{"kind":"Article","sha256":"64e332b489b5abf1eb4e628ab0ae70ab6587d3dce1c1ac6fe40434512794becf","slug":"entanglement-0intro","location":"/entanglement/4entanglement_0intro.md","dependencies":[],"frontmatter":{"title":"Quantifying entanglement of two-mode Gaussian states","short_title":"Probing entanglement","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"}},"exports":[{"format":"md","filename":"4entanglement_0intro.md","url":"/~gondret/phd_manuscript/build/4entanglement_0intro-63df5bb5d507babf884dec69169bf32f.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":6,"column":1},"end":{"line":6,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"children":[{"type":"text","value":"Central to this thesis is the detection of entanglement. In the last chapter, we introduced ","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"sW7skV628l"},{"type":"inlineMath","value":"\\Delta_k=\\sqrt{n_kn_{-k}} - |\\braket{\\hat{a}_k\\hat{a}_{-k}}|","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"html":"Δk=nkn−k−∣⟨a^ka^−k⟩∣\\Delta_k=\\sqrt{n_kn_{-k}} - |\\braket{\\hat{a}_k\\hat{a}_{-k}}|Δk​=nk​n−k​​−∣⟨a^k​a^−k​⟩∣","key":"EtuOmGB0oR"},{"type":"text","value":" and explained that negativity of this quantity assesses entanglement. In the experiment, we do not have access to the anomalous correlation term ","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"gVTwE2g4zQ"},{"type":"inlineMath","value":"\\braket{\\hat{a}_k\\hat{a}_{-k}}","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"html":"⟨a^ka^−k⟩\\braket{\\hat{a}_k\\hat{a}_{-k}}⟨a^k​a^−k​⟩","key":"HW9nC55tDH"},{"type":"text","value":": this quantity does not conserve the number of particles. On the other hand, we measure any ","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"soR1G5AF3a"},{"type":"inlineMath","value":"N","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"html":"NNN","key":"tbZgdcv22Z"},{"type":"text","value":"-body correlation function. When the state is Gaussian, the measurement of these correlation functions can be expanded as a sum of two-field correlation functions. The theory we used to describe our system involves time-dependant second order in creation and annihilation operator Hamiltonian that preserves Gaussianity of the state. Here we therefore discuss how to probe entanglement of ","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"XwRtQhF65e"},{"type":"emphasis","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"children":[{"type":"text","value":"Gaussian","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"PI838PT67w"}],"key":"AN50heAAPv"},{"type":"text","value":" states with a single particle detector (or at least a detector that can resolve many-body correlation functions). In particular, if the Gaussian two-mode state is centered, the normalized two-body correlation function is given by","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"key":"SJ9xZ3ZvwA"}],"key":"Ns0cpkSF8N"},{"type":"math","value":"g_{k,-k}^{(2)} = \\frac{\\braket{\\hat{a}_k^\\dagger\\hat{a}_{-k}^\\dagger\\hat{a}_k\\hat{a}_{-k}}}{n_kn_{-k}} = 1 + \\frac{|\\braket{\\hat{a}_k\\hat{a}_{-k}}|^2}{n_kn_{-k}}+\\frac{|\\braket{\\hat{a}_k^\\dagger\\hat{a}_{-k}}|^2}{n_kn_{-k}}","tight":true,"html":"gk,−k(2)=⟨a^k†a^−k†a^ka^−k⟩nkn−k=1+∣⟨a^ka^−k⟩∣2nkn−k+∣⟨a^k†a^−k⟩∣2nkn−kg_{k,-k}^{(2)} = \\frac{\\braket{\\hat{a}_k^\\dagger\\hat{a}_{-k}^\\dagger\\hat{a}_k\\hat{a}_{-k}}}{n_kn_{-k}} = 1 + \\frac{|\\braket{\\hat{a}_k\\hat{a}_{-k}}|^2}{n_kn_{-k}}+\\frac{|\\braket{\\hat{a}_k^\\dagger\\hat{a}_{-k}}|^2}{n_kn_{-k}}gk,−k(2)​=nk​n−k​⟨a^k†​a^−k†​a^k​a^−k​⟩​=1+nk​n−k​∣⟨a^k​a^−k​⟩∣2​+nk​n−k​∣⟨a^k†​a^−k​⟩∣2​","enumerator":"1","key":"m0PRGKL4Kx"},{"type":"paragraph","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"which was expanded using Wick theorem. If we assume that the last term in this equation is zero, it means that the second order correlation function is in one-to-one correspondence with the measurement of the pure correlation term ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"aOwcUCbuOd"},{"type":"inlineMath","value":"|\\braket{\\hat{a}_k\\hat{a}_{-k}}|","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"html":"∣⟨a^ka^−k⟩∣|\\braket{\\hat{a}_k\\hat{a}_{-k}}|∣⟨a^k​a^−k​⟩∣","key":"VvN0t6FzLf"},{"type":"text","value":". This chapter aims to discuss such assumption. In fact, we will show that this assumption might not be needed to detect entanglement. We also aim to discuss other correlation witnesses that have been used to claim entanglement: the violation of the classical Cauchy-Schwarz inequality and the observation of relative number squeezing. For our discussion, we will use Gaussian state formalism.","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"JDgw1NLskt"}],"key":"nUUoJbizED"},{"type":"comment","value":"Much experimental progress were made during the late 90's, demonstrating further capability in manipulating individual quantum systems. This is referred as *the second quantum revolution* [@bell2004speakable]. Among them, one can cite the first experimental demonstration of quantum teleportation by @furusawa_unconditional_1998 or the integers factorization with Shor's algorithm of 15 by @vandersypen_experimental_2001. Entanglement plays a crucial role in quantum technologies but the violation of Bell inequality by a state is not a necessary condition for it to be useful. This problem was addressed by @popescu_bells_1994 in \"*Bell’s inequalities versus teleportation: What is nonlocality?