta_+, \\beta_-)\\geq0","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"html":"P+(n1,n2,β+,β−)≥0\\mathcal{P}_+(n_1, n_2, \\beta_+, \\beta_-)\\geq0P+​(n1​,n2​,β+​,β−​)≥0","key":"h442Z6LOOY"},{"type":"text","value":" and ","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"key":"VSu9SNrmYO"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)<0","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"html":"P+(n1,n2,β−,β+)<0\\mathcal{P}_+(n_1, n_2, \\beta_-, \\beta_+)<0P+​(n1​,n2​,β−​,β+​)<0","key":"yJM6Phbo6l"},{"type":"text","value":". It means that the only solution is the state defined by ","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"key":"GYDJ6uy9wh"},{"type":"inlineMath","value":"|c|=\\beta_+","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"html":"∣c∣=β+|c|=\\beta_+∣c∣=β+​","key":"Z23v4MJF8m"},{"type":"text","value":" and ","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"key":"XfVuc73HAV"},{"type":"inlineMath","value":"|d|=\\beta_-","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"html":"∣d∣=β−|d|=\\beta_-∣d∣=β−​","key":"e4LOyCDYz6"},{"type":"text","value":". The density matrix of its partial transposed is also negative, which means the state is entangled.","position":{"start":{"line":306,"column":1},"end":{"line":306,"column":1}},"key":"Mz4RcoTHeJ"}],"key":"cLZ5bauOih"},{"type":"listItem","spread":true,"position":{"start":{"line":307,"column":1},"end":{"line":308,"column":1}},"children":[{"type":"text","value":"If both ","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"mRJC1Hj6FN"},{"type":"inlineMath","value":"\\mathcal{P}_+","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"html":"P+\\mathcal{P}_+P+​","key":"t6Hq9CHhQE"},{"type":"text","value":" and ","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"L3v4TXGaEf"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"bLZNw6x51C"},{"type":"text","value":" are negative, there are no solutions. Either the state is non-Gaussian or one of the hypothesis is not satisfied (the state is slightly displaced or not purely thermal).","position":{"start":{"line":307,"column":1},"end":{"line":307,"column":1}},"key":"ZCqREv0ZyU"}],"key":"fTNVQfKliV"}],"key":"mlg9qjjl5R"},{"type":"paragraph","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"children":[{"type":"text","value":"The last option is impossible with our hypothesis: it means that the sign of ","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"key":"w1g19TkqSE"},{"type":"inlineMath","value":"\\mathcal{P}_+(n_1,","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"html":"P+(n1,\\mathcal{P}_+(n_1,P+​(n1​,","key":"dahlebrALq"},{"type":"text","value":" ","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"key":"eZfIOuWe95"},{"type":"inlineMath","value":" n_2, \\beta_-,","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"html":"n2,β−, n_2, \\beta_-,n2​,β−​,","key":"KnE7fbuKm3"},{"type":"text","value":" ","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"key":"knfguqlpjL"},{"type":"inlineMath","value":" \\beta_+)","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"html":"β+) \\beta_+)β+​)","key":"hsXrskhVIB"},{"type":"text","value":" completely determines the separability of the state, which ends the demonstration.","position":{"start":{"line":309,"column":1},"end":{"line":309,"column":1}},"key":"vnR4oTJ0HL"}],"key":"zQr0JMun30"}],"key":"UWEzbduGgF"},{"type":"heading","depth":3,"position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"children":[{"type":"text","value":"Quantifying entanglement","position":{"start":{"line":312,"column":1},"end":{"line":312,"column":1}},"key":"Tt2wJ7bc0Y"}],"identifier":"quantifying-entanglement","label":"Quantifying entanglement","html_id":"quantifying-entanglement","implicit":true,"key":"V5befzZgiv"},{"type":"paragraph","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"children":[{"type":"text","value":"Some authors discussed the need to better quantify entanglement rather than just a yes/no answer ","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"r3lExvpOsl"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"children":[{"type":"cite","identifier":"isoard_2021_bipartite","label":"isoard_2021_bipartite","kind":"parenthetical","position":{"start":{"line":313,"column":99},"end":{"line":313,"column":121}},"children":[{"type":"text","value":"Isoard ","key":"jYfDQkteD1"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ewVD48EPej"}],"key":"u3Rp5diuHx"},{"type":"text","value":", 2021","key":"DEKj3ZBXsC"}],"enumerator":"11","key":"v2cfj8HFeK"}],"key":"uhhI2kd3fK"},{"type":"text","value":". Here we briefly explain that with our method, we have access to all the symplectic invariants that allows us to compute the logarithmic negativity, and therefore to quantify entanglement","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"CFlffFKhfu"},{"type":"footnoteReference","identifier":"note_discord","label":"note_discord","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"number":4,"enumerator":"4","key":"FHqd5zENpC"},{"type":"text","value":". From the values of ","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"InSti9x06S"},{"type":"inlineMath","value":"(g^{(2)}, g^{(4)})","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"html":"(g(2),g(4))(g^{(2)}, g^{(4)})(g(2),g(4))","key":"d95WU7nMNL"},{"type":"text","value":", we compute ","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"S7jvtalhyZ"},{"type":"inlineMath","value":"\\beta_\\pm","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"html":"β±\\beta_\\pmβ±​","key":"txVreHeouo"},{"type":"text","value":". Using the ","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"n0zOlV8FXD"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"html":"g(2)/g(4)g^{(2)}/g^{(4)}g(2)/g(4)","key":"ovv3puXAEb"},{"type":"text","value":" ","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"SfGXtAWArT"},{"type":"crossReference","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"children":[{"type":"text","value":"criterion","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"InMGjatWWT"}],"identifier":"g2g4criterion","label":"g2g4criterion","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"g2g4criterion","key":"PACnGoyWcP"},{"type":"text","value":", we know that:","position":{"start":{"line":313,"column":1},"end":{"line":313,"column":1}},"key":"NYMBAsLSY0"}],"key":"jwrIMXFRs7"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":315,"column":1},"end":{"line":318,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"children":[{"type":"text","value":"Either the state is separable, which means that the logarithmic negativity is null: ","position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"key":"W0q5xCpVJw"},{"type":"inlineMath","value":"E_\\mathcal{N}=0","position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"html":"EN=0E_\\mathcal{N}=0EN​=0","key":"w5wWHcuiZU"},{"type":"text","value":".","position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"key":"t11pxQybvi"}],"key":"Dc3JGfRTbO"},{"type":"listItem","spread":true,"position":{"start":{"line":316,"column":1},"end":{"line":318,"column":1}},"children":[{"type":"text","value":"Or the state is entangled and is characterized by ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"cRpLm7WBLf"},{"type":"inlineMath","value":"|c|=\\beta_+","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"∣c∣=β+|c|=\\beta_+∣c∣=β+​","key":"NhKJdtvZlj"},{"type":"text","value":" and ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"O9coXiv1Nw"},{"type":"inlineMath","value":"|d|=\\beta_-","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"∣d∣=β−|d|=\\beta_-∣d∣=β−​","key":"E5FNgYhmd5"},{"type":"text","value":". We therefore have the values of all the local symplectic invariants of the partially transposed covariance matrix: ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"hrzGc3MQOR"},{"type":"inlineMath","value":"\\text{det}\\boldsymbol{A}","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"detA\\text{det}\\boldsymbol{A}detA","key":"licULoCOxj"},{"type":"text","value":", ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"vpacxK7Mys"},{"type":"inlineMath","value":"\\text{det}\\boldsymbol{B}","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"detB\\text{det}\\boldsymbol{B}detB","key":"KoBc15eyAK"},{"type":"text","value":", ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"l8E86loHYh"},{"type":"inlineMath","value":"\\text{det}\\boldsymbol{C}","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"detC\\text{det}\\boldsymbol{C}detC","key":"n7a8Zr62sv"},{"type":"text","value":", and ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"Mu1DcEX2El"},{"type":"inlineMath","value":"\\text{det}\\boldsymbol{\\sigma}","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"detσ\\text{det}\\boldsymbol{\\sigma}detσ","key":"ghXXoD9GfZ"},{"type":"text","value":". We can compute the value of its smallest eigenvalue ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"l0jMTBHvO3"},{"type":"inlineMath","value":"\\tilde{\\nu}_-","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"ν~−\\tilde{\\nu}_-ν~−​","key":"OXawOR1ILV"},{"type":"text","value":" using eq. ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"oRfpfnu8fk"},{"type":"crossReference","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"children":[{"type":"text","value":"(","key":"Sm9nMzw0Tx"},{"type":"text","value":"14","key":"h9S9lgyYFf"},{"type":"text","value":")","key":"sOnsvcazWe"}],"identifier":"eigen_val_small","label":"eigen_val_small","kind":"equation","template":"(%s)","enumerator":"14","resolved":true,"html_id":"eigen-val-small","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"myYi8zpPGj"},{"type":"text","value":" and, consequently, the logarithmic negativity ","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"Bgf2diyoTq"},{"type":"inlineMath","value":"E_\\mathcal{N}(\\boldsymbol{\\sigma}) = - \\log_2 \\tilde{\\nu}_-","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"html":"EN(σ)=−log⁡2ν~−E_\\mathcal{N}(\\boldsymbol{\\sigma}) = - \\log_2 \\tilde{\\nu}_-EN​(σ)=−log2​ν~−​","key":"vN2CjH5fzD"},{"type":"text","value":".","position":{"start":{"line":316,"column":1},"end":{"line":316,"column":1}},"key":"G3D5z8s9vU"}],"key":"a73U3ZFCBq"}],"key":"VkMzDeybGm"},{"type":"heading","depth":3,"position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"children":[{"type":"text","value":"Relating the ","position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"key":"q4Bo5hytRs"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"html":"g(2)/g(4)g^{(2)}/g^{(4)}g(2)/g(4)","key":"RJo4Q0TliU"},{"type":"text","value":" criterion to the ","position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"key":"Zoboj3xEyp"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"zYecPXs6Ug"},{"type":"text","value":" witness","position":{"start":{"line":321,"column":1},"end":{"line":321,"column":1}},"key":"yHe6wDeAII"}],"identifier":"relating-the-g-2-g-4-criterion-to-the-g-2-witness","label":"Relating the g^{(2)}/g^{(4)} criterion to the g^{(2)} witness","html_id":"relating-the-g-2-g-4-criterion-to-the-g-2-witness","implicit":true,"key":"JlwHyjnisA"},{"type":"paragraph","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"children":[{"type":"text","value":"In the proof of the ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"tnCVWhPYF3"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"TzGIc1iKjY"},{"type":"text","value":" theorem, we wrote the expression of ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"tBeTQ4BTpn"},{"type":"inlineMath","value":"\\beta_\\pm","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"β±\\beta_\\pmβ±​","key":"BPv0GP0hIv"},{"type":"text","value":" in Eq. ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"Yrv37qtIMR"},{"type":"crossReference","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"children":[{"type":"text","value":"(","key":"sndSba4Mus"},{"type":"text","value":"22","key":"jKXZ4nnaqP"},{"type":"text","value":")","key":"qGFmyg2O1u"}],"identifier":"cd_values_g2g4","label":"cd_values_g2g4","kind":"equation","template":"(%s)","enumerator":"22","resolved":true,"html_id":"cd-values-g2g4","key":"XHtH8ZRKsA"},{"type":"text","value":". This quantity is the value of ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"nnY2z1LMnz"},{"type":"inlineMath","value":"|c|^2","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"∣c∣2|c|^2∣c∣2","key":"s5irKIs7sk"},{"type":"text","value":" and ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"ML15zdYmz6"},{"type":"inlineMath","value":"|d|^2","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"∣d∣2|d|^2∣d∣2","key":"DoweMNLbXk"},{"type":"text","value":" hence ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"Jy9T6LH23T"},{"type":"inlineMath","value":"\\beta_\\pm","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"β±\\beta_\\pmβ±​","key":"EQq2DJevKI"},{"type":"text","value":" must be real and positive. This implies that the fourth order correlation function is bounded from above so that ","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"key":"ZF7vyA8n3o"},{"type":"inlineMath","value":"\\beta_\\pm\\in \\mathbb{R}","position":{"start":{"line":322,"column":1},"end":{"line":322,"column":1}},"html":"β±∈R\\beta_\\pm\\in \\mathbb{R}β±​∈R","key":"P1uHz38wxk"}],"key":"uN9PO8QVYc"},{"type":"math","identifier":"max_boundg4","label":"max_boundg4","value":"g^{(4)}_{12} \\leq 16g^{(2)}_{12} +6\\left(g^{(2)}_{12}-1\\right)^2 -12 .","tight":true,"html":"g12(4)≤16g12(2)+6(g12(2)−1)2−12.g^{(4)}_{12} \\leq 16g^{(2)}_{12} +6\\left(g^{(2)}_{12}-1\\right)^2 -12 .g12(4)​≤16g12(2)​+6(g12(2)​−1)2−12.","enumerator":"24","html_id":"max-boundg4","key":"X7X651tMHO"},{"type":"paragraph","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"children":[{"type":"text","value":"and from below to ensure ","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"key":"wjWW4SvuFf"},{"type":"inlineMath","value":"\\beta_-\\geq0","position":{"start":{"line":327,"column":1},"end":{"line":327,"column":1}},"html":"β−≥0\\beta_-\\geq0β−​≥0","key":"uJKmTJg06k"}],"key":"LOPV02p7Gg"},{"type":"math","identifier":"minboundg4_eq","label":"minboundg4_eq","value":"g^{(4)}_{12} \\geq 16g^{(2)}_{12} +4\\left(g^{(2)}_{12}-1\\right)^2 -12","tight":true,"html":"g12(4)≥16g12(2)+4(g12(2)−1)2−12g^{(4)}_{12} \\geq 16g^{(2)}_{12} +4\\left(g^{(2)}_{12}-1\\right)^2 -12g12(4)​≥16g12(2)​+4(g12(2)​−1)2−12","enumerator":"25","html_id":"minboundg4-eq","key":"lToll8ueFG"},{"type":"paragraph","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"children":[{"type":"text","value":"The two later conditions are needed to ensure that ","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"key":"ckWRJ2ex3K"},{"type":"inlineMath","value":"\\beta_\\pm","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"html":"β±\\beta_\\pmβ±​","key":"lNLyXbgnyS"},{"type":"text","value":" exists. A deviation from of these bounds would imply that either the state is non-Gaussian state or the local correlation is not exactly 2","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"key":"p0D6knY3QS"},{"type":"footnoteReference","identifier":"footnote_g2egal2","label":"footnote_g2egal2","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"number":5,"enumerator":"5","key":"yrWNjyV5WM"},{"type":"text","value":", which invalidates expression ","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"key":"AKlHwHgTqF"},{"type":"crossReference","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"children":[{"type":"text","value":"(","key":"rb66oT28Su"},{"type":"text","value":"17","key":"xbO7ArleH1"},{"type":"text","value":")","key":"LFKQZmZEoj"}],"identifier":"fourth_order_correlation_expression","label":"fourth_order_correlation_expression","kind":"equation","template":"(%s)","enumerator":"17","resolved":true,"html_id":"fourth-order-correlation-expression","key":"BgMXaJ4cDW"},{"type":"text","value":" for the fourth order correlation function.","position":{"start":{"line":332,"column":1},"end":{"line":332,"column":1}},"key":"NsLcJLXHnO"}],"key":"jDOs65JVsm"},{"type":"paragraph","position":{"start":{"line":336,"column":1},"end":{"line":336,"column":1}},"children":[{"type":"text","value":"Because of these bounds, it is better to parametrize the fourth order correlation function with ","position":{"start":{"line":336,"column":1},"end":{"line":336,"column":1}},"key":"Df9D10yNgr"},{"type":"inlineMath","value":"\\theta \\in[0,1]","position":{"start":{"line":336,"column":1},"end":{"line":336,"column":1}},"html":"θ∈[0,1]\\theta \\in[0,1]θ∈[0,1]","key":"b9omTKm4Cf"},{"type":"text","value":" so that we can write the fourth order correlation function as","position":{"start":{"line":336,"column":1},"end":{"line":336,"column":1}},"key":"kYpjaApi7g"}],"key":"nsuZXFPKGg"},{"type":"math","identifier":"g4theta","label":"g4theta","value":"g^{(4)}_{12} = 16g^{(2)}_{12} +4\\left(g^{(2)}_{12}-1\\right)^2 -12 +\\left(g^{(2)}_{12}-1\\right)^2\\times 2\\theta","tight":true,"html":"g12(4)=16g12(2)+4(g12(2)−1)2−12+(g12(2)−1)2×2θg^{(4)}_{12} = 16g^{(2)}_{12} +4\\left(g^{(2)}_{12}-1\\right)^2 -12 +\\left(g^{(2)}_{12}-1\\right)^2\\times 2\\thetag12(4)​=16g12(2)​+4(g12(2)​−1)2−12+(g12(2)​−1)2×2θ","enumerator":"26","html_id":"g4theta","key":"gZZcsRv9dO"},{"type":"paragraph","position":{"start":{"line":341,"column":1},"end":{"line":341,"column":1}},"children":[{"type":"text","value":"so that the ","position":{"start":{"line":341,"column":1},"end":{"line":341,"column":1}},"key":"xX8JgWnthI"},{"type":"inlineMath","value":"\\beta_\\pm","position":{"start":{"line":341,"column":1},"end":{"line":341,"column":1}},"html":"β±\\beta_\\pmβ±​","key":"Ru9F5aRRLK"},{"type":"text","value":" is given by","position":{"start":{"line":341,"column":1},"end":{"line":341,"column":1}},"key":"QEl6d0JRuU"}],"key":"MWw4vAZxXm"},{"type":"math","identifier":"beta_theta_expr","label":"beta_theta_expr","value":"\\beta_\\pm^2 = n_1n_2(g_{12}^{(2)} - 1)\\frac{ 1\\pm \\sqrt{1 - \\theta }}{2}.","tight":"before","html":"β±2=n1n2(g12(2)−1)1±1−θ2.\\beta_\\pm^2 = n_1n_2(g_{12}^{(2)} - 1)\\frac{ 1\\pm \\sqrt{1 - \\theta }}{2}.β±2​=n1​n2​(g12(2)​−1)21±1−θ​​.","enumerator":"27","html_id":"beta-theta-expr","key":"fMhG1dSh68"},{"type":"paragraph","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"fpAzeGpzOF"},{"type":"inlineMath","value":"\\theta=0","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"θ=0\\theta=0θ=0","key":"zMRqm49QD8"},{"type":"text","value":", that is when the fourth order correlation function takes its minimum value, this implies that either ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"uduZ3XYCW1"},{"type":"inlineMath","value":"c","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"ccc","key":"SpqqsJfwKg"},{"type":"text","value":" or ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"NosIY2jDEJ"},{"type":"inlineMath","value":"d","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"ddd","key":"XtKqp5wKuB"},{"type":"text","value":" is null. On the opposite, when the fourth order correlation function takes its maximum value, this means that ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"lwfsBTE9d7"},{"type":"inlineMath","value":"|c|=|d|","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"∣c∣=∣d∣|c|=|d|∣c∣=∣d∣","key":"WBjXQucTSF"},{"type":"text","value":". We now fix the value of the population to 0.4 and 0.9 to study in ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"VFQ142uugW"},{"type":"crossReference","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"children":[{"type":"text","value":"Figure ","key":"leFEjZ3qir"},{"type":"text","value":"3","key":"PwOK0ZRaRV"}],"identifier":"g2g4_map_image","label":"g2g4_map_image","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"g2g4-map-image","key":"soYmjP6acr"},{"type":"text","value":" the entanglement in the ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"nzVE10SInT"},{"type":"inlineMath","value":"(g^{(2)}, \\theta)","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"(g(2),θ)(g^{(2)}, \\theta)(g(2),θ)","key":"t1ypnBKqOl"},{"type":"text","value":" plane. First we observe there are still un-physical regions on these maps. Indeed, the bounds on ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"CfFK6zjaCe"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"KEdsxpYbZ4"},{"type":"text","value":" that we derived are just algebraic bounds that imply consistency of our reasoning. They do not take into account the ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"RorThmgXo8"},{"type":"emphasis","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"ijc1iJmCeJ"}],"key":"r8IvhsiyUe"},{"type":"text","value":" condition. The grey regions in ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"vHYvslaj7O"},{"type":"crossReference","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"children":[{"type":"text","value":"Figure ","key":"In9Qn1GcJP"},{"type":"text","value":"3","key":"PFVNi8r2rJ"}],"identifier":"g2g4_map_image","label":"g2g4_map_image","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"g2g4-map-image","key":"rm6tTc7yfD"},{"type":"text","value":" is a consequence of this ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"p4e4thLFrK"},{"type":"emphasis","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"pmPVhNPGP1"}],"key":"vdl7AsimGJ"},{"type":"text","value":" condition. It emphasizes that the bounds on ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"BdWgUruOZR"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"HROuTqKiZQ"},{"type":"text","value":" are stricter. In fact, it is these stricter bounds that permitted us to derive the ","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"AXPS9pC2nU"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"piVatiu0wv"},{"type":"text","value":" bound to witness entanglement.","position":{"start":{"line":347,"column":1},"end":{"line":347,"column":1}},"key":"zyQNy7fuyp"}],"key":"Dt5exV3AcI"},{"type":"container","kind":"figure","identifier":"g2g4_map_image","label":"g2g4_map_image","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/g2g4_map-19ceb280d46148382e28aae5eb6d4862.png","alt":"Entanglement as a function","width":"100%","align":"center","key":"pV4WhpnXCU","urlSource":"images/g2g4_map.png","urlOptimized":"/~gondret/phd_manuscript/build/g2g4_map-19ceb280d46148382e28aae5eb6d4862.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"g2g4_map_image","identifier":"g2g4_map_image","html_id":"g2g4-map-image","enumerator":"3","children":[{"type":"text","value":"Figure ","key":"G79U65XcES"},{"type":"text","value":"3","key":"H5qUUO3Xlx"},{"type":"text","value":":","key":"IiDMzvihbt"}],"template":"Figure %s:","key":"gxV98X61PU"},{"type":"text","value":"Entanglement 2D plane of the state for a state with an average population of 0.4 (left) and 0.9 (right). On the ","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"EJgR9DPwyx"},{"type":"inlineMath","value":"x","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"html":"xxx","key":"Gt6DOitv59"},{"type":"text","value":"-axis lies the second order correlation function and on the ","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"YOF7wnsyGx"},{"type":"inlineMath","value":"y","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"html":"yyy","key":"DTVuUOiG8A"},{"type":"text","value":" axis the ","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"is1EkR8coE"},{"type":"text","value":"θ","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"KcoeMCpV1Y"},{"type":"text","value":" parameter defined in eq ","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"HrB4WP8Gmd"},{"type":"crossReference","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"children":[{"type":"text","value":"(","key":"uqfd4uDkCO"},{"type":"text","value":"26","key":"ipxYev6gHN"},{"type":"text","value":")","key":"h4mxBrdqua"}],"identifier":"g4theta","label":"g4theta","kind":"equation","template":"(%s)","enumerator":"26","resolved":true,"html_id":"g4theta","key":"ENDe9m5k9Y"},{"type":"text","value":". The separable and non-separable regions are split by the g","key":"UH6YHhaV99"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"GGtrwLzqGE"}],"key":"Rkh0ctZfpM"},{"type":"text","value":" criterion (dotted black line). The solid green (dashed brown) vertical line represents the entanglement (separable) ","key":"p20DhjzCtm"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"qAiDw2aojv"},{"type":"text","value":" threshold. The un-physical region is shown in grey.","position":{"start":{"line":357,"column":1},"end":{"line":357,"column":1}},"key":"n6lebKGVb1"}],"key":"CbWxDaaHsK"}],"key":"HTIhLRLAHa"}],"enumerator":"3","html_id":"g2g4-map-image","key":"CsckDirktQ"},{"type":"paragraph","position":{"start":{"line":360,"column":1},"end":{"line":360,"column":1}},"children":[{"type":"text","value":"As previously, we need to discriminate the case where the population is higher than ","position":{"start":{"line":360,"column":1},"end":{"line":360,"column":1}},"key":"UjwZs5n5wT"},{"type":"text","value":"0.7","position":{"start":{"line":360,"column":1},"end":{"line":360,"column":1}},"key":"phCOruT9do"},{"type":"text","value":" or smaller.","position":{"start":{"line":360,"column":1},"end":{"line":360,"column":1}},"key":"jdFGkbSBJN"}],"key":"durdAXXXoY"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":361,"column":1},"end":{"line":364,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"x6SsdPFkRt"},{"type":"inlineMath","value":"n_1n_2 < 0.5","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"n1n2<0.5n_1n_2 < 0.5n1​n2​<0.5","key":"mLoZ4GmwO0"},{"type":"text","value":" (left subplot of ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"ee3ycZy5zo"},{"type":"crossReference","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"children":[{"type":"text","value":"Figure ","key":"pD3qa98QfO"},{"type":"text","value":"3","key":"WAhkmAkiJq"}],"identifier":"g2g4_map_image","label":"g2g4_map_image","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"g2g4-map-image","key":"oPwqz5ETD0"},{"type":"text","value":" ), the ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"MM8kC2sojV"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"I5spen75pr"},{"type":"text","value":" threshold to assess entanglement is higher than 2. For a 0.4 population it is 2.21. Second, we know also that for such low population, a second order correlation function below 2 implies separability of the state. Those two lines are respectively sketched on the plane as solid and dashed vertical lines. In the tight region where the ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"pGWbyNoDis"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"SVhG2P0dVH"},{"type":"text","value":" witness is not sufficient, the fourth order correlation function gives the answer. We represented the border between the two regions with a dotted black line. In this case (low population)e, we see that at fixed ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"nR0r2vlgGu"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"xH7DkjWuiC"},{"type":"text","value":" a lower ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"akD6BNO5dW"},{"type":"text","value":"θ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"hH8H9HxEyH"},{"type":"text","value":" ables one to jump from the separable region to the entangled region. This means that the value of ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"NkttWHE9sZ"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"∣d∣|d|∣d∣","key":"reCtfUVhY2"},{"type":"text","value":" must be as small as possible. In this case, ","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"JnhkvOunQi"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"html":"∣d∣|d|∣d∣","key":"opmiCg8and"},{"type":"text","value":" does not contribute to the bi-partite