Quantum entanglement in phase space
Summarize this article with:
AbstractWhile commonly used entanglement criteria for continuous variable systems are based on quadrature measurements, here we study entanglement detection from measurements of the Wigner function. These are routinely performed in platforms such as trapped ions and circuit QED, where homodyne measurements are difficult to be implemented. We provide complementary criteria which we show to be tight for a variety of experimentally relevant Gaussian and non-Gaussian states. Our results show novel approaches to detect entanglement in continuous variable systems and shed light on interesting connections between known criteria and the Wigner function.Featured image: Full information of a bipartite continuous-variable system is contained in a joint Wigner function $W_{AB}(x_A,p_A,x_B,p_B)$ defined in a four-dimensional phase space. Revealing entanglement from measurements of this function is in general a difficult task. To overcome this problem, we present entanglement criteria based on measurements on a two-dimensional slice $W_{AB}(x,p,x',p')$, where $x'$ and $p'$ are linear functions of $x,p$, representing a coordinate transformation.Popular summaryDetecting entanglement in continuous-variable quantum systems typically relies on quadrature measurements. In many experimental platforms, including trapped ions, cavity and circuit quantum electrodynamics, and circuit quantum acoustodynamics, such measurements are difficult to implement. In contrast, the Wigner function can be measured naturally in these systems through displaced parity measurements. It is therefore valuable to develop entanglement criteria that work directly with Wigner function data, ideally matching or even surpassing the power of quadrature-based criteria. In this work, we derive a family of phase-space entanglement criteria based on a carefully chosen two-dimensional slice of the joint Wigner function, avoiding the need to reconstruct the full four-dimensional phase-space distribution of two modes. By optimizing the coordinate system used for this slice, the criteria can be tailored to different states. For Gaussian states, our criteria are necessary and sufficient, coinciding with the Duan-Simon criteria. For certain non-Gaussian states, they can outperform Duan-Simon, and are also tight for representative examples such as Werner states in the Fock basis and entangled cat states. The violations provide quantitative bounds on the entanglement negativity and on the degree of violation of the realignment criterion. In practice, the required data can be collected through a randomized measurement protocol that is statistically equivalent to measuring the Wigner function at one or two (depending on the chosen criterion) phase-space points. This offers a substantially more efficient route to entanglement detection in platforms where Wigner function measurements are native.► BibTeX data@article{Liu2026quantumentanglement, doi = {10.22331/q-2026-03-23-2040}, url = {https://doi.org/10.22331/q-2026-03-23-2040}, title = {Quantum entanglement in phase space}, author = {Liu, Shuheng and Guo, Jiajie and He, Qiongyi and Fadel, Matteo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2040}, month = mar, year = {2026} }► References [1] R. Simon. ``Peres-horodecki separability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2726–2729 (2000). https://doi.org/10.1103/PhysRevLett.84.2726 [2] Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller. ``Inseparability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2722–2725 (2000). https://doi.org/10.1103/PhysRevLett.84.2722 [3] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland. ``Quantum dynamics of single trapped ions''. Rev. Mod. Phys. 75, 281–324 (2003). https://doi.org/10.1103/RevModPhys.75.281 [4] J. M. Raimond, M. Brune, and S. Haroche. ``Manipulating quantum entanglement with atoms and photons in a cavity''. Rev. Mod. Phys. 73, 565–582 (2001). https://doi.org/10.1103/RevModPhys.73.565 [5] Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff. ``Circuit quantum electrodynamics''. Rev. Mod. Phys. 93, 025005 (2021). https://doi.org/10.1103/RevModPhys.93.025005 [6] Uwe von Lüpke, Yu Yang, Marius Bild, Laurent Michaud, Matteo Fadel, and Yiwen Chu. ``Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics''. Nat. Phys. 18, 794–799 (2022). https://doi.org/10.1038/s41567-022-01591-2 [7] L. G. Lutterbach and L. Davidovich. ``Method for direct measurement of the wigner function in cavity qed and ion traps''. Phys. Rev. Lett. 78, 2547–2550 (1997). https://doi.org/10.1103/PhysRevLett.78.2547 [8] Konrad Banaszek and Krzysztof Wódkiewicz. ``Testing quantum nonlocality in phase space''. Phys. Rev. Lett. 82, 2009–2013 (1999). https://doi.org/10.1103/PhysRevLett.82.2009 [9] Konrad Banaszek and Krzysztof Wódkiewicz. ``Nonlocality of the einstein-podolsky-rosen state in the wigner representation''. Phys. Rev. A 58, 4345–4347 (1998). https://doi.org/10.1103/PhysRevA.58.4345 [10] Mattia Walschaers and Nicolas Treps. ``Remote generation of wigner negativity through einstein-podolsky-rosen steering''. Phys. Rev. Lett. 124, 150501 (2020). https://doi.org/10.1103/PhysRevLett.124.150501 [11] Pooja Jayachandran, Lin Htoo Zaw, and Valerio Scarani. ``Dynamics-based entanglement witnesses for non-gaussian states of harmonic oscillators''. Phys. Rev. Lett. 130, 160201 (2023). https://doi.org/10.1103/PhysRevLett.130.160201 [12] Lin Htoo Zaw. ``Certifiable lower bounds of wigner negativity volume and non-gaussian entanglement with conditional displacement gates''. Phys. Rev. Lett. 133, 050201 (2024). https://doi.org/10.1103/PhysRevLett.133.050201 [13] Chengjie Zhang, Sixia Yu, Qing Chen, and C. H. Oh. ``Detecting and estimating continuous-variable entanglement by local orthogonal observables''. Phys. Rev. Lett. 111, 190501 (2013). https://doi.org/10.1103/PhysRevLett.111.190501 [14] R. F. Bishop and A. Vourdas. ``Displaced and squeezed parity operator: Its role in classical mappings of quantum theories''. Phys. Rev. A 50, 4488–4501 (1994). https://doi.org/10.1103/PhysRevA.50.4488 [15] Héctor Moya-Cessa and Peter L. Knight. ``Series representation of quantum-field quasiprobabilities''. Phys. Rev. A 48, 2479–2481 (1993). https://doi.org/10.1103/PhysRevA.48.2479 [16] Todd Tilma, Mark J. Everitt, John H. Samson, William J. Munro, and Kae Nemoto. ``Wigner functions for arbitrary quantum systems''. Phys. Rev. Lett. 117, 180401 (2016). https://doi.org/10.1103/PhysRevLett.117.180401 [17] Chen Wang, Yvonne Y. Gao, Philip Reinhold, and et. al. ``A schrödinger cat living in two boxes''. Science 352, 1087 (2016). https://doi.org/10.1126/science.aaf2941 [18] A. Ferraro, S. Olivares, and Matteo G. A. Paris. ``Gaussian states in quantum information''. Napoli Series on physics and Astrophysics. Bibliopolis. (2005). [19] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouzé. ``Convergence rates for the quantum central limit theorem''. Commun. in Math. Phys. 383, 223–279 (2021). https://doi.org/10.1007/s00220-021-03988-1 [20] M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. ``Bell's inequality violation with non-negative wigner functions''. Phys. Rev. A 71, 022103 (2005). https://doi.org/10.1103/PhysRevA.71.022103 [21] Eugene Wigner. ``Group theory: and its application to the quantum mechanics of atomic spectra''. Volume 5, page 233–236. Elsevier. (2012). [22] ``See supplemental material for details.''