*\". In his work, he indeed shows that \"*there are mixed states which do not violate any Bell type inequality, but still can be used for teleportation*\". The question of how much a system cannot behave like a classical one, or wether this system be useful in quantum technologies is closely related to the notion of entanglement [@schrodinger_discussion_1935], even if this notion is not so easy to define.","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"EQmZWryVtM"},{"type":"paragraph","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"The ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"SZEiwPjbXh"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"first","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"tvediqxOMz"}],"identifier":"subsection_gaussian_state","label":"subsection_gaussian_state","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"subsection-gaussian-state","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"Zoa5ag8ioK"},{"type":"text","value":" section of this chapter will be devoted to the introduction to the Gaussian state formalism, which is a convenient toolbox to describe our state ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"Pabl7A1sUb"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"cite","identifier":"cerf_quantum_2007","label":"cerf_quantum_2007","kind":"parenthetical","position":{"start":{"line":16,"column":187},"end":{"line":16,"column":205}},"children":[{"type":"text","value":"Cerf ","key":"pq3wkcW8yT"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Pf4j8VyJ2v"}],"key":"c8yC7LZSSG"},{"type":"text","value":", 2007","key":"k2Ye51fQO8"}],"enumerator":"1","key":"f2JHahhQaZ"}],"key":"EG1ar5HA6A"},{"type":"text","value":". We will then devote a section to ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"GOYeANMazd"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"entanglement","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"UzCEMXXS0c"}],"identifier":"separability_def_section","label":"separability_def_section","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"separability-def-section","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"kE9QWJeNMj"},{"type":"text","value":", starting with a discussion on the difference between ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"gJMgDuZxEX"},{"type":"emphasis","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"mode entanglement","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"e7c4Q7LebY"}],"key":"w8zALkOhfp"},{"type":"text","value":" and ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"fOjhzsI7Of"},{"type":"emphasis","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"particle entanglement","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"ewwXJxYswp"}],"key":"PrAZasfmRx"},{"type":"text","value":". We will define the ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"I9qFj4kKyi"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"PPT","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"pdpMbtdWQX"}],"identifier":"ppt_criterion_section","label":"ppt_criterion_section","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"ppt-criterion-section","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"Hn9jToLUox"},{"type":"text","value":" criterion and its generalization by ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"DuO2Qx7j1u"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"Simon","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"GEobL7Q0dz"}],"identifier":"simon_subsection","label":"simon_subsection","kind":"heading","template":"{name}","resolved":true,"html_id":"simon-subsection","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"y9ct1fQyBp"},{"type":"text","value":", the ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"oI3Ilzr109"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"logarithmic negativity","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"oM2GSADgDU"}],"identifier":"log_neg_section","label":"log_neg_section","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"log-neg-section","remote":true,"url":"/entanglement-2criteria","dataUrl":"/entanglement-2criteria.json","key":"kdeIKxTMTr"},{"type":"text","value":" and ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"WZc894cJLx"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"other","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"sRKu4J7EmN"}],"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":"QH2XYfHK0m"},{"type":"text","value":" entanglement witnesses. The third section is devoted to relative number ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"COiDvydTY3"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"squeezing","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"G3NCnU5OSm"}],"identifier":"squeezing_section","label":"squeezing_section","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"squeezing-section","remote":true,"url":"/entanglement-3particle","dataUrl":"/entanglement-3particle.json","key":"Q4NpNwuuKT"},{"type":"text","value":" and the ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"jtYFIs0xtW"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"Cauchy-Schwarz","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"CfR2wkzELv"}],"identifier":"cauchy_schwarz_particle","label":"cauchy_schwarz_particle","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"cauchy-schwarz-particle","remote":true,"url":"/entanglement-3particle","dataUrl":"/entanglement-3particle.json","key":"PSmRgqT7aV"},{"type":"text","value":" inequality. Those two quantities quantify correlations (entanglement?) and are widely used within the community. We aim here to investigate under which conditions they can assess mode entanglement. The last section of this chapter is devoted to the