entanglement.","position":{"start":{"line":361,"column":1},"end":{"line":361,"column":1}},"key":"GUko4vZBTC"}],"key":"lJHurevj7W"},{"type":"listItem","spread":true,"position":{"start":{"line":362,"column":1},"end":{"line":364,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"p23C6N292R"},{"type":"inlineMath","value":"n_1n_2 > 0.5","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"html":"n1n2>0.5n_1n_2 > 0.5n1​n2​>0.5","key":"oGYfWgHEMS"},{"type":"text","value":", (right subplot of ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"cNRSMQEqB6"},{"type":"crossReference","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"children":[{"type":"text","value":"Figure ","key":"vB3x6UB4E9"},{"type":"text","value":"3","key":"ltjUVxIIE3"}],"identifier":"g2g4_map_image","label":"g2g4_map_image","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"g2g4-map-image","key":"kSP66yMtZG"},{"type":"text","value":"), the ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"m6krJE9PtX"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"gE2Nhap9qW"},{"type":"text","value":" threshold to assess entanglement equal to 2. The reasoning here is different from the previous one: a higher value of ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"dWQTOnbHBL"},{"type":"text","value":"θ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"sh0jtDrRbW"},{"type":"text","value":", ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"lCWAMB37H8"},{"type":"emphasis","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"vvtPPrLQjL"}],"key":"FThE6bdYbY"},{"type":"text","value":" a higher value of ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"qfSsoOQzZF"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"QraFmFlF9f"},{"type":"text","value":" ables to pass from the separable to the entangled region. In this case, ","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"OAmRezjHbo"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"html":"∣d∣|d|∣d∣","key":"uL4HC4Xy38"},{"type":"text","value":" contributes to the non-separability.","position":{"start":{"line":362,"column":1},"end":{"line":362,"column":1}},"key":"pX2yGHSxSf"}],"key":"jiRbt1Vq0Q"}],"key":"Gj8syTjJiO"},{"type":"paragraph","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"YC16K59YdJ"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"R0l0a096jd"},{"type":"text","value":" lies above the non-separability threshold ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"al0hW0vI3z"},{"type":"crossReference","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"children":[{"type":"text","value":"(","key":"KXujDM6bJK"},{"type":"text","value":"11","key":"YWNr7o4Se7"},{"type":"text","value":")","key":"KtxlDQ7Avb"}],"identifier":"fg2","label":"Fg2","kind":"equation","template":"(%s)","enumerator":"11","resolved":true,"html_id":"fg2","key":"mgLDp6r2LK"},{"type":"text","value":", ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"zZ9lbWtBot"},{"type":"text","value":"θ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"uewaotI9cO"},{"type":"text","value":" cannot take any value in the ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"p8Ha3OBhqH"},{"type":"inlineMath","value":"[0,1]","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"html":"[0,1][0,1][0,1]","key":"BtfYzY071w"},{"type":"text","value":" interval. It is bounded from above by the un-physical region. As the second correlation function increases, it goes to 0 and vanishes in the limiting case where ","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"aKk4gGsDxR"},{"type":"inlineMath","value":"g^{(2)} = 2+1/n","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"html":"g(2)=2+1/ng^{(2)} = 2+1/ng(2)=2+1/n","key":"G4k8rrq49n"},{"type":"text","value":": this corresponds to a pure two-mode squeezed vacuum state. In this case, the fourth order correlation function must take its lower value.","position":{"start":{"line":365,"column":1},"end":{"line":365,"column":1}},"key":"ZiqTbxisYu"}],"key":"rHXX2bOW7l"},{"type":"heading","depth":2,"position":{"start":{"line":372,"column":1},"end":{"line":372,"column":1}},"children":[{"type":"text","value":"Finite efficiency effects","position":{"start":{"line":372,"column":1},"end":{"line":372,"column":1}},"key":"wIsqm9tEtB"}],"identifier":"finite-efficiency-effects","label":"Finite efficiency effects","html_id":"finite-efficiency-effects","implicit":true,"enumerator":"6","key":"ChygtSQJYn"},{"type":"paragraph","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"children":[{"type":"text","value":"So far, I have not discussed the effect of pure losses, which might be significant given that a micro-channel plate has a rather small detection efficiency (approximately 50%, see the fourth chapter’s ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"USowTGvHEt"},{"type":"crossReference","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"EdiDO5vL5l"}],"identifier":"efficiency_mcp","label":"efficiency_mcp","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"efficiency-mcp","remote":true,"url":"/mcp-physics","dataUrl":"/mcp-physics.json","key":"xPYCmOZlk9"},{"type":"text","value":" ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"tRYUw3ejep"},{"type":"crossReference","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"children":[{"type":"text","value":"4","key":"RusjCQLQuk"}],"identifier":"efficiency_mcp","label":"efficiency_mcp","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"efficiency-mcp","remote":true,"url":"/mcp-physics","dataUrl":"/mcp-physics.json","key":"v91HHheQte"},{"type":"text","value":"). As we saw in ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"YqIN0uQ0Ds"},{"type":"crossReference","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"ZUmlZlWFh5"}],"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","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"YHhLjLuxW0"},{"type":"text","value":" ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"j3KrGK7KyO"},{"type":"crossReference","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"children":[{"type":"text","value":"5","key":"KUvPzEdZ6g"}],"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","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"qjNpGcnh28"},{"type":"text","value":", losses are represented by a beam splitter with transmittance ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"UZy0ULqlo0"},{"type":"inlineMath","value":"\\sqrt{\\eta}","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"html":"η\\sqrt{\\eta}η​","key":"vhacdAZ3Qr"},{"type":"text","value":" that mixes each mode with the vacuum, where ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"ItahMLERV3"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"html":"η\\eta η","key":"R3NiBk01Hg"},{"type":"text","value":" is the efficiency of the detector.","position":{"start":{"line":374,"column":1},"end":{"line":374,"column":1}},"key":"vfAIs5o1b7"}],"key":"n4wUMt8f8N"},{"type":"paragraph","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"children":[{"type":"text","value":"It must be emphasized that a beam splitter, being a passive transformation, cannot produce entanglement ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"GDXaBFyp1y"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"children":[{"type":"cite","identifier":"weedbrook_gaussian_2012","label":"weedbrook_gaussian_2012","kind":"parenthetical","position":{"start":{"line":377,"column":106},"end":{"line":377,"column":130}},"children":[{"type":"text","value":"Weedbrook ","key":"SrgJJ1lJ07"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lzjjEM8Si6"}],"key":"p73DjUrlsh"},{"type":"text","value":", 2012","key":"FqaPOR6sRu"}],"enumerator":"12","key":"MoxPDH3w6I"}],"key":"Ia79Xbuhz8"},{"type":"text","value":". Consequently, if the state after the beam splitter is entangled, the state before it must also be entangled. In our case, we are interested in determining whether the state before the detector is entangled, and therefore, we need to correct for the effects of finite efficiency. For thermal states, the effect of non-perfect detection efficiency results in multiplying ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"pBunasOaaT"},{"type":"inlineMath","value":"n_1","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"html":"n1n_1n1​","key":"uX1zoWSmY0"},{"type":"text","value":", ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"I5ZgZjkwWZ"},{"type":"inlineMath","value":"n_2","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"html":"n2n_2n2​","key":"ouEL4PQurm"},{"type":"text","value":", ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"E5kAIb1brR"},{"type":"inlineMath","value":"|c|","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"html":"∣c∣|c|∣c∣","key":"FQEsgwUlii"},{"type":"text","value":", and ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"KPhS5jTXoQ"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"html":"∣d∣|d|∣d∣","key":"X1aEVhRinh"},{"type":"text","value":" by ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"BF0RYVywds"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"html":"η\\eta η","key":"Ynz3waUpG6"},{"type":"text","value":".","position":{"start":{"line":377,"column":1},"end":{"line":377,"column":1}},"key":"mYAJNc43Xg"}],"key":"YGyJWNCXWX"},{"type":"paragraph","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"text","value":"How losses affect entanglement depends strongly on the nature of the entanglement: it is known for example that a two-mode squeezed thermal state remains entangled after a pure loss channel ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"PxZP9nlti9"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"cite","identifier":"scheel_entanglement_2001b","label":"scheel_entanglement_2001b","kind":"parenthetical","position":{"start":{"line":381,"column":192},"end":{"line":381,"column":218}},"children":[{"type":"text","value":"Scheel ","key":"kdtiqFUuaz"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wuv96WcWHM"}],"key":"JYYXS3TnqJ"},{"type":"text","value":", 2001","key":"jcQo3GPEyn"}],"enumerator":"13","key":"umy6XKfH5L"}],"key":"KmES4ewQZZ"},{"type":"text","value":". In general, any state that violates the ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"rVUCNnVmAy"},{"type":"crossReference","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"text","value":"Campo-Parentani","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"A7piSbFzjt"}],"identifier":"busch","label":"busch","kind":"proof:theorem","template":"{name}","resolved":true,"html_id":"busch","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"TgLsKc6jiN"},{"type":"text","value":" witness ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"T7ueDwop7I"},{"type":"inlineMath","value":"\\Delta ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"html":"Δ\\Delta Δ","key":"r69x2NUW4L"},{"type":"text","value":" before a pure loss channel remains non-separable after (simply because ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"mcfwB1dd17"},{"type":"inlineMath","value":"\\Delta ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"html":"Δ\\Delta Δ","key":"mkxBQkKORI"},{"type":"text","value":" is proportional to the detection efficiency hence it does not change its sign). Some other states are however more fragile. In the literature, ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"DncYNQoYC8"},{"type":"cite","identifier":"barbosa_disentanglement_2011","label":"barbosa_disentanglement_2011","kind":"narrative","position":{"start":{"line":381,"column":529},"end":{"line":381,"column":558}},"children":[{"type":"text","value":"Barbosa ","key":"ypZYdL8U1u"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OH8fM0Hmy5"}],"key":"L9ytaaQcKU"},{"type":"text","value":" (2011)","key":"SWcnHIFMMK"}],"enumerator":"14","key":"YCL3RzZjnv"},{"type":"text","value":" divide states in three categories : ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"ZOQOIPNPbu"},{"type":"emphasis","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"text","value":"robust","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"TtBZLrSFNW"}],"key":"MLGlzlMAs1"},{"type":"text","value":", ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"YOBvwQBCBA"},{"type":"emphasis","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"text","value":"partially robust","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"nHYq2AvPL0"}],"key":"dJdfDbHtU1"},{"type":"text","value":" and ","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"q81xKFvh8x"},{"type":"emphasis","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"children":[{"type":"text","value":"fragile","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"EXrGPpTskr"}],"key":"HZW4unYQf6"},{"type":"text","value":" states, showing that the effect of losses is not trivial.","position":{"start":{"line":381,"column":1},"end":{"line":381,"column":1}},"key":"rx0H9DPjU7"}],"key":"coP8leXOs2"},{"type":"paragraph","position":{"start":{"line":383,"column":1},"end":{"line":383,"column":1}},"children":[{"type":"text","value":"In this manuscript, we are interested in the physics before the detector, eventually by removing it influence","position":{"start":{"line":383,"column":1},"end":{"line":383,"column":1}},"key":"EmSxrJ4mEW"},{"type":"footnoteReference","identifier":"note_fiber","label":"note_fiber","position":{"start":{"line":383,"column":1},"end":{"line":383,"column":1}},"number":6,"enumerator":"6","key":"PRYctZDBrk"},{"type":"text","value":".","position":{"start":{"line":383,"column":1},"end":{"line":383,"column":1}},"key":"lpTY8b0Ysb"}],"key":"tZcI4PE9Y5"},{"type":"heading","depth":3,"position":{"start":{"line":385,"column":1},"end":{"line":385,"column":1}},"children":[{"type":"text","value":"Taking into account losses on the ","position":{"start":{"line":385,"column":1},"end":{"line":385,"column":1}},"key":"jErmyWiMfj"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":385,"column":1},"end":{"line":385,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"ihb5HIOzxE"},{"type":"text","value":" witness","position":{"start":{"line":385,"column":1},"end":{"line":385,"column":1}},"key":"jLSLWen9BJ"}],"identifier":"taking-into-account-losses-on-the-g-2-witness","label":"Taking into account losses on the g^{(2)} witness","html_id":"taking-into-account-losses-on-the-g-2-witness","implicit":true,"key":"zdx25TcP3F"},{"type":"paragraph","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"children":[{"type":"text","value":"The threshold on ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"vO5CQAdJOa"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"W3yFOOrgKH"},{"type":"text","value":" depends on the population, hence the detection efficiency. The measure of the second order correlation function does not depend on losses while the measured population does. This means that if we measure a population ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"rcBuWdxtJf"},{"type":"inlineMath","value":"n_{det}","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"html":"ndetn_{det}ndet​","key":"nU0ufptRdU"},{"type":"text","value":", we should compare the criterion to the threshold value for ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"lFR1gGo4vR"},{"type":"inlineMath","value":"n_{det}/\\eta","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"html":"ndet/ηn_{det}/\\etandet​/η","key":"AyGna8NUl0"},{"type":"text","value":", where ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"pJH7bgH2wM"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"html":"η\\eta η","key":"uonJTZdMI5"},{"type":"text","value":" is the detector efficiency. Geometrically, this corresponds to a shift to the right on the ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"AAxmxYvnqm"},{"type":"inlineMath","value":"x","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"html":"xxx","key":"RmY95aDSvH"},{"type":"text","value":" axis of ","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"XHgFGNmVKU"},{"type":"crossReference","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"children":[{"type":"text","value":"Figure ","key":"Gnl9QPfF0y"},{"type":"text","value":"1","key":"vZpbs36GNZ"}],"identifier":"g2_criterion_witness","label":"g2_criterion_witness","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"g2-criterion-witness","key":"w0jcdhI9kC"},{"type":"text","value":". One could say that taking into account efficiency facilitate the probe of entanglement: but I would answer that a pure loss channel does reduce entanglement and can even destroy it.","position":{"start":{"line":386,"column":1},"end":{"line":386,"column":1}},"key":"wpZvFAxsHh"}],"key":"oK7dJx1fr1"},{"type":"paragraph","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"children":[{"type":"text","value":"For a two-mode squeezed vacuum state, the second order non-normalized correlation function is ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"L6ate0H8IM"},{"type":"inlineMath","value":"G^{(2)}_{12} = 2n_{0}^2+n_{0}","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"G12(2)=2n02+n0G^{(2)}_{12} = 2n_{0}^2+n_{0}G12(2)​=2n02​+n0​","key":"IGM2nA6whI"},{"type":"text","value":" where ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"z2NhKPQpgs"},{"type":"inlineMath","value":"n_{0}","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"n0n_{0}n0​","key":"TC0Blxf8SG"},{"type":"text","value":" is the number of particles in the state. After detection, we measure the same value of the normalized correlation function, but we measure a population ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"tgamS4oYsN"},{"type":"inlineMath","value":"n_{det} = \\eta n_{0}","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"ndet=ηn0n_{det} = \\eta n_{0}ndet​=ηn0​","key":"zONEsldvVC"},{"type":"text","value":". This means that the non-normalized correlation function reads ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"DtEtmf5a1B"},{"type":"inlineMath","value":"G^{(2)}_{12} = 2n_{det}^2+n_{det}/n_0","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"G12(2)=2ndet2+ndet/n0G^{(2)}_{12} = 2n_{det}^2+n_{det}/n_0G12(2)​=2ndet2​+ndet​/n0​","key":"DTq6oQtec4"},{"type":"text","value":". If we define the correlation strength as ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"xuaa82kWpE"},{"type":"inlineMath","value":"G^{(2)}_{12} - n_1n_2","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"G12(2)−n1n2G^{(2)}_{12} - n_1n_2G12(2)​−n1​n2​","key":"giIQGr5wiM"},{"type":"text","value":", this means that the initial correlation of the state is ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"yCTKNc5GGD"},{"type":"inlineMath","value":"n_0","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"n0n_0n0​","key":"FnpGuWE6Ot"},{"type":"text","value":" while the measured one is only ","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"UMgLxnj6vn"},{"type":"inlineMath","value":"\\eta n_0","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"html":"ηn0\\eta n_0ηn0​","key":"KP14BCEptE"},{"type":"text","value":": a pure loss channel decreases the correlation.","position":{"start":{"line":392,"column":1},"end":{"line":392,"column":1}},"key":"MqC5S9xQCt"}],"key":"dew5VPTUwK"},{"type":"heading","depth":3,"position":{"start":{"line":396,"column":1},"end":{"line":396,"column":1}},"children":[{"type":"text","value":"Taking into account losses on the ","position":{"start":{"line":396,"column":1},"end":{"line":396,"column":1}},"key":"UWfSXP6kNa"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":396,"column":1},"end":{"line":396,"column":1}},"html":"g(2)/g(4)g^{(2)}/g^{(4)}g(2)/g(4)","key":"M3AbPRqy0o"},{"type":"text","value":" criterion","position":{"start":{"line":396,"column":1},"end":{"line":396,"column":1}},"key":"wD0cE2KlJf"}],"identifier":"taking-into-account-losses-on-the-g-2-g-4-criterion","label":"Taking into account losses on the g^{(2)}/g^{(4)} criterion","html_id":"taking-into-account-losses-on-the-g-2-g-4-criterion","implicit":true,"key":"Shb7eJRvan"},{"type":"paragraph","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"children":[{"type":"text","value":"The effect of losses on the ","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"key":"f41C4K4CTz"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"html":"g(2)/g(4)g^{(2)}/g^{(4)}g(2)/g(4)","key":"tSLDKfIhr8"},{"type":"text","value":" criterion is more subtle than for the ","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"key":"vNmfuDPDHp"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"zxLcPNRcmP"},{"type":"text","value":" witness. Note however that the measurement of both correlation functions do not depend on the efficiency of the detector. In the general case, as for the other witness, taking into account the efficiency can be necessary to reveal entanglement. As I did not find a simple way to illustrate the dependance of the criterion with the efficiency, I will just provide an example.","position":{"start":{"line":397,"column":1},"end":{"line":397,"column":1}},"key":"b8ytBfUhl3"}],"key":"PWlTT8Wg41"},{"type":"paragraph","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"children":[{"type":"text","value":"Let’s assume we measure ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"VN3xZLfAEe"},{"type":"inlineMath","value":"g^{(2)}=1.8","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"g(2)=1.8g^{(2)}=1.8g(2)=1.8","key":"VxQrRaAUSI"},{"type":"text","value":". To ensure ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"BlrjWQFWzU"},{"type":"inlineMath","value":"\\beta_\\pm\\in \\mathbb{R}^+","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"β±∈R+\\beta_\\pm\\in \\mathbb{R}^+β±​∈R+","key":"Bzexujeyhs"},{"type":"text","value":", the fourth order correlation function must lie in the interval ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"zI2ic0gXle"},{"type":"inlineMath","value":"[19.36, 20.64]","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"[19.36,20.64][19.36, 20.64][19.36,20.64]","key":"SDJauWhw0S"},{"type":"footnoteReference","identifier":"footnote_experi_rmk","label":"footnote_experi_rmk","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"number":7,"enumerator":"7","key":"Q4S8E0pCTm"},{"type":"text","value":". On ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"pPo9KObyUm"},{"type":"crossReference","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"children":[{"type":"text","value":"Figure ","key":"Mb6tpGZpSf"},{"type":"text","value":"4","key":"qD2bQdc2hP"}],"identifier":"g2_withg4_population","label":"g2_withg4_population","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"g2-withg4-population","key":"LgFFNVtRxr"},{"type":"text","value":", we consider the case where we measured 19.37 (top left), 20 (top right), 20.3 (bottom left) and 20.63 (bottom right). Each plot represents the logarithmic negativity of the state as a function of the population. As in the ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"Q5RAMLfGCt"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"W1eRRMbcJu"},{"type":"text","value":" entanglement witness, when correcting the measured population, we shift on the curve, and we might reveal entanglement. Note however that for a fixed value of the fourth order correlation function, the population cannot be too large. The x-axis reflects the population of the state that is compatible with (","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"eyCLMPs7xD"},{"type":"inlineMath","value":"g^{(2)}, g^{(4)}","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"g(2),g(4)g^{(2)}, g^{(4)}g(2),g(4)","key":"tWTjps9WTy"},{"type":"text","value":") : the x-axis of each subplot is different. The limiting case is when ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"HhPMakALcu"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"AqtjwobSsf"},{"type":"text","value":" goes to its minimal value which means that ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"w3zY29YLFX"},{"type":"inlineMath","value":"\\beta_-=0","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"β−=0\\beta_-=0β−​=0","key":"v2kXEyiqfR"},{"type":"text","value":" (top left). In this case either ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"GhIL7fm3UK"},{"type":"inlineMath","value":"c","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"ccc","key":"c9fmtkp82s"},{"type":"text","value":" or ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"zne8Fwvm29"},{"type":"inlineMath","value":"d","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"ddd","key":"XRamIOuUCb"},{"type":"text","value":" is null, and the state is never entangled (but is always physical) as ","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"T58nOJDkWg"},{"type":"inlineMath","value":"g^{(2)}=1.8 < 2","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"html":"g(2)=1.8<2g^{(2)}=1.8 < 2g(2)=1.8<2","key":"h6BgXcxIbD"},{"type":"text","value":".","position":{"start":{"line":399,"column":1},"end":{"line":399,"column":1}},"key":"GGfGB3zmTf"}],"key":"z77GknHMZP"},{"type":"container","kind":"figure","identifier":"g2_withg4_population","label":"g2_withg4_population","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/range_g2g4_bis-8516d929e068337dc52c7ad76913c55d.png","alt":"Entanglement as a function","width":"100%","align":"center","key":"DmKqrZa7vA","urlSource":"images/range_g2g4_bis.png","urlOptimized":"/~gondret/phd_manuscript/build/range_g2g4_bis-8516d929e068337dc52c7ad76913c55d.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"g2_withg4_population","identifier":"g2_withg4_population","html_id":"g2-withg4-population","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"pKX70WaF5F"},{"type":"text","value":"4","key":"RE4YMhAvkd"},{"type":"text","value":":","key":"rxSZK7Lw94"}],"template":"Figure %s:","key":"ISSnZoSy0j"},{"type":"text","value":"Logarithmic negativity of the state as a function of the population for a fixed value of ","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"key":"uWL4KCglu6"},{"type":"inlineMath","value":"g^{(2)}=1.8","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"html":"g(2)=1.8g^{(2)}=1.8g(2)=1.8","key":"iPFUk3aYGl"},{"type":"text","value":" and ","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"key":"wXRgm802pK"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"T9foQL4SCA"},{"type":"text","value":" (see legend). The state is entangled when the logarithmic negativity is strictly positive. At fixed value of ","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"key":"jtuzNzyegb"},{"type":"inlineMath","value":"(g^{(2)},\\, g^{(4)})","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"html":"(g(2), g(4))(g^{(2)},\\, g^{(4)})(g(2),g(4))","key":"fKwsogg8Zo"},{"type":"text","value":", the population cannot be arbitrary large: the length of the curve of each panel (see the span of the ","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"key":"FtZ0wprwp7"},{"type":"inlineMath","value":"x","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"html":"xxx","key":"SlIQ3n9JkF"},{"type":"text","value":"-axis) is different.","position":{"start":{"line":411,"column":1},"end":{"line":411,"column":1}},"key":"Rd3wgmHwez"}],"key":"W9J11ZCWzj"}],"key":"HRondnT4Gd"}],"enumerator":"4","html_id":"g2-withg4-population","key":"IehafeQvz9"},{"type":"paragraph","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"children":[{"type":"text","value":"We now assume we measure a population of 0.8, with a 50% efficiency detector: the population of the state is therefore 1.6. According to the ","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"key":"J0UvVhx7d0"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"JEeFxzHnKM"},{"type":"text","value":" entanglement witness, this means that we are in the question mark zone of ","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"key":"veVxr0aFH5"},{"type":"crossReference","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"children":[{"type":"text","value":"Figure ","key":"L6OHqtGYck"},{"type":"text","value":"1","key":"tIhUbKKQ67"}],"identifier":"g2_criterion_witness","label":"g2_criterion_witness","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"g2-criterion-witness","key":"hvPmkyNmP4"},{"type":"text","value":". The fourth order correlation function measurement will give us the answer.","position":{"start":{"line":415,"column":1},"end":{"line":415,"column":1}},"key":"E3kn9lKrLO"}],"key":"mtdVCCj0v8"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":417,"column":1},"end":{"line":420,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"children":[{"type":"text","value":"Case where ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"KtFkbeUbc3"},{"type":"inlineMath","value":"g^{(4)}=19.37","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"g(4)=19.37g^{(4)}=19.37g(4)=19.37","key":"XPPJ7jDwHZ"},{"type":"text","value":" : with such low fourth order correlation function, this means that ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"SVqBScsIad"},{"type":"inlineMath","value":"c","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"ccc","key":"EARlvN7HvT"},{"type":"text","value":" or ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"fWaM3yv7YJ"},{"type":"inlineMath","value":"d","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"ddd","key":"h1eW1ztZKi"},{"type":"text","value":" is small and that no matter the population of the state, the state is (almost) never entangled. In the limit where ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"GYKLf4trdk"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"UV9KDHq2KX"},{"type":"text","value":" takes it smaller value, this also mean that the population of the state can be arbitrary large (in this plot, we restricted the population to ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"WlJuxzJvxl"},{"type":"span","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"children":[{"type":"text","value":"10","key":"XWjotJCbYI"},{"type":"superscript","children":[{"type":"text","value":"5","key":"w6cQN3oeug"}],"key":"Hh7K6RGk5X"}],"key":"I8soYfs1Vi"},{"type":"text","value":" but it is not the limit). Here we can safely say that the state is separable: we cannot distinguish ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"snXjdsVno9"},{"type":"inlineMath","value":"c","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"ccc","key":"jD48fOFQDN"},{"type":"text","value":" and ","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"kNa85K0Adl"},{"type":"inlineMath","value":"d","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"html":"ddd","key":"CdPrcN4XtS"},{"type":"text","value":".","position":{"start":{"line":417,"column":1},"end":{"line":417,"column":1}},"key":"Vs6eOimDPF"}],"key":"GlthfEuyZf"},{"type":"listItem","spread":true,"position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"children":[{"type":"text","value":"Case where ","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"xCOywcMKKG"},{"type":"inlineMath","value":"g^{(4)}=20","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"html":"g(4)=20g^{(4)}=20g(4)=20","key":"KapJKnkRNO"},{"type":"text","value":" : the fourth order correlation function is now higher, which means that ","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"xAXjpp5vtp"},{"type":"inlineMath","value":"c","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"html":"ccc","key":"QdwVxhyaln"},{"type":"text","value":" and ","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"vhgcvfQeyw"},{"type":"inlineMath","value":"d","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"html":"ddd","key":"kWqvQnk24x"},{"type":"text","value":" are closer, still quite different. Here we observe that for such correlation, a state big enough (population larger than 2) would be entangled. For a population of ","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"fe3l0u0Vmc"},{"type":"text","value":"1.6","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"snpEauH4z4"},{"type":"text","value":", the logarithmic negativity is zero here which means the state is separable.","position":{"start":{"line":418,"column":1},"end":{"line":418,"column":1}},"key":"VtqSIBC8Te"}],"key":"NHqpeRQKXJ"},{"type":"listItem","spread":true,"position":{"start":{"line":419,"column":1},"end":{"line":419,"column":1}},"children":[{"type":"text","value":"Case where ","position":{"start":{"line":419,"column":1},"end":{"line":419,"column":1}},"key":"BJt0YnB6J3"},{"type":"inlineMath","value":"g^{(4)}=20.3","position":{"start":{"line":419,"column":1},"end":{"line":419,"column":1}},"html":"g(4)=20.3g^{(4)}=20.3g(4)=20.3","key":"NuWfbDHcvD"},{"type":"text","value":" : with a higher value of the fourth order correlation function, the population threshold for non-separability shifts. If the population of the state is 0.8, the state is separable while for a population of 1.6, it ise entangled. Here we see that the pure loss channel of the detector destroyed the entanglement. If one takes it into account, this means that the state before the detector (the one we want to characterize) is entangled.","position":{"start":{"line":419,"column":1},"end":{"line":419,"column":1}},"key":"Nqlj6UJDkg"}],"key":"i83h9oYPnx"},{"type":"listItem","spread":true,"position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"children":[{"type":"text","value":"Case where ","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"key":"k5UsiKgAIL"},{"type":"inlineMath","value":"g^{(4)}=20.63","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"html":"g(4)=20.63g^{(4)}=20.63g(4)=20.63","key":"ql5sGmr12m"},{"type":"text","value":" : the value of ","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"key":"ElgUvjgeBj"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"rerYW1Q0bv"},{"type":"text","value":" approaches its highest value which means that ","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"key":"MaZ0Qdtb5Y"},{"type":"inlineMath","value":"|c|\\sim |d|","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"html":"∣c∣∼∣d∣|c|\\sim |d|∣c∣∼∣d∣","key":"MvGSLWpjBP"},{"type":"text","value":". The population cannot be arbitrary large here and it is not possible for a state to exhibits this correlation and to have a population higher than 1.4.","position":{"start":{"line":420,"column":1},"end":{"line":420,"column":1}},"key":"ABXKiW60aE"}],"key":"exaYliqXSY"}],"key":"gv16rl8EeD"},{"type":"comment","value":"A different way to see that is at fixed population and second order correlation function, the values that can take $c$ and $d$ is relatively narrow.","position":{"start":{"line":421,"column":1},"end":{"line":421,"column":1}},"key":"Sij9kfs0IL"},{"type":"paragraph","position":{"start":{"line":423,"column":1},"end":{"line":423,"column":1}},"children":[{"type":"text","value":"We therefore saw that the measurement of the fourth order correlation function allows to distinguish the problematic cases where the ","position":{"start":{"line":423,"column":1},"end":{"line":423,"column":1}},"key":"KnnADV3KzR"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":423,"column":1},"end":{"line":423,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"NeroripRaG"},{"type":"text","value":" witness cannot. However, this function lies in a rather small interval which might be difficult to resolve experimentally.","position":{"start":{"line":423,"column":1},"end":{"line":423,"column":1}},"key":"WxazdbuHGn"}],"key":"g3hSKUOC7w"},{"type":"heading","depth":2,"position":{"start":{"line":426,"column":1},"end":{"line":426,"column":1}},"children":[{"type":"text","value":"Conclusion","position":{"start":{"line":426,"column":1},"end":{"line":426,"column":1}},"key":"ZxgY3DtWYz"}],"identifier":"ccl_entanglement","label":"ccl_entanglement","html_id":"ccl-entanglement","enumerator":"7","key":"KOfIL8n5c4"},{"type":"comment","value":": measuring entanglement while measuring commutating observable ?","position":{"start":{"line":427,"column":1},"end":{"line":427,"column":1}},"key":"IY0a4YuOrD"},{"type":"paragraph","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"children":[{"type":"text","value":"In this section, we showed that it is possible to demonstrate the entanglement of thermal Gaussian states with 2- and 4-body correlation functions. In this sense, our reasoning is in line with ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"vcUiwDzeU8"},{"type":"cite","identifier":"schweigler_experimental_2017","label":"schweigler_experimental_2017","kind":"narrative","position":{"start":{"line":428,"column":194},"end":{"line":428,"column":223}},"children":[{"type":"text","value":"Schweigler ","key":"WEki17bADl"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"HithxFMz1x"}],"key":"Xnvmr7foEE"},{"type":"text","value":" (2017)","key":"nrjhAccwHj"}],"enumerator":"15","key":"u3t0Ovebrr"},{"type":"text","value":", who characterize their system via correlation functions. They showed a deviation of the fourth order correlation function from its connected part. To make a long story short, they emphasize a non-Gaussian state using ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"sHjylLNZ55"},{"type":"inlineMath","value":" g^{(4)} ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"html":"g(4) g^{(4)} g(4)","key":"hmTHsy1PAB"},{"type":"text","value":". Our work is somehow a complement as we use ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"H2H1VgVb6S"},{"type":"inlineMath","value":"g^{(4)} ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"html":"g(4)g^{(4)} g(4)","key":"ElnmoQ8zMU"},{"type":"text","value":" to assess the non-separability of the ","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"xe9W5X7UC2"},{"type":"emphasis","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"children":[{"type":"text","value":"Gaussian","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"QspqeN1dZZ"}],"key":"Sx4zxdIW9r"},{"type":"text","value":" state.","position":{"start":{"line":428,"column":1},"end":{"line":428,"column":1}},"key":"loDoOUvVIc"}],"key":"Tx51SQ0dAG"},{"type":"paragraph","position":{"start":{"line":430,"column":1},"end":{"line":430,"column":1}},"children":[{"type":"text","value":"Other theoretical works suggested different strategies to assess and quantify entanglement in analog gravity setups. ","position":{"start":{"line":430,"column":1},"end":{"line":430,"column":1}},"key":"YP9uSXYiyP"},{"type":"cite","identifier":"finazzi_entangled_2014","label":"finazzi_entangled_2014","kind":"narrative","position":{"start":{"line":430,"column":118},"end":{"line":430,"column":141}},"children":[{"type":"text","value":"Finazzi & Carusotto (2014)","key":"GZopcVo48f"}],"enumerator":"16","key":"lPH2OzMqUu"},{"type":"text","value":" proposed an experimental scheme to measure all the terms of the covariance matrix. The idea is to measure the state by coupling it to a cavity and measuring the photon field at the output. This work was motivated by the system set up by ","position":{"start":{"line":430,"column":1},"end":{"line":430,"column":1}},"key":"qvH6aCPMKp"},{"type":"cite","identifier":"brennecke_cavity_2008","label":"brennecke_cavity_2008","kind":"narrative","position":{"start":{"line":430,"column":379},"end":{"line":430,"column":401}},"children":[{"type":"text","value":"Brennecke ","key":"H9oyShS4vD"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"NNs5gJtb97"}],"key":"sMWUN7wSDM"},{"type":"text","value":" (2008)","key":"TW3PXXyHBA"}],"enumerator":"17","key":"bf8AzYP6r2"},{"type":"text","value":". This is also the stance taken by ","position":{"start":{"line":430,"column":1},"end":{"line":430,"column":1}},"key":"HB8DRVYJer"},{"type":"cite","identifier":"finke_observation_2016","label":"finke_observation_2016","kind":"narrative","position":{"start":{"line":430,"column":436},"end":{"line":430,"column":459}},"children":[{"type":"text","value":"Finke ","key":"nmjzJd7G5x"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"FlWtpn28vr"}],"key":"tHCAGywgI8"},{"type":"text","value":" (2016)","key":"p3RNPGl4wI"}],"enumerator":"18","key":"pxJvBoD2dk"},{"type":"text","value":", who propose to measure “additional observables, including noncommuting ones, e.g., density and phase fluctuations.”","position":{"start":{"line":430,"column":1},"end":{"line":430,"column":1}},"key":"K6rt2TPTWG"}],"key":"Gg8yXgMCsB"},{"type":"paragraph","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"children":[{"type":"text","value":"In the case of a stationary field","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"ZXCLHKCJVF"},{"type":"footnoteReference","identifier":"footnote_stationary","label":"footnote_stationary","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"number":8,"enumerator":"8","key":"poftpHxMoU"},{"type":"text","value":", the measurement of the density-density correlation function was shown by ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"DHjnALO267"},{"type":"cite","identifier":"robertson_controlling_2017","label":"robertson_controlling_2017","kind":"narrative","position":{"start":{"line":434,"column":131},"end":{"line":434,"column":158}},"children":[{"type":"text","value":"Robertson ","key":"zGNWP7erb5"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"kz5GhauLNW"}],"key":"ISVojo0E1N"},{"type":"text","value":" (2017)","key":"XzdgWeEC6E"}],"enumerator":"19","key":"QBMQn1nuPu"},{"type":"text","value":" to be sufficient to assess the non-separability of the state. This correlation function oscillates in time, and the non-separability criterion lies in its amplitude. In this case, the “noncommuting” paradox was solved elegantly: if the density perturbations ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"kbcgk8kpJI"},{"type":"inlineMath","value":"\\delta \\hat{n}_{-k}(t)","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"html":"δn^−k(t)\\delta \\hat{n}_{-k}(t)δn^−k​(t)","key":"c6WBpSDGLz"},{"type":"text","value":" and ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"URy5yY3y2C"},{"type":"inlineMath","value":"\\delta \\hat{n}_k(t)","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"html":"δn^k(t)\\delta \\hat{n}_k(t)δn^k​(t)","key":"VqHkrfYqZQ"},{"type":"text","value":" commute, it is not the case for ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"KZPQky6eUv"},{"type":"inlineMath","value":"\\delta \\hat{n}_{-k}(t)","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"html":"δn^−k(t)\\delta \\hat{n}_{-k}(t)δn^−k​(t)","key":"iinu4RJYth"},{"type":"text","value":" and ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"ldGPqAn0dH"},{"type":"inlineMath","value":"\\delta \\hat{n}_k(t')","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"html":"δn^k(t′)\\delta \\hat{n}_k(t')δn^k​(t′)","key":"rTRcjVqjmD"},{"type":"text","value":" ","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"lF0AuqiLhp"},{"type":"emphasis","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"u6XNlSEVQx"}],"key":"hzJ95pqcsC"},{"type":"text","value":" at different times. To measure the amplitude of the oscillation, one needs to measure the density-density correlation at different times, hence noncommuting observables.","position":{"start":{"line":434,"column":1},"end":{"line":434,"column":1}},"key":"m9svUZANiw"}],"key":"sKh55jAlCs"},{"type":"comment","value":"presents also a stationary field\nand then stopped or a inhomogeneous background that would create quasi-particles. he field is measured *in situ*. This is not the case for us as the field","position":{"start":{"line":437,"column":1},"end":{"line":438,"column":1}},"key":"nqKEdjAoZJ"},{"type":"paragraph","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"children":[{"type":"text","value":"Here, in line with ","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"dSTYxl2Q7I"},{"type":"cite","identifier":"robertson_assessing_2017","label":"robertson_assessing_2017","kind":"narrative","position":{"start":{"line":440,"column":20},"end":{"line":440,"column":45}},"children":[{"type":"text","value":"Robertson ","key":"cLyV7gfZTN"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Q9jbOxDOFu"}],"key":"Baat1eyLdX"},{"type":"text","value":" (2017)","key":"JMQ6DT8CSn"}],"enumerator":"20","key":"VgNFoiEBs0"},{"type":"text","value":", our claim is that we do not need to measure noncommuting observables to establish the non-separability of thermal Gaussian states. However, it must be clearly stated that there are drawbacks to our reasoning: we ","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"A9oOpdDJdH"},{"type":"emphasis","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"children":[{"type":"text","value":"assume","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"pCTFBfJKHZ"}],"key":"uMcUVUz1i6"},{"type":"text","value":" the state to be Gaussian. It is this fundamental assumption that allows us to derive such a simple entanglement witness. Without measuring the Wigner function of the state, this assumption is hard (impossible) to prove experimentally. One would need to show that ","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"lqrN6BPWXr"},{"type":"emphasis","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"children":[{"type":"text","value":"all","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"dvTtJy2YGm"}],"key":"TIw7T4YRDy"},{"type":"text","value":" correlation functions have vanishing non-connected contributions. Such a hypothesis can, however, be checked for consistency within the error bars and the range accessible by the experiment.","position":{"start":{"line":440,"column":1},"end":{"line":440,"column":1}},"key":"ssuMvhXOIA"}],"key":"O1Q21FMymo"},{"type":"comment","value":"@dall_ideal_2013;@perrier_thermal_2019;@herce_full_2023]","position":{"start":{"line":441,"column":1},"end":{"line":441,"column":1}},"key":"hZm1nxewwk"},{"type":"paragraph","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"children":[{"type":"text","value":"Still assuming a centered Gaussian function, we also considered un-squeezed thermal states, ","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"key":"A1SUhTfzFn"},{"type":"emphasis","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"key":"Vxe3wKlGEB"}],"key":"vUG0jGMyQ8"},{"type":"text","value":", states whose local second-order correlation functions ","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"key":"YWMEwK2P3h"},{"type":"crossReference","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"children":[{"type":"text","value":"(","key":"a6A3kXkXl0"},{"type":"text","value":"5","key":"thYyGLLbXh"},{"type":"text","value":")","key":"S3lOtwyaqj"}],"identifier":"fct_g2_local","label":"fct_g2_local","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"fct-g2-local","key":"WGHMezdYRF"},{"type":"text","value":" are equal to 2. If this is not observed experimentally (or assumed), the consequences are the following:","position":{"start":{"line":444,"column":1},"end":{"line":444,"column":1}},"key":"oMpJHiBlhG"}],"key":"NkYWSvVeNi"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":445,"column":1},"end":{"line":448,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"children":[{"type":"text","value":"It invalidates the ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"zFuk7hJMyQ"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"StFh6i9awQ"},{"type":"text","value":" witness. Indeed, the expression of the quantity ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"NBvR0aCF7v"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"fm56GStLAt"},{"type":"text","value":" that we wrote in ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"pE9UCiFVNt"},{"type":"crossReference","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"children":[{"type":"text","value":"(","key":"FnshpIjwOS"},{"type":"text","value":"12","key":"nXGJrHfTPW"},{"type":"text","value":")","key":"pppBHwFFPx"}],"identifier":"pminus_demonstration","label":"pminus_demonstration","kind":"equation","template":"(%s)","enumerator":"12","resolved":true,"html_id":"pminus-demonstration","key":"jTbQCa84nm"},{"type":"text","value":" is no longer correct. It will now contain a non-trivial dependence on both the amplitude and the phase ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"BtmmXXaYtK"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i}^2","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"html":"⟨a^i⟩2\\braket{\\hat{a}_i}^2⟨a^i​⟩2","key":"qRp3V5zBlJ"},{"type":"text","value":". With a single-particle detector, this amplitude can be measured but not the phase. This means that one must write ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"FnTQA1NHTp"},{"type":"inlineMath","value":"\\mathcal{P}_-","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"html":"P−\\mathcal{P}_-P−​","key":"x7Vmm41lpf"},{"type":"text","value":" from scratch, maximize it with respect to the unknown phase, and bound it with respect to the coherence term ","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"Gg7SHlnfy4"},{"type":"inlineMath","value":"|d|","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"html":"∣d∣|d|∣d∣","key":"YdoxrS99Lp"},{"type":"text","value":".","position":{"start":{"line":445,"column":1},"end":{"line":445,"column":1}},"key":"mYuAAI9Ei4"}],"key":"Q1akrLDy6t"},{"type":"listItem","spread":true,"position":{"start":{"line":446,"column":1},"end":{"line":448,"column":1}},"children":[{"type":"text","value":"It invalidates the expression of the fourth-order correlation function ","position":{"start":{"line":446,"column":1},"end":{"line":446,"column":1}},"key":"E4w8ien7xz"},{"type":"crossReference","position":{"start":{"line":446,"column":1},"end":{"line":446,"column":1}},"children":[{"type":"text","value":"(","key":"JxVrT3gfeE"},{"type":"text","value":"17","key":"ShCmXA6zYZ"},{"type":"text","value":")","key":"CVhBQmIxrA"}],"identifier":"fourth_order_correlation_expression","label":"fourth_order_correlation_expression","kind":"equation","template":"(%s)","enumerator":"17","resolved":true,"html_id":"fourth-order-correlation-expression","key":"IGFSJqiW19"},{"type":"text","value":" that contains these terms. For example, one term is ","position":{"start":{"line":446,"column":1},"end":{"line":446,"column":1}},"key":"Y2fUvlj811"},{"type":"inlineMath","value":"\\braket{(\\hat{a}_1^\\dagger)^2}\\braket{(\\hat{a}_2^\\dagger)^2}\\braket{\\hat{a}_1\\hat{a}_2}^2","position":{"start":{"line":446,"column":1},"end":{"line":446,"column":1}},"html":"⟨(a^1†)2⟩⟨(a^2†)2⟩⟨a^1a^2⟩2\\braket{(\\hat{a}_1^\\dagger)^2}\\braket{(\\hat{a}_2^\\dagger)^2}\\braket{\\hat{a}_1\\hat{a}_2}^2⟨(a^1†​)2⟩⟨(a^2†​)2⟩⟨a^1​a^2​⟩2","key":"XCTML5xd0Y"},{"type":"text","value":", which contains the relative phase between the covariance matrix terms.","position":{"start":{"line":446,"column":1},"end":{"line":446,"column":1}},"key":"Xy1OUQ38Kc"}],"key":"xNsNkf3mBO"}],"key":"uRm1fPcUe2"},{"type":"admonition","kind":"note","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Physical meaning of ","position":{"start":{"line":449,"column":1},"end":{"line":449,"column":1}},"key":"jQPXneCQ0i"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i^2}\\neq 0","position":{"start":{"line":449,"column":1},"end":{"line":449,"column":1}},"html":"⟨a^i2⟩≠0\\braket{\\hat{a}_i^2}\\neq 0⟨a^i2​⟩=0","key":"QO5vjvpnob"}],"key":"Dcntj4ZaAy"},{"type":"paragraph","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"children":[{"type":"text","value":"A thermal state for which ","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"key":"kZ17V3fdfd"},{"type":"inlineMath","value":"\\braket{\\hat{a}_i^2}\\neq 0","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"html":"⟨a^i2⟩≠0\\braket{\\hat{a}_i^2}\\neq 0⟨a^i2​⟩=0","key":"biABkBz4z2"},{"type":"text","value":" is a ","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"key":"NcSRTv3kcO"},{"type":"emphasis","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"children":[{"type":"text","value":"squeezed thermal state","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"key":"XjNuBieG38"}],"key":"CsCGRzmElC"},{"type":"text","value":". In the context of analog gravity, it is for example proposed by ","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"key":"KRhHMj07GR"},{"type":"cite","identifier":"agullo_quantum_2022","label":"agullo_quantum_2022","kind":"narrative","position":{"start":{"line":451,"column":151},"end":{"line":451,"column":171}},"children":[{"type":"text","value":"Agullo ","key":"cTFAf0izpA"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"T4L28SrU9Z"}],"key":"uA6Ua5HbOU"},{"type":"text","value":" (2022)","key":"Bm2WhAJ4Vh"}],"enumerator":"21","key":"FsGVbo6AiD"},{"type":"text","value":" to send squeezed thermal light on an optical analog to a white-black hole pair. They show that the entanglement generated by the analog black-hole horizon was enhanced hence easier to detect.","position":{"start":{"line":451,"column":1},"end":{"line":451,"column":1}},"key":"rCCca62QeP"}],"key":"JgqaxyMqfc"},{"type":"comment","value":"Physical meaning of $ \\langle 1 \\rangle $","position":{"start":{"line":452,"column":1},"end":{"line":452,"column":1}},"key":"KyUlE9FcJb"}],"key":"fbRwJMGbWr"},{"type":"comment","value":"*Why this state could be non-zero*: a thermal state exhibiting such\nIn other words, observation of entanglement here requires many hypotheses. Turning to different setups allows richer measurements. With polariton superfluids, @delhom_entanglement_2024 propose to use homodyne detection to demonstrate and quantify entanglement of Hawking pairs.","position":{"start":{"line":454,"column":1},"end":{"line":455,"column":1}},"key":"Y7yfP1clni"},{"type":"comment","value":"We then explained that it was possible to check experimentally that this thermal state is not squeezed: one must check that the .","position":{"start":{"line":457,"column":1},"end":{"line":457,"column":1}},"key":"HDBO12ht15"},{"type":"container","kind":"figure","identifier":"distrib_coherent_vs_th","label":"distrib_coherent_vs_th","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/distribution_coheren-280595403b3fb243deed2d06ac3c488a.png","alt":"Distribution of coherent versus thermal for fixed mean","width":"100%","align":"center","key":"DuJnEfFyYa","urlSource":"images/distribution_coherent_versus_thermal.png","urlOptimized":"/~gondret/phd_manuscript/build/distribution_coheren-280595403b3fb243deed2d06ac3c488a.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":468,"column":1},"end":{"line":468,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"distrib_coherent_vs_th","identifier":"distrib_coherent_vs_th","html_id":"distrib-coherent-vs-th","enumerator":"5","children":[{"type":"text","value":"Figure ","key":"tDe3FfmAZi"},{"type":"text","value":"5","key":"kPTb1vjuPU"},{"type":"text","value":":","key":"vHg5tT5it2"}],"template":"Figure %s:","key":"MsnL6eX28O"},{"type":"text","value":"Left: Probability distribution of coherent (violet dots) versus thermal (green crosses) states with a fixed mean number of particles of 4. Each symbol and color represent a different combination of the displacement (coherent part) and width (thermal part) of the Gaussian Wigner function, ranging from a relative fraction of thermal state of zero (purple dots, fully coherent state) to 1 (green crosses, fully thermal state). Middle: same as on the left but with a logarithmic ","position":{"start":{"line":468,"column":1},"end":{"line":468,"column":1}},"key":"DCjrrLWjkZ"},{"type":"inlineMath","value":"y","position":{"start":{"line":468,"column":1},"end":{"line":468,"column":1}},"html":"yyy","key":"C8CUJKqerh"},{"type":"text","value":"-scale. Right: second-order correlation function as a function of the fraction of thermal particles. The symbols of the left subplots are also shown on this plot.","position":{"start":{"line":468,"column":1},"end":{"line":468,"column":1}},"key":"Zyt3F51ZnE"}],"key":"BtT20DY0Py"}],"key":"UvWmCaUndS"}],"enumerator":"5","html_id":"distrib-coherent-vs-th","key":"rDrIDgbwpj"},{"type":"paragraph","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"Another assumption of this work should be emphasized: we assumed working with ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"N9S6btVh5q"},{"type":"emphasis","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"thermal","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"UHyqu8rr3K"}],"key":"UB1d68SW9g"},{"type":"text","value":" Gaussian states, ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"AxHI9C0RIM"},{"type":"emphasis","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"cdojScqCyJ"}],"key":"o1jBLSLQbf"},{"type":"text","value":", states with zero mean. ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"LwGo2yZxnZ"},{"type":"cite","identifier":"finke_observation_2016","label":"finke_observation_2016","kind":"narrative","position":{"start":{"line":474,"column":137},"end":{"line":474,"column":160}},"children":[{"type":"text","value":"Finke ","key":"F3SZ6pxM55"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"HSaRuKoQUf"}],"key":"GUDol2Pf1n"},{"type":"text","value":" (2016)","key":"xCBV4qy99q"}],"enumerator":"18","key":"Fzey0VKPY6"},{"type":"text","value":" set up a classical (wave) model that mimics the correlation (variance and Cauchy-Schwarz) observed in experiments. The probability distribution of each mode can, in principle, discriminate between a coherent state and a thermal state, and therefore the probability distribution of a displaced thermal state. In ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"O4nOeNi7hI"},{"type":"crossReference","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"Figure ","key":"BvrzrMxUzX"},{"type":"text","value":"5","key":"Jr3VErwjhc"}],"identifier":"distrib_coherent_vs_th","label":"distrib_coherent_vs_th","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"distrib-coherent-vs-th","key":"eaVuySQkab"},{"type":"text","value":", we represent the probability distribution of four states with a mean number of particles of 4. We parametrize those states by the thermal fraction (see legend), setting the displacement to keep the number of particles fixed. The fully coherent state (0/3) is represented in purple dots and the fully thermal state in green crosses (3/3). In between, states are thermal displaced states. The middle subplot is the same as the left one with a logarithmic ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"u5x0QXDqMt"},{"type":"inlineMath","value":"y","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"html":"yyy","key":"x7BbsVZ2C6"},{"type":"text","value":"-scale. In these two plots, it is easy to distinguish a fully coherent state from a thermal state. However, the difference between a state that is mostly thermal but with still a bit of coherence (2/3 of thermal population, “+” symbols) is not very different from a state that is fully thermal (“x” symbols): the two curves overlap, and we cannot distinguish one from the other. On the right subplot of ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"J1yvBWsyTZ"},{"type":"crossReference","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"Figure ","key":"be68ZiNCYs"},{"type":"text","value":"5","key":"A1MnQ3MrfJ"}],"identifier":"distrib_coherent_vs_th","label":"distrib_coherent_vs_th","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"distrib-coherent-vs-th","key":"zYIPPKbQwD"},{"type":"text","value":", the second-order correlation function is shown as a function of the fraction of thermal particles. As expected, it ranges from 1, the value for a fully coherent state, to 2 for a fully thermal state. The imperceptible difference between the two “most thermal” states of the right subplot (2/3 and 1 in the legend) is more visible. The second-order correlation function is a better witness to ensure the state is thermal. This is not surprising: rare events (number of particles larger than 20) are not represented on ","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"u9ioxrNMxu"},{"type":"crossReference","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"children":[{"type":"text","value":"Figure ","key":"MnAe1I6Oeh"},{"type":"text","value":"5","key":"zfphHXyr4U"}],"identifier":"distrib_coherent_vs_th","label":"distrib_coherent_vs_th","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"distrib-coherent-vs-th","key":"jEGjw18L2X"},{"type":"text","value":" but have a strong weight on the calculation of the second-order correlation function.","position":{"start":{"line":474,"column":1},"end":{"line":474,"column":1}},"key":"js7WNWY3JA"}],"key":"YTxOmgY4OE"},{"type":"comment","value":" :::{note} Note\nThis is not so surprising: rare events (number of particles larger than 20) are not represented on [](#distrib_coherent_vs_th) but have a strong weight on the second order correlation function. \n::: ","key":"w7vwSidLn6"},{"type":"comment","value":"looking at the probability distribution are When looking at the (local) second order correlation function, the difference are larger:","position":{"start":{"line":480,"column":1},"end":{"line":480,"column":1}},"key":"todqkxeRu3"},{"type":"comment","value":"Here, we assumed from scratch the state to be centered *i.e.* we assumed that $\\braket{\\hat{a}_i} = \\braket{\\hat{a}_i^\\dagger} = 0$.","position":{"start":{"line":485,"column":1},"end":{"line":485,"column":1}},"key":"CyhnYeltYp"},{"type":"comment","value":"also on the fact The proposal to measure entanglement by\nTo overcome this issue, @finazzi_entangled_2014 proposed an experimental scheme to measure all the terms of the covariance matrix. The idea is to measure the state by coupling it to a cavity, as in the system developed by @brennecke_cavity_2008. In @finke_observation_2016, the authors derived a classical model in which they obtained sub-Poissonian variance, highlighting the fact that the observation of sub-shot noise variance is not a proof of entanglement. They also showed that it was possible for a classical system to violate the Cauchy-Schwarz inequality when the normal ordering is applied *by hand*, that is, when the local correlation function is defined as","position":{"start":{"line":487,"column":1},"end":{"line":488,"column":1}},"key":"GS8O4gjOvt"},{"type":"comment","value":"Concerning the $g^{(2)}/g^{(4)}$ criterion, I have no feeling for now about the effect of losses. I think one should take into account the losses by multiplying all values ($n_1, n_2, c, d$) by $1/\\eta$ but the consequences on the Heisenberg function $\\mathcal{H}$ is not entirely clear. For example, @martin_comparing_2023 provides a rescaled squeezing parameter and purity for a general two-mode thermal state. However, states that do not violate this witness are subject to \"de-entangled\" after a pure loss channel.","position":{"start":{"line":490,"column":1},"end":{"line":490,"column":1}},"key":"Hw8wjyg9MB"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Summary","position":{"start":{"line":494,"column":1},"end":{"line":494,"column":1}},"key":"OrwFWmVNfv"}],"key":"edg8sxOyZ2"},{"type":"paragraph","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"children":[{"type":"text","value":"In this section, we showed that it is possible to probe the non-separability of a zero-mean thermal Gaussian state. Measuring ","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"G5Id6mIIOq"},{"type":"crossReference","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"children":[{"type":"text","value":"both","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"i8lE8r2C33"}],"identifier":"fourth_order_corr_func","label":"fourth_order_corr_func","kind":"heading","template":"Section %s","enumerator":"5","resolved":true,"html_id":"fourth-order-corr-func","key":"CdfG8yYh8n"},{"type":"text","value":" the fourth and second order correlation function allows to completely quantify entanglement. The measurement of the second order correlation function can be sufficient to only ","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"ss86ZYaAsu"},{"type":"crossReference","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"children":[{"type":"text","value":"witness","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"Tn1lAv03dN"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","key":"KYpE6BxN0n"},{"type":"text","value":" non-separability. As a result, the classical Cauchy-Schwarz threshold is shifted depending on the population of the state if the nullity of the coherence term between the mode is not assumed. We finally ","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"bDfhp6EvBZ"},{"type":"crossReference","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"children":[{"type":"text","value":"discussed","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"CkKVcxosc3"}],"identifier":"ccl_entanglement","label":"ccl_entanglement","kind":"heading","template":"Section %s","enumerator":"7","resolved":true,"html_id":"ccl-entanglement","key":"ZYG7wT5daA"},{"type":"text","value":" the drawbacks of this result that only applies for purely thermal Gaussian states.","position":{"start":{"line":495,"column":1},"end":{"line":495,"column":1}},"key":"TRWDDrcWOK"}],"key":"w8S9krV4V1"},{"type":"comment","value":" that the local correlation functions reach 2, an *observation of a sub-shot-noise variance or violation of the Cauchy-Schwarz inequality or $g^{(2)}$ was a signature of mode entanglement*, provided that the population is not too low ($n_1n_2>1/2$). I insist here that the fact that normalized variance implies entanglement is not general but is accompanied by the measure of the local correlations. If not, our discussion fails. If the population are smaller, the threshold for the second order correlation function is slightly changed according to equation [](#Fg2) and goes to 3 in the limit of vanishing populations. We also derived a way to assess entanglement and not only witnessing it using the second and fourth order correlation function. ","key":"I3tOKwytBo"}],"key":"YHCevsKHj8"},{"type":"comment","value":" \nOne could argue that, by taking into account finite efficiency, one *helps* to witness entanglement. I would therefore answer back that, because $g^{(2)}$ is bounded by $2+1/n$, if we measure a quite high value for the correlation function while the population \n ","key":"vaGlvFDGVS"},{"type":"comment","value":" \n\n\n* In the criterion we derived, the absolute number of atoms is important. However, a state that is entangled should not be disantangled by passing through a beam splitter (see the appendix of @martin_comparing_2023 for the derivation). We know that the second order correlation function is not changed after a beam-splitter but it is the case of the fourth order correlation function ? And does the final criterion depends on the efficiency of the detector ?\n* One could also add noise on one channel and see the robustness of the criterion. If the finite efficiency test is passed, I would intuitively say that the criterion should be robust enough.\n* There might be something I missed: in @busch_quantum_2014, they insists on the fact that it is important that the state is stationary for there criterion to be sufficient. As far as I understood, this is to assume that $d=0$, which I do not assume therefore I guess I do not need that assumption. Or is this assumption needed to assume that $\\braket{\\hat{a}_i^2}=0$ ?\n* The Logarithmic Negativity [@vidal_computable_2002] can be also used to assess entanglement. It was shown also to be a really good entanglement quantifier because it increases with squeezing, see @plenio_logarithmic_2005. \n* An entanglement witness might not be the *best* (see @guhne_entanglement_2009) and one could optimize it. @lewenstein_optimization_2000 explains how to proceed to the \"Optimization of entanglement witnesses\". Can this be applied to $g^{(2)}$ ? I have doubts and I am not really used to the formalism they use. However, it might worth it to have a look (even to prove that, if $g^{(2)}$ is our only knowledge, the witness bound derived here is the best). \n\n## Some random references\n\n\n\nHow the criterion obehaves when we consider a lossy beam splitter ? Sheel et al sutdy the [Entanglement degradation of a two-mode squeezed vacuum in absorbing and amplifying optical fibers](https://arxiv.org/abs/quant-ph/0006026) whose ref was found in @10.1103/PhysRevA.64.063811.\n@scheel_entanglement_2001\n@10.1103/PhysRevA.84.052330 \nNote also that @10.1103/PhysRevA.84.052330 shows that the gPPT criterion does not depend on the efficiency $\\eta_A$ and $\\eta_B$.