. [23] Asher Peres. ``Separability criterion for density matrices''. Phys. Rev. Lett. 77, 1413–1415 (1996). https://doi.org/10.1103/PhysRevLett.77.1413 [24] Pawel Horodecki. ``Separability criterion and inseparable mixed states with positive partial transposition''. Physics Letters A 232, 333–339 (1997). https://doi.org/10.1016/S0375-9601(97)00416-7 [25] Wolfgang P. Schleich. ``Wigner function''. Chapter 3, pages 67–98. John Wiley & Sons, Ltd. (2001). https://doi.org/10.1002/3527602976.ch3 [26] Paweł Horodecki, J. Ignacio Cirac, and Maciej Lewenstein. ``Bound entanglement for continuous variables is a rare phenomenon''. Pages 211–228. Springer Netherlands. Dordrecht (2003). https://doi.org/10.1007/978-94-015-1258-9_17 [27] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. ``Volume of the set of separable states''. Phys. Rev. A 58, 883–892 (1998). https://doi.org/10.1103/PhysRevA.58.883 [28] Jens Eisert and Martin B. Plenio. ``A comparison of entanglement measures''. J. Mod. Opt. 46, 145–154 (1999). https://doi.org/10.1080/09500349908231260 [29] G. Vidal and R. F. Werner. ``Computable measure of entanglement''. Phys. Rev. A 65, 032314 (2002). https://doi.org/10.1103/PhysRevA.65.032314 [30] Y. J.
Park Jinhyoung Lee, M. S. Kim and S. Lee. ``Partial teleportation of entanglement in a noisy environment''. J. Mod. Opt. 47, 2151–2164 (2000). https://doi.org/10.1080/09500340008235138 [31] M. Khasin, R. Kosloff, and D. Steinitz. ``Negativity as a distance from a separable state''. Phys. Rev. A 75, 052325 (2007). https://doi.org/10.1103/PhysRevA.75.052325 [32] Oliver Rudolph. ``A separability criterion for density operators''. J. Phys. A Math. Gen. 33, 3951 (2000). https://doi.org/10.1088/0305-4470/33/21/308 [33] Oliver Rudolph. ``Further results on the cross norm criterion for separability''. Quantum Inf. Process. 4, 219–239 (2005). [34] David Pérez-García. ``Deciding separability with a fixed error''. Phys. Lett. A 330, 149–154 (2004). https://doi.org/10.1016/j.physleta.2004.07.059 [35] Oliver Rudolph. ``Some properties of the computable cross-norm criterion for separability''. Phys. Rev. A 67, 032312 (2003). https://doi.org/10.1103/PhysRevA.67.032312 [36] Ali Asadian, Paul Erker, Marcus Huber, and Claude Klöckl. ``Heisenberg-weyl observables: Bloch vectors in phase space''. Phys. Rev. A 94, 010301 (2016). https://doi.org/10.1103/PhysRevA.94.010301 [37] Sixia Yu and Nai-le Liu. ``Entanglement detection by local orthogonal observables''. Phys. Rev. Lett. 95, 150504 (2005). https://doi.org/10.1103/PhysRevLett.95.150504 [38] Yu Xiang, Buqing Xu, Ladislav Mišta, Tommaso Tufarelli, Qiongyi He, and Gerardo Adesso. ``Investigating einstein-podolsky-rosen steering of continuous-variable bipartite states by non-gaussian pseudospin measurements''. Phys. Rev. A 96, 042326 (2017). https://doi.org/10.1103/PhysRevA.96.042326 [39] Jérôme Martin and Vincent Vennin. ``Obstructions to bell cmb experiments''. Phys. Rev. D 96, 063501 (2017). https://doi.org/10.1103/PhysRevD.96.063501 [40] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Separability of mixed states: necessary and sufficient conditions''. Physics Letters A 223, 1–8 (1996). https://doi.org/10.1016/S0375-9601(96)00706-2 [41] Manuel Gessner, Luca Pezzè, and Augusto Smerzi. ``Efficient entanglement criteria for discrete, continuous, and hybrid variables''. Phys. Rev. A 94, 020101 (2016). https://doi.org/10.1103/PhysRevA.94.020101 [42] Mark Hillery and M. Suhail Zubairy. ``Entanglement conditions for two-mode states''. Phys. Rev. Lett. 96, 050503 (2006). https://doi.org/10.1103/PhysRevLett.96.050503 [43] Helmut Vogel. ``A better way to construct the sunflower head''. Mathematical Biosciences 44, 179–189 (1979). https://doi.org/10.1016/0025-5564(79)90080-4 [44] Jonathan P. Dowling, G. S. Agarwal, and Wolfgang P. Schleich. ``Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms''. Phys. Rev. A 49, 4101–4109 (1994). https://doi.org/10.1103/PhysRevA.49.4101 [45] Yumang Jing, Matteo Fadel, Valentin Ivannikov, and Tim Byrnes. ``Split spin-squeezed bose–einstein condensates''. New J. of Phys. 21, 093038 (2019). https://doi.org/10.1088/1367-2630/ab3fcf [46] Matteo Fadel and Manuel Gessner. ``Relating spin squeezing to multipartite entanglement criteria for particles and modes''. Phys. Rev. A 102, 012412 (2020). https://doi.org/10.1103/PhysRevA.102.012412 [47] Jonas Kitzinger, Xin Meng, Matteo Fadel, Valentin Ivannikov, Kae Nemoto, William J. Munro, and Tim Byrnes. ``Bell correlations in a split two-mode-squeezed bose-einstein condensate''. Phys. Rev. A 104, 043323 (2021). https://doi.org/10.1103/PhysRevA.104.043323 [48] Jiajie Guo, Feng-Xiao Sun, Daoquan Zhu, Manuel Gessner, Qiongyi He, and Matteo Fadel. ``Detecting einstein-podolsky-rosen steering in non-gaussian spin states from conditional spin-squeezing parameters''. Phys. Rev. A 108, 012435 (2023). https://doi.org/10.1103/PhysRevA.108.012435 [49] Matteo Fadel, Benjamin Yadin, Yuping Mao, Tim Byrnes, and Manuel Gessner. ``Multiparameter quantum metrology and mode entanglement with spatially split nonclassical spin ensembles''. New Journal of Physics 25, 073006 (2023). https://doi.org/10.1088/1367-2630/ace1a0 [50] Ioannis Kogias, Paul Skrzypczyk, Daniel Cavalcanti, Antonio Acín, and Gerardo Adesso. ``Hierarchy of steering criteria based on moments for all bipartite quantum systems''. Phys. Rev. Lett. 115, 210401 (2015). https://doi.org/10.1103/PhysRevLett.115.210401 [51] M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. ``Bell's inequality violation with non-negative wigner functions''. Phys. Rev. A 71, 022103 (2005). https://doi.org/10.1103/PhysRevA.71.022103Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-23 12:50:27: Could not fetch cited-by data for 10.22331/q-2026-03-23-2040 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-23 12:50:27: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWhile commonly used entanglement criteria for continuous variable systems are based on quadrature measurements, here we study entanglement detection from measurements of the Wigner function. These are routinely performed in platforms such as trapped ions and circuit QED, where homodyne measurements are difficult to be implemented. We provide complementary criteria which we show to be tight for a variety of experimentally relevant Gaussian and non-Gaussian states. Our results show novel approaches to detect entanglement in continuous variable systems and shed light on interesting connections between known criteria and the Wigner function.Featured image: Full information of a bipartite continuous-variable system is contained in a joint Wigner function $W_{AB}(x_A,p_A,x_B,p_B)$ defined in a four-dimensional phase space. Revealing entanglement from measurements of this function is in general a difficult task. To overcome this problem, we present entanglement criteria based on measurements on a two-dimensional slice $W_{AB}(x,p,x',p')$, where $x'$ and $p'$ are linear functions of $x,p$, representing a coordinate transformation.Popular summaryDetecting entanglement in continuous-variable quantum systems typically relies on quadrature measurements. In many experimental platforms, including trapped ions, cavity and circuit quantum electrodynamics, and circuit quantum acoustodynamics, such measurements are difficult to implement. In contrast, the Wigner function can be measured naturally in these systems through displaced parity measurements. It is therefore valuable to develop entanglement criteria that work directly with Wigner function data, ideally matching or even surpassing the power of quadrature-based criteria. In this work, we derive a family of phase-space entanglement criteria based on a carefully chosen two-dimensional slice of the joint