Estimating the best separable approximation of non-pure spin-squeezed states
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AbstractWe discuss the estimation of the distance of a given mixed many-body quantum state to the set of fully separable states, applied to the concrete scenario of collective spin states. Concretely, we discuss lower bounds to distances from the set of fully separable states based on entanglement criteria and upper bounds to those distances using an iterative algorithm to find the optimal separable state closest to the target. Focusing on collective states of $N$ spin-$1/2$ particles, we consider spin-squeezing inequalities (SSIs), which provide a complete set of nonlinear entanglement criteria based on collective spin variances. First, we find a lower bound to distance-based entanglement monotones, specifically the so-called best separable approximation (BSA) from the complete set of SSIs, thereby bypassing entirely a numerical optimization over a (potentially very large) set of linear entanglement witnesses. Then, we improve current algorithms to iteratively find the closest separable state to a given target state, exploiting the symmetry of the system. These results allow us to study entanglement quantitatively on thermal states of spin systems on fully-connected graphs at nonzero temperature, as well as potentially similar states arising in out-of-equilibrium situations. We thus apply our methods to investigate entanglement across different phases of a fully-connected XXZ model. We observe that our lower bound becomes often tight for zero temperature as well as for the temperature at which entanglement disappears, both of which are thus precisely captured by the SSIs. We further observe, among other things, that entanglement can arise at nonzero temperature even in the ordered phase, where the ground state is separable, revealing the potential usefulness of entanglement quantification also beyond the ground state paradigm.Featured image: We estimate the best separable approximation for an $N$-qubit state (top right), by computing lower and upper bounds (left). Lower bounds are computed from the negative expectation value of an entanglement witness W that is illustrated as blue lines not crossing the set of separable states (shaded ellipse). Upper bounds can be computed from the distance to any given separable state (red/purple lines), minimized over as many separable states as possible. Optimal bounds are illustrated as straight or dotted lines, non-optimal bounds as dashed lines. For states featuring symmetries, the problem can be formulated in a symmetric subspace (bottom right: example matrix for $N = 4$, blocks of different color are a result of permutational invariance, while diagonal matrices have an additional rotation symmetry). Popular summaryEntanglement in many-body systems is usually analyzed for pure ground states, but realistic systems are often mixed because of temperature, noise, or nonequilibrium dynamics. In such cases, even deciding whether a state is entangled can be difficult, let alone quantifying how much entanglement it contains. In this work, we study this problem for collective spin states by asking how far a given mixed state is from the set of fully separable states. This distance is quantified by the best separable approximation, which tells us how well the state can still be described by a classical-like mixture of unentangled particles. We derive a lower bound on this quantity from spin-squeezing inequalities built from standard collective observables, and an upper bound from an iterative algorithm that searches for the closest separable state while exploiting the symmetries of the system. Applying these tools to thermal states of fully connected spin models, we show that this method can provide insightful quantitative information about mixed-state entanglement across different phases. In particular, we find that entanglement may appear at nonzero temperature even in regimes where the ground state itself is separable, highlighting that relevant quantum correlations can emerge beyond the usual ground-state picture. Our results also suggest that entanglement quantification may refine the usual phase diagram by revealing a finer structure within conventional phases, where states with different amounts of entanglement define distinct regimes.► BibTeX data@article{Mathe2026estimatingbest, doi = {10.22331/q-2026-04-21-2078}, url = {https://doi.org/10.22331/q-2026-04-21-2078}, title = {Estimating the best separable approximation of non-pure spin-squeezed states}, author = {Math{\'{e}}, Julia and Usui, Ayaka and G{\"{u}}hne, Otfried and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2078}, month = apr, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). arXiv:quant-ph/0702225. https://doi.org/10.1103/RevModPhys.81.865 arXiv:quant-ph/0702225 [2] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. ``Entanglement in many-body systems''. Rev. Mod. Phys. 80, 517–576 (2008). arXiv:quant-ph/0703044. https://doi.org/10.1103/RevModPhys.80.517 arXiv:quant-ph/0703044 [3] Nicolas Laflorencie. ``Quantum entanglement in condensed matter systems''. Phys. Rep. 646, 1–59 (2016). arXiv:1512.03388. https://doi.org/10.1016/j.physrep.2016.06.008 arXiv:1512.03388 [4] Subir Sachdev. ``Quantum Phase Transitions''.
Cambridge University Press. (2011). https://doi.org/10.1017/CBO9780511973765 [5] Steven M. Girvin and Kun Yang. ``Modern Condensed Matter Physics''.