\n ","key":"PUms6NGKvn"},{"type":"footnoteDefinition","identifier":"foot_glauber","label":"foot_glauber","position":{"start":{"line":505,"column":1},"end":{"line":505,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"children":[{"type":"text","value":"We describe the measure with a micro-channel plate reproducing Glauber’s work in the fourth ","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"key":"k0WJPXMDis"},{"type":"crossReference","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"children":[{"type":"text","value":"chapter","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"key":"P3PiBsnGo5"}],"identifier":"description_glauber","label":"description_glauber","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"description-glauber","remote":true,"url":"/mcp-physics","dataUrl":"/mcp-physics.json","key":"PlAzYpuj7t"},{"type":"text","value":".","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"key":"DWj3cfLphR"}],"key":"hLof5qsPwK"}],"number":1,"enumerator":"1","key":"hpkgmmsnn7"},{"type":"footnoteDefinition","identifier":"footnote_measure_not_assume","label":"footnote_measure_not_assume","position":{"start":{"line":84,"column":1},"end":{"line":84,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":124,"column":1},"end":{"line":124,"column":1}},"children":[{"type":"text","value":"Here again I want to stress that this is not only an assumption: it can be checked experimentally.","position":{"start":{"line":124,"column":1},"end":{"line":124,"column":1}},"key":"SHZ9h7UfEY"}],"key":"N6ed9v9f4N"}],"number":2,"enumerator":"2","key":"D9LiR9lU14"},{"type":"footnoteDefinition","identifier":"note_phase_thermal_states","label":"note_phase_thermal_states","position":{"start":{"line":124,"column":1},"end":{"line":124,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"text","value":"Note that this is no longer true if the covariance matrix of subsystem ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"u16WxGwdBU"},{"type":"inlineMath","value":"A","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"AAA","key":"IbkQj824SC"},{"type":"text","value":" or subsystem ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"nlURdju31v"},{"type":"inlineMath","value":"B","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"BBB","key":"J3e5ZM2jEh"},{"type":"text","value":" is not diagonal, ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"YMbkmZ8Avm"},{"type":"emphasis","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"lthB0G47oG"}],"key":"rb3Vo73u6M"},{"type":"text","value":" if ","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"Xo0MglQllx"},{"type":"inlineMath","value":" \\braket{\\hat{a}_i^2}\\neq 0","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"html":"⟨a^i2⟩≠0 \\braket{\\hat{a}_i^2}\\neq 0⟨a^i2​⟩=0","key":"jnJ9zqA3WJ"},{"type":"text","value":".","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"key":"FJGNQC4oNc"}],"key":"ciOFlXN3QY"}],"number":3,"enumerator":"3","key":"SrQLmUtxke"},{"type":"footnoteDefinition","identifier":"note_discord","label":"note_discord","position":{"start":{"line":298,"column":1},"end":{"line":298,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"children":[{"type":"text","value":"Note that the symplectic spectrum also allows to compute another entanglement quantifier, the Quantum Discord, see ","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"hxehX3HTaM"},{"type":"cite","identifier":"adesso_2010_quantum","label":"adesso_2010_quantum","kind":"narrative","position":{"start":{"line":319,"column":116},"end":{"line":319,"column":136}},"children":[{"type":"text","value":"Adesso & Datta (2010)","key":"r4dZdFvhem"}],"enumerator":"22","key":"Ozfb0Cg0Bd"},{"type":"text","value":" for example.","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"key":"okpRzpH7lZ"}],"key":"FIFovM4z6R"}],"number":4,"enumerator":"4","key":"zCKuuJm3oM"},{"type":"footnoteDefinition","identifier":"footnote_g2egal2","label":"footnote_g2egal2","position":{"start":{"line":319,"column":1},"end":{"line":319,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"children":[{"type":"text","value":"This is because we used the fact that ","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"key":"Huu969xFMb"},{"type":"inlineMath","value":"g^{(2)}_{11}=g^{(2)}_{22}=2","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"html":"g11(2)=g22(2)=2g^{(2)}_{11}=g^{(2)}_{22}=2g11(2)​=g22(2)​=2","key":"uXkjsK596B"},{"type":"text","value":" to derive the equation on ","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"key":"snkChCtLVg"},{"type":"inlineMath","value":"g^{(4)}_{12}","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"html":"g12(4)g^{(4)}_{12}g12(4)​","key":"O7Ov0hamUo"},{"type":"text","value":".","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"key":"QBo0B8RX3S"}],"key":"OetBO0VkQP"}],"number":5,"enumerator":"5","key":"soks2KF9kX"},{"type":"footnoteDefinition","identifier":"note_fiber","label":"note_fiber","position":{"start":{"line":334,"column":1},"end":{"line":334,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"children":[{"type":"text","value":"This is not the case for example of people that want to distribute quantum states across long distances. In their case, they want to quantify entanglement after fiber transmission.","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"key":"EcGkfePpEG"}],"key":"pUu7vZIGA5"}],"number":6,"enumerator":"6","key":"yVpShAmWbv"},{"type":"footnoteDefinition","identifier":"footnote_experi_rmk","label":"footnote_experi_rmk","position":{"start":{"line":389,"column":1},"end":{"line":389,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":402,"column":1},"end":{"line":402,"column":1}},"children":[{"type":"text","value":"As an experimentalist, I must say that this is interval (frightfully) small.","position":{"start":{"line":402,"column":1},"end":{"line":402,"column":1}},"key":"TemanHilLc"}],"key":"xxyYpL5flk"}],"number":7,"enumerator":"7","key":"AuJUy7ywb2"},{"type":"footnoteDefinition","identifier":"footnote_stationary","label":"footnote_stationary","position":{"start":{"line":402,"column":1},"end":{"line":402,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":436,"column":1},"end":{"line":436,"column":1}},"children":[{"type":"text","value":"One can think of a squeezing Hamiltonian that could be turned on and off for a given amount of time. It is the case for example for Hawking radiation in an inhomogeneous fluid.","position":{"start":{"line":436,"column":1},"end":{"line":436,"column":1}},"key":"mQf5R6Jfbw"}],"key":"HCsYMjAQeW"}],"number":8,"enumerator":"8","key":"nAGHuXxFXZ"}],"key":"zHk5HznAqr"}],"key":"u8oAtXyTL9"},"references":{"cite":{"order":["wick_evaluation_1950","isserlis_formula_1918","castin_mecanique_2011","glauber_quantum_1963","dall_ideal_2013","perrier_thermal_2019","herce_full_2023","robertson_notes_2021","simon_peres_horodecki_2000","clement_fourth_2022","isoard_2021_bipartite","weedbrook_gaussian_2012","scheel_entanglement_2001b","barbosa_disentanglement_2011","schweigler_experimental_2017","finazzi_entangled_2014","brennecke_cavity_2008","finke_observation_2016","robertson_controlling_2017","robertson_assessing_2017","agullo_quantum_2022","adesso_2010_quantum"],"data":{"wick_evaluation_1950":{"label":"wick_evaluation_1950","enumerator":"1","doi":"10.1103/PhysRev.80.268","html":"Wick, G. C. (1950). The Evaluation of the Collision Matrix. Physical Review, 80(2), 268–272. 10.1103/PhysRev.80.268","url":"https://doi.org/10.1103/PhysRev.80.268"},"isserlis_formula_1918":{"label":"isserlis_formula_1918","enumerator":"2","html":"Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12(1/2), 134–139."},"castin_mecanique_2011":{"label":"castin_mecanique_2011","enumerator":"3","html":"Castin, Y. (2011). Mécanique quantique : seconde quantification et résolvante. Cours de M2 Concepts fondamentaux de la physique. https://www.phys.ens.fr/~castin/cours.pdf","url":"https://www.phys.ens.fr/~castin/cours.pdf"},"glauber_quantum_1963":{"label":"glauber_quantum_1963","enumerator":"4","doi":"10.1103/PhysRev.130.2529","html":"Glauber, R. J. (1963). The Quantum Theory of Optical Coherence. Physical Review, 130(6), 2529–2539. 10.1103/PhysRev.130.2529","url":"https://doi.org/10.1103/PhysRev.130.2529"},"dall_ideal_2013":{"label":"dall_ideal_2013","enumerator":"5","doi":"10.1038/nphys2632","html":"Dall, R. G., Manning, A. G., Hodgman, S. S., RuGway, W., Kheruntsyan, K. V., & Truscott, A. G. (2013). Ideal n-body correlations with massive particles. Nature Physics, 9(6), 341–344. 10.1038/nphys2632","url":"https://doi.org/10.1038/nphys2632"},"perrier_thermal_2019":{"label":"perrier_thermal_2019","enumerator":"6","doi":"10.21468/SciPostPhys.7.1.002","html":"Perrier, M., Amodjee, Z., Dussarrat, P., Dareau, A., Aspect, A., Cheneau, M., Boiron, D., & Westbrook, C. (2019). Thermal counting statistics in an atomic two-mode squeezed vacuum state. SciPost Physics, 7(1), 002. 10.21468/SciPostPhys.7.1.002","url":"https://doi.org/10.21468/SciPostPhys.7.1.002"},"herce_full_2023":{"label":"herce_full_2023","enumerator":"7","doi":"10.1103/PhysRevResearch.5.L012037","html":"Hercé, G., Bureik, J.-P., Ténart, A., Aspect, A., Dareau, A., & Clément, D. (2023). Full counting statistics of interacting lattice gases after an expansion: The role of condensate depletion in many-body coherence. Physical Review Research, 5(1), L012037. 10.1103/PhysRevResearch.5.L012037","url":"https://doi.org/10.1103/PhysRevResearch.5.L012037"},"robertson_notes_2021":{"label":"robertson_notes_2021","enumerator":"8","html":"Robertson, S. (2021). Notes on COSQUA: Separability, g2, all that stuff. Private communication."},"simon_peres_horodecki_2000":{"label":"simon_peres_horodecki_2000","enumerator":"9","doi":"10.1103/PhysRevLett.84.2726","html":"Simon, R. (2000). Peres-Horodecki Separability Criterion for Continuous Variable Systems. Physical Review Letters, 84(12), 2726–2729. 10.1103/PhysRevLett.84.2726","url":"https://doi.org/10.1103/PhysRevLett.84.2726"},"clement_fourth_2022":{"label":"clement_fourth_2022","enumerator":"10","html":"Clément, D. (2022). Experimental schemes to certify entanglement associated with pair creation. Private communication."},"isoard_2021_bipartite":{"label":"isoard_2021_bipartite","enumerator":"11","doi":"10.1103/PhysRevA.104.063302","html":"Isoard, M., Milazzo, N., Pavloff, N., & Giraud, O. (2021). Bipartite and tripartite entanglement in a Bose-Einstein acoustic black hole. Phys. Rev. A, 104(6), 063302. 10.1103/PhysRevA.104.063302","url":"https://doi.org/10.1103/PhysRevA.104.063302"},"weedbrook_gaussian_2012":{"label":"weedbrook_gaussian_2012","enumerator":"12","doi":"10.1103/RevModPhys.84.621","html":"Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N. J., Ralph, T. C., Shapiro, J. H., & Lloyd, S. (2012). Gaussian quantum information. Reviews of Modern Physics, 84(2), 621–669. 10.1103/RevModPhys.84.621","url":"https://doi.org/10.1103/RevModPhys.84.621"},"scheel_entanglement_2001b":{"label":"scheel_entanglement_2001b","enumerator":"13","doi":"10.1134/1.1405221","html":"Scheel, S., Opatrny, T., & Welsch, D. G. (2001). Entanglement degradation of a two-mode squeezed vacuum in absorbing and amplifying optical fibers. Optics and Spectroscopy, 91(3), 411–417. 10.1134/1.1405221","url":"https://doi.org/10.1134/1.1405221"},"barbosa_disentanglement_2011":{"label":"barbosa_disentanglement_2011","enumerator":"14","doi":"10.1103/PhysRevA.84.052330","html":"Barbosa, F. A. S., De Faria, A. J., Coelho, A. S., Cassemiro, K. N., Villar, A. S., Nussenzveig, P., & Martinelli, M. (2011). Disentanglement in bipartite continuous-variable systems. Physical Review A, 84(5), 052330. 10.1103/PhysRevA.84.052330","url":"https://doi.org/10.1103/PhysRevA.84.052330"},"schweigler_experimental_2017":{"label":"schweigler_experimental_2017","enumerator":"15","doi":"10.1038/nature22310","html":"Schweigler, T., Kasper, V., Erne, S., Mazets, I., Rauer, B., Cataldini, F., Langen, T., Gasenzer, T., Berges, J., & Schmiedmayer, J. (2017). Experimental characterization of a quantum many-body system via higher-order correlations. Nature, 545(7654), 323–326. 10.1038/nature22310","url":"https://doi.org/10.1038/nature22310"},"finazzi_entangled_2014":{"label":"finazzi_entangled_2014","enumerator":"16","doi":"10.1103/PhysRevA.90.033607","html":"Finazzi, S., & Carusotto, I. (2014). Entangled phonons in atomic Bose-Einstein condensates. Physical Review A, 90(3), 033607. 10.1103/PhysRevA.90.033607","url":"https://doi.org/10.1103/PhysRevA.90.033607"},"brennecke_cavity_2008":{"label":"brennecke_cavity_2008","enumerator":"17","doi":"10.1126/science.1163218","html":"Brennecke, F., Ritter, S., Donner, T., & Tilman Esslinger. (2008). Cavity Optomechanics with a Bose-Einstein Condensate. Science, 322(5899), 235–238. 10.1126/science.1163218","url":"https://doi.org/10.1126/science.1163218"},"finke_observation_2016":{"label":"finke_observation_2016","enumerator":"18","doi":"10.1088/1367-2630/18/11/113017","html":"Finke, A., Jain, P., & Weinfurtner, S. (2016). On the observation of nonclassical excitations in Bose–Einstein condensates. New Journal of Physics, 18(11), 113017. 10.1088/1367-2630/18/11/113017","url":"https://doi.org/10.1088/1367-2630/18/11/113017"},"robertson_controlling_2017":{"label":"robertson_controlling_2017","enumerator":"19","doi":"10.1103/PhysRevD.95.065020","html":"Robertson, S., Michel, F., & Parentani, R. (2017). Controlling and observing nonseparability of phonons created in time-dependent 1D atomic Bose condensates. Physical Review D, 95(6), 065020. 10.1103/PhysRevD.95.065020","url":"https://doi.org/10.1103/PhysRevD.95.065020"},"robertson_assessing_2017":{"label":"robertson_assessing_2017","enumerator":"20","doi":"10.1103/PhysRevD.96.045012","html":"Robertson, S., Michel, F., & Parentani, R. (2017). Assessing degrees of entanglement of phonon states in atomic Bose gases through the measurement of commuting observables. Physical Review D, 96(4), 045012. 10.1103/PhysRevD.96.045012","url":"https://doi.org/10.1103/PhysRevD.96.045012"},"agullo_quantum_2022":{"label":"agullo_quantum_2022","enumerator":"21","doi":"10.1103/PhysRevLett.128.091301","html":"Agullo, I., Brady, A. J., & Kranas, D. (2022). Quantum Aspects of Stimulated Hawking Radiation in an Optical Analog White-Black Hole Pair. Physical Review Letters, 128(9), 091301. 10.1103/PhysRevLett.128.091301","url":"https://doi.org/10.1103/PhysRevLett.128.091301"},"adesso_2010_quantum":{"label":"adesso_2010_quantum","enumerator":"22","doi":"10.1103/PhysRevLett.105.030501","html":"Adesso, G., & Datta, A. (2010). Quantum versus Classical Correlations in Gaussian States. Phys. Rev. Lett., 105(3), 030501. 10.1103/PhysRevLett.105.030501","url":"https://doi.org/10.1103/PhysRevLett.105.030501"}}}},"footer":{"navigation":{"prev":{"title":"On the Cauchy-Schwarz inequality and the normalized variance","short_title":"Particle entanglement","url":"/entanglement-3particle","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Bose-Einstein condensation of metastable Helium","short_title":"Bose-Einstein condensation of metastable Helium","url":"/bec-intro","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}