Wigner function, avoiding the need to reconstruct the full four-dimensional phase-space distribution of two modes. By optimizing the coordinate system used for this slice, the criteria can be tailored to different states. For Gaussian states, our criteria are necessary and sufficient, coinciding with the Duan-Simon criteria. For certain non-Gaussian states, they can outperform Duan-Simon, and are also tight for representative examples such as Werner states in the Fock basis and entangled cat states. The violations provide quantitative bounds on the entanglement negativity and on the degree of violation of the realignment criterion. In practice, the required data can be collected through a randomized measurement protocol that is statistically equivalent to measuring the Wigner function at one or two (depending on the chosen criterion) phase-space points. This offers a substantially more efficient route to entanglement detection in platforms where Wigner function measurements are native.► BibTeX data@article{Liu2026quantumentanglement, doi = {10.22331/q-2026-03-23-2040}, url = {https://doi.org/10.22331/q-2026-03-23-2040}, title = {Quantum entanglement in phase space}, author = {Liu, Shuheng and Guo, Jiajie and He, Qiongyi and Fadel, Matteo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2040}, month = mar, year = {2026} }► References [1] R. Simon. ``Peres-horodecki separability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2726–2729 (2000). https://doi.org/10.1103/PhysRevLett.84.2726 [2] Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller. ``Inseparability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2722–2725 (2000). https://doi.org/10.1103/PhysRevLett.84.2722 [3] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland. ``Quantum dynamics of single trapped ions''. Rev. Mod. Phys. 75, 281–324 (2003). https://doi.org/10.1103/RevModPhys.75.281 [4] J. M. Raimond, M. Brune, and S. Haroche. ``Manipulating quantum entanglement with atoms and photons in a cavity''. Rev. Mod. Phys. 73, 565–582 (2001). https://doi.org/10.1103/RevModPhys.73.565 [5] Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff. ``Circuit quantum electrodynamics''. Rev. Mod. Phys. 93, 025005 (2021). https://doi.org/10.1103/RevModPhys.93.025005 [6] Uwe von Lüpke, Yu Yang, Marius Bild, Laurent Michaud, Matteo Fadel, and Yiwen Chu. ``Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics''. Nat. Phys. 18, 794–799 (2022). https://doi.org/10.1038/s41567-022-01591-2 [7] L. G. Lutterbach and L. Davidovich. ``Method for direct measurement of the wigner function in cavity qed and ion traps''. Phys. Rev. Lett. 78, 2547–2550 (1997). https://doi.org/10.1103/PhysRevLett.78.2547 [8] Konrad Banaszek and Krzysztof Wódkiewicz. ``Testing quantum nonlocality in phase space''. Phys. Rev. Lett. 82, 2009–2013 (1999). https://doi.org/10.1103/PhysRevLett.82.2009 [9] Konrad Banaszek and Krzysztof Wódkiewicz. ``Nonlocality of the einstein-podolsky-rosen state in the wigner representation''. Phys. Rev. A 58, 4345–4347 (1998). https://doi.org/10.1103/PhysRevA.58.4345 [10] Mattia Walschaers and Nicolas Treps. ``Remote generation of wigner negativity through einstein-podolsky-rosen steering''. Phys. Rev. Lett. 124, 150501 (2020). https://doi.org/10.1103/PhysRevLett.124.150501 [11] Pooja Jayachandran, Lin Htoo Zaw, and Valerio Scarani. ``Dynamics-based entanglement witnesses for non-gaussian states of harmonic oscillators''. Phys. Rev. Lett. 130, 160201 (2023). https://doi.org/10.1103/PhysRevLett.130.160201 [12] Lin Htoo Zaw. ``Certifiable lower bounds of wigner negativity volume and non-gaussian entanglement with conditional displacement gates''. Phys. Rev. Lett. 133, 050201 (2024). https://doi.org/10.1103/PhysRevLett.133.050201 [13] Chengjie Zhang, Sixia Yu, Qing Chen, and C. H. Oh. ``Detecting