Cambridge University Press. (2019). https://doi.org/10.1017/9781316480649 [6] Ulrich Schollwöck. ``The density-matrix renormalization group''. Rev. Mod. Phys. 77, 259–315 (2005). arXiv:cond-mat/0409292. https://doi.org/10.1103/RevModPhys.77.259 arXiv:cond-mat/0409292 [7] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Ann. Phys. 326, 96–192 (2011). arXiv:1008.3477. https://doi.org/10.1016/j.aop.2010.09.012 arXiv:1008.3477 [8] Román Orús. ``Tensor networks for complex quantum systems''. Nat. Rev. Phys. 1, 538–550 (2019). arXiv:1812.04011. https://doi.org/10.1038/s42254-019-0086-7 arXiv:1812.04011 [9] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. Rep. 474, 1–75 (2009). arXiv:0811.2803. https://doi.org/10.1016/j.physrep.2009.02.004 arXiv:0811.2803 [10] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber. ``Entanglement certification from theory to experiment''. Nat. Rev. Phys. 1, 72–87 (2018). arXiv:1906.10929. https://doi.org/10.1038/s42254-018-0003-5 arXiv:1906.10929 [11] V. Vedral and M. B. Plenio. ``Entanglement measures and purification procedures''. Phys. Rev. A 57, 1619–1633 (1998). arXiv:quant-ph/9707035. https://doi.org/10.1103/physreva.57.1619 arXiv:quant-ph/9707035 [12] Martin B. Plenio and Shashank Virmani. ``An introduction to entanglement measures''. Quant. Inf. Comput. 7, 1–51 (2007). arXiv:quant-ph/0504163. https://doi.org/10.26421/QIC7.1-2-1 arXiv:quant-ph/0504163 [13] William K. Wootters. ``Entanglement of Formation of an Arbitrary State of Two Qubits''. Phys. Rev. Lett. 80, 2245–2248 (1998). arXiv:quant-ph/9709029. https://doi.org/10.1103/PhysRevLett.80.2245 arXiv:quant-ph/9709029 [14] Karl Gerd H. Vollbrecht and Reinhard F. Werner. ``Entanglement measures under symmetry''. Phys. Rev. A 64, 062307 (2001). arXiv:quant-ph/0010095. https://doi.org/10.1103/PhysRevA.64.062307 arXiv:quant-ph/0010095 [15] Armin Uhlmann. ``Fidelity and concurrence of conjugated states''. Phys. Rev. A 62, 032307 (2000). arXiv:quant-ph/9909060. https://doi.org/10.1103/PhysRevA.62.032307 arXiv:quant-ph/9909060 [16] Robert Lohmayer, Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Entangled Three-Qubit States without Concurrence and Three-Tangle''. Phys. Rev. Lett. 97, 260502 (2006). arXiv:quant-ph/0606071. https://doi.org/10.1103/PhysRevLett.97.260502 arXiv:quant-ph/0606071 [17] Christopher Eltschka, Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Three-tangle for mixtures of generalized GHZ and generalized W states''. New J. Phys. 10, 043014 (2008). arXiv:0711.4477. https://doi.org/10.1088/1367-2630/10/4/043014 arXiv:0711.4477 [18] Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Tangles of superpositions and the convex-roof extension''. Phys. Rev. A 77, 032310 (2008). arXiv:0710.5909. https://doi.org/10.1103/PhysRevA.77.032310 arXiv:0710.5909 [19] Jens Siewert and Christopher Eltschka. ``Quantifying Tripartite Entanglement of Three-Qubit Generalized Werner States''. Phys. Rev. Lett. 108, 230502 (2012). https://doi.org/10.1103/PhysRevLett.108.230502 [20] Koenraad Audenaert, Jens Eisert, Martin B. Plenio, and Reinhard F. Werner. ``Entanglement properties of the harmonic chain''. Phys. Rev. A 66, 042327 (2002). arXiv:quant-ph/0205025. https://doi.org/10.1103/PhysRevA.66.042327 arXiv:quant-ph/0205025 [21] Jens Eisert, Marcus Cramer, and Martin B. Plenio. ``Colloquium: Area laws for the entanglement entropy''. Rev. Mod. Phys. 82, 277 (2010). arXiv:0808.3773. https://doi.org/10.1103/RevModPhys.82.277 arXiv:0808.3773 [22] Pasquale Calabrese, John Cardy, and Erik Tonni. ``Entanglement negativity in extended systems: A field theoretical approach''. J. Stat. Mech.: Theory Exp. 2013, P02008 (2013). arXiv:1210.5359. https://doi.org/10.1088/1742-5468/2013/02/P02008 arXiv:1210.5359 [23] Viktor Eisler and Zoltán Zimborás. ``Entanglement negativity in the harmonic chain out of equilibrium''. New J. Phys. 16, 123020 (2014). arXiv:1406.5474. https://doi.org/10.1088/1367-2630/16/12/123020 arXiv:1406.5474 [24] Pasquale Calabrese, John Cardy, and Erik Tonni. ``Finite temperature entanglement negativity in conformal field theory''. J. Phys. A: Math. Theor. 