application of the generalized ","key":"O6wCICUrhr"},{"type":"abbreviation","title":"Positive Partial Transpose","children":[{"type":"text","value":"PPT","key":"mcCDn0ol66"}],"key":"m4PBnJGVZ0"},{"type":"text","value":" criterion for thermal Gaussian states. Central to our journey, we demonstrate that the measurement of the ","key":"YOZQfS0aK4"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"second","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"IbSUj1xw8x"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"kZvsrr04to"},{"type":"text","value":" and ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"cqxDP3v5pW"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"fourth","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"QbH4EQQIma"}],"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","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"aXQIkEj5DC"},{"type":"text","value":" order correlation functions allow one not only to assess entanglement but also to quantify it.","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"GBGdGUxlAs"}],"key":"yD9ZgCtjOT"}],"data":{"part":"abstract"},"key":"Ndb6o3fzmi"},{"type":"block","position":{"start":{"line":17,"column":1},"end":{"line":17,"column":1}},"children":[{"type":"comment","value":" \n\nFurthermore, violation of Bell inequalities with spin entangled photons [@aspect_experimental_1981;@aspect_experimental_1982] and in momentum [@rarity_two_photon_1990] did \n\n\n\n\nDans les expériences de Spin: démonstration d'intrication EPR like @peise_satisfying_2015 mais pas Bell qui a été fait après. \n@lange_entanglement_2018 @fadel_spatial_2018 @kunkel_spatially_2018\n@shin_bell_2019\n\n\n Depending on the setup, this quantity might not be easy to The \"observation of simultaneity in parametric production of optical photon pairs\" by @burnham_observation_1970 was performed using \n\nLet me first consider the case of photons. For a two-mode squeezed state, \nIn the case of photons, the number difference variance is hard to \n\n ","key":"vSsF3VpVIM"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"What we knew, what is new ?","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"key":"wcLmVkbnUp"}],"key":"UTDpzYvAlh"},{"type":"paragraph","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"children":[{"type":"text","value":"The first and second sections of this chapter are a literature review: we introduce Gaussian states and some entanglement criteria/witnesses. The third part of this work discusses the notion of particle and mode entanglement, as well as the range of applicability of the classical Cauchy-Schwarz inequality and relative number squeezing. It does not contain “new” contributions and the discussion might seem trivial; however I did not find in the literature a clear explanation why those quantities could or could not witness mode entanglement. In that sense, the discussion is original. The ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"BAACq4Gcur"},{"type":"crossReference","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"children":[{"type":"text","value":"last section","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"ymp2eP1vMb"}],"identifier":"what_info_cov_matrix","label":"what_info_cov_matrix","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"what-info-cov-matrix","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"R8puNsVzGy"},{"type":"text","value":" is the major theoretical contribution of this thesis. It demonstrates how 2- and 4-body correlation functions can be used to quantify the entanglement of thermal Gaussian states.","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"PFGtkfErJH"}],"key":"DTAcUhzPMB"},{"type":"comment","value":"As explained in the introduction, the first section of this chapter is devoted to Gaussian states. Although there is nothing *new* here for the community, I aimed to introduce them in a straightforward manner, as our team does not have extensive experience in this field. This work was prompted by our need to better understand how to capture entanglement with our single-particle detector. It was also motivated by many discussions, seminars, and conferences. Does the violation of the Cauchy-Schwarz inequality implies entanglement? Does a second-order correlation function above 2 implies it as well? If not, why do we measure it to probe entanglement? And if it is useless, why did the ANR found the project?","position":{"start":{"line":42,"column":1},"end":{"line":42,"column":1}},"key":"klkNCAFpfX"},{"type":"comment","value":"This final section of this chapter contains the major theoretical contribution of this thesis. We demonstrate that it is possible to assess the non-separability of some Gaussian states with a single-particle detector.","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"key":"MYYWXZCh3x"}],"key":"Xr8XX1b7De"}],"key":"j6uYm2Jrv5"}],"key":"Zn57WzawSC"},"references":{"cite":{"order":["cerf_quantum_2007"],"data":{"cerf_quantum_2007":{"label":"cerf_quantum_2007","enumerator":"1","doi":"10.1142/p489","html":"Cerf, N. J., Leuchs, G., & Polzik, E. S. (2007). Quantum Information with Continuous Variables of Atoms and Light. Imperial College Press. 10.1142/p489","url":"https://doi.org/10.1142/p489"}}}},"footer":{"navigation":{"prev":{"title":"Parametric creation of quasi-particles in a BEC","short_title":"Parametric creation of quasi particles in a BEC","url":"/dce-bogoliubov","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Gaussian states","short_title":"Gaussian states","url":"/entanglement-1gaussian","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}