and estimating continuous-variable entanglement by local orthogonal observables''. Phys. Rev. Lett. 111, 190501 (2013). https://doi.org/10.1103/PhysRevLett.111.190501 [14] R. F. Bishop and A. Vourdas. ``Displaced and squeezed parity operator: Its role in classical mappings of quantum theories''. Phys. Rev. A 50, 4488–4501 (1994). https://doi.org/10.1103/PhysRevA.50.4488 [15] Héctor Moya-Cessa and Peter L. Knight. ``Series representation of quantum-field quasiprobabilities''. Phys. Rev. A 48, 2479–2481 (1993). https://doi.org/10.1103/PhysRevA.48.2479 [16] Todd Tilma, Mark J. Everitt, John H. Samson, William J. Munro, and Kae Nemoto. ``Wigner functions for arbitrary quantum systems''. Phys. Rev. Lett. 117, 180401 (2016). https://doi.org/10.1103/PhysRevLett.117.180401 [17] Chen Wang, Yvonne Y. Gao, Philip Reinhold, and et. al. ``A schrödinger cat living in two boxes''. Science 352, 1087 (2016). https://doi.org/10.1126/science.aaf2941 [18] A. Ferraro, S. Olivares, and Matteo G. A. Paris. ``Gaussian states in quantum information''. Napoli Series on physics and Astrophysics. Bibliopolis. (2005). [19] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouzé. ``Convergence rates for the quantum central limit theorem''. Commun. in Math. Phys. 383, 223–279 (2021). https://doi.org/10.1007/s00220-021-03988-1 [20] M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. ``Bell's inequality violation with non-negative wigner functions''. Phys. Rev. A 71, 022103 (2005). https://doi.org/10.1103/PhysRevA.71.022103 [21] Eugene Wigner. ``Group theory: and its application to the quantum mechanics of atomic spectra''. Volume 5, page 233–236. Elsevier. (2012). [22] ``See supplemental material for details.''. [23] Asher Peres. ``Separability criterion for density matrices''. Phys. Rev. Lett. 77, 1413–1415 (1996). https://doi.org/10.1103/PhysRevLett.77.1413 [24] Pawel Horodecki. ``Separability criterion and inseparable mixed states with positive partial transposition''. Physics Letters A 232, 333–339 (1997). https://doi.org/10.1016/S0375-9601(97)00416-7 [25] Wolfgang P. Schleich. ``Wigner function''. Chapter 3, pages 67–98. John Wiley & Sons, Ltd. (2001). https://doi.org/10.1002/3527602976.ch3 [26] Paweł Horodecki, J. Ignacio Cirac, and Maciej Lewenstein. ``Bound entanglement for continuous variables is a rare phenomenon''. Pages 211–228. Springer Netherlands. Dordrecht (2003). https://doi.org/10.1007/978-94-015-1258-9_17 [27] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. ``Volume of the set of separable states''. Phys. Rev. A 58, 883–892 (1998). https://doi.org/10.1103/PhysRevA.58.883 [28] Jens Eisert and Martin B. Plenio. ``A comparison of entanglement measures''. J. Mod. Opt. 46, 145–154 (1999). https://doi.org/10.1080/09500349908231260 [29] G. Vidal and R. F. Werner. ``Computable measure of entanglement''. Phys. Rev. A 65, 032314 (2002). https://doi.org/10.1103/PhysRevA.65.032314 [30] Y. J.
Park Jinhyoung Lee, M. S. Kim and S. Lee. ``Partial teleportation of entanglement in a noisy environment''. J. Mod. Opt. 47, 2151–2164 (2000). https://doi.org/10.1080/09500340008235138 [31] M. Khasin, R. Kosloff, and D. Steinitz. ``Negativity as a distance from a separable state''. Phys. Rev. A 75, 052325 (2007). https://doi.org/10.1103/PhysRevA.75.052325 [32] Oliver Rudolph. ``A separability criterion for density operators''. J. Phys. A Math. Gen. 33, 3951 (2000). https://doi.org/10.1088/0305-4470/33/21/308 [33] Oliver Rudolph. ``Further results on the cross norm criterion for separability''. Quantum Inf. Process. 