48, 015006 (2015). arXiv:1408.3043. https://doi.org/10.1088/1751-8113/48/1/015006 arXiv:1408.3043 [25] Viktor Eisler and Zoltán Zimborás. ``On the partial transpose of fermionic Gaussian states''. New J. Phys. 17, 053048 (2015). arXiv:1502.01369. https://doi.org/10.1088/1367-2630/17/5/053048 arXiv:1502.01369 [26] Viktor Eisler and Zoltán Zimborás. ``Entanglement negativity in two-dimensional free lattice models''. Phys. Rev. B 93, 115148 (2016). arXiv:1511.08819. https://doi.org/10.1103/PhysRevB.93.115148 arXiv:1511.08819 [27] Jens Eisert, Viktor Eisler, and Zoltán Zimborás. ``Entanglement negativity bounds for fermionic gaussian states''. Phys. Rev. B 97 (2018). arXiv:1611.08007. https://doi.org/10.1103/physrevb.97.165123 arXiv:1611.08007 [28] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera. ``Symmetric quantum states: a review of recent progress''. Rep. Prog. Phys. 89, 024001 (2025). arXiv:2506.10185. https://doi.org/10.1088/1361-6633/ae440a arXiv:2506.10185 [29] Alessandro Ferraro, Daniel Cavalcanti, Artur García-Saez, and Antonio Acín. ``Thermal Bound Entanglement in Macroscopic Systems and Area Law''. Phys. Rev. Lett. 100 (2008). arXiv:0804.4867. https://doi.org/10.1103/physrevlett.100.080502 arXiv:0804.4867 [30] Daniel Cavalcanti, Alessandro Ferraro, Artur García-Saez, and Antonio Acín. ``Distillable entanglement and area laws in spin and harmonic-oscillator systems''. Phys. Rev. A 78 (2008). arXiv:0705.3762. https://doi.org/10.1103/physreva.78.012335 arXiv:0705.3762 [31] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Spin squeezing and entanglement''. Phys. Rev. A 79, 042334 (2009). arXiv:0806.1048. https://doi.org/10.1103/PhysRevA.79.042334 arXiv:0806.1048 [32] Giuseppe Vitagliano, Otfried Gühne, and Géza Tóth. ``$su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems''. Quantum 9, 1844 (2025). arXiv:2406.13338. https://doi.org/10.22331/q-2025-09-03-1844 arXiv:2406.13338 [33] Fernando G. S. L. Brandão. ``Quantifying entanglement with witness operators''. Phys. Rev. A 72, 022310 (2005). arXiv:quant-ph/0503152. https://doi.org/10.1103/PhysRevA.72.022310 arXiv:quant-ph/0503152 [34] Koenraad M. R. Audenaert and Martin B. Plenio. ``When are correlations quantum?—verification and quantification of entanglement by simple measurements''. New J. Phys. 8, 266–266 (2006). arXiv:quant-ph/0608067. https://doi.org/10.1088/1367-2630/8/11/266 arXiv:quant-ph/0608067 [35] Jens Eisert, Fernando G. S. L. Brandão, and Koenraad M. R. Audenaert. ``Quantitative entanglement witnesses''. New J. Phys. 9, 46 (2007). arXiv:quant-ph/0607167. https://doi.org/10.1088/1367-2630/9/3/046 arXiv:quant-ph/0607167 [36] Otfried Gühne, Michael Reimpell, and Reinhard F. Werner. ``Estimating Entanglement Measures in Experiments''. Phys. Rev. Lett. 98, 110502 (2007). arXiv:quant-ph/0607163. https://doi.org/10.1103/PhysRevLett.98.110502 arXiv:quant-ph/0607163 [37] Otfried Gühne, Michael Reimpell, and Reinhard F. Werner. ``Lower bounds on entanglement measures from incomplete information''. Phys. Rev. A 77, 052317 (2008). arXiv:0802.1734. https://doi.org/10.1103/PhysRevA.77.052317 arXiv:0802.1734 [38] Matteo Fadel, Ayaka Usui, Marcus Huber, Nicolai Friis, and Giuseppe Vitagliano. ``Entanglement Quantification in Atomic Ensembles''. Phys. Rev. Lett. 127, 010401 (2021). arXiv:2103.15730. https://doi.org/10.1103/PhysRevLett.127.010401 arXiv:2103.15730 [39] Géza Tóth. ``Entanglement witnesses in spin models''. Phys. Rev. A 71, 010301 (2005). arXiv:quant-ph/0406061. https://doi.org/10.1103/PhysRevA.71.010301 arXiv:quant-ph/0406061 [40] Janet Anders. ``Thermal state entanglement in harmonic lattices''. Phys. Rev. A 77, 062102 (2008). arXiv:0803.1102. https://doi.org/10.1103/PhysRevA.77.062102 arXiv:0803.1102 [41] Oliver Marty, Michael Epping, Hermann Kampermann, Dagmar Bruß, Martin B. Plenio, and M. Cramer. ``Quantifying entanglement with scattering experiments''. Phys. Rev. B 89, 125117 (2014). arXiv:1310.0929. https://doi.org/10.1103/PhysRevB.89.125117 arXiv:1310.0929 [42] Oliver Marty, Marcus Cramer, and Martin B. Plenio. ``Practical Entanglement Estimation for Spin-System Quantum Simulators''. Phys. Rev. Lett. 116, 105301 (2016). arXiv:1504.03572. https://doi.org/10.1103/PhysRevLett.116.105301 arXiv:1504.03572 [43] Silvia Pappalardi, Angelo