4, 219–239 (2005). [34] David Pérez-García. ``Deciding separability with a fixed error''. Phys. Lett. A 330, 149–154 (2004). https://doi.org/10.1016/j.physleta.2004.07.059 [35] Oliver Rudolph. ``Some properties of the computable cross-norm criterion for separability''. Phys. Rev. A 67, 032312 (2003). https://doi.org/10.1103/PhysRevA.67.032312 [36] Ali Asadian, Paul Erker, Marcus Huber, and Claude Klöckl. ``Heisenberg-weyl observables: Bloch vectors in phase space''. Phys. Rev. A 94, 010301 (2016). https://doi.org/10.1103/PhysRevA.94.010301 [37] Sixia Yu and Nai-le Liu. ``Entanglement detection by local orthogonal observables''. Phys. Rev. Lett. 95, 150504 (2005). https://doi.org/10.1103/PhysRevLett.95.150504 [38] Yu Xiang, Buqing Xu, Ladislav Mišta, Tommaso Tufarelli, Qiongyi He, and Gerardo Adesso. ``Investigating einstein-podolsky-rosen steering of continuous-variable bipartite states by non-gaussian pseudospin measurements''. Phys. Rev. A 96, 042326 (2017). https://doi.org/10.1103/PhysRevA.96.042326 [39] Jérôme Martin and Vincent Vennin. ``Obstructions to bell cmb experiments''. Phys. Rev. D 96, 063501 (2017). https://doi.org/10.1103/PhysRevD.96.063501 [40] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Separability of mixed states: necessary and sufficient conditions''. Physics Letters A 223, 1–8 (1996). https://doi.org/10.1016/S0375-9601(96)00706-2 [41] Manuel Gessner, Luca Pezzè, and Augusto Smerzi. ``Efficient entanglement criteria for discrete, continuous, and hybrid variables''. Phys. Rev. A 94, 020101 (2016). https://doi.org/10.1103/PhysRevA.94.020101 [42] Mark Hillery and M. Suhail Zubairy. ``Entanglement conditions for two-mode states''. Phys. Rev. Lett. 96, 050503 (2006). https://doi.org/10.1103/PhysRevLett.96.050503 [43] Helmut Vogel. ``A better way to construct the sunflower head''. Mathematical Biosciences 44, 179–189 (1979). https://doi.org/10.1016/0025-5564(79)90080-4 [44] Jonathan P. Dowling, G. S. Agarwal, and Wolfgang P. Schleich. ``Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms''. Phys. Rev. A 49, 4101–4109 (1994). https://doi.org/10.1103/PhysRevA.49.4101 [45] Yumang Jing, Matteo Fadel, Valentin Ivannikov, and Tim Byrnes. ``Split spin-squeezed bose–einstein condensates''. New J. of Phys. 21, 093038 (2019). https://doi.org/10.1088/1367-2630/ab3fcf [46] Matteo Fadel and Manuel Gessner. ``Relating spin squeezing to multipartite entanglement criteria for particles and modes''. Phys. Rev. A 102, 012412 (2020). https://doi.org/10.1103/PhysRevA.102.012412 [47] Jonas Kitzinger, Xin Meng, Matteo Fadel, Valentin Ivannikov, Kae Nemoto, William J. Munro, and Tim Byrnes. ``Bell correlations in a split two-mode-squeezed bose-einstein condensate''. Phys. Rev. A 104, 043323 (2021). https://doi.org/10.1103/PhysRevA.104.043323 [48] Jiajie Guo, Feng-Xiao Sun, Daoquan Zhu, Manuel Gessner, Qiongyi He, and Matteo Fadel. ``Detecting einstein-podolsky-rosen steering in non-gaussian spin states from conditional spin-squeezing parameters''. Phys. Rev. A 108, 012435 (2023). https://doi.org/10.1103/PhysRevA.108.012435 [49] Matteo Fadel, Benjamin Yadin, Yuping Mao, Tim Byrnes, and Manuel Gessner. ``Multiparameter quantum metrology and mode entanglement with spatially split nonclassical spin ensembles''. New Journal of Physics 25, 073006 (2023). https://doi.org/10.1088/1367-2630/ace1a0 [50] Ioannis Kogias, Paul Skrzypczyk, Daniel Cavalcanti, Antonio Acín, and Gerardo Adesso. ``Hierarchy of steering criteria based on moments for all bipartite quantum systems''. Phys. Rev. Lett. 115, 210401 (2015). https://doi.org/10.1103/PhysRevLett.115.210401 [51] M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. ``Bell's inequality violation with non-negative wigner functions''. Phys. Rev. A 71, 022103 (2005). https://doi.org/10.1103/PhysRevA.71.022103Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-23 12:50:27: Could not fetch cited-by data for 10.22331/q-2026-03-23-2040 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-23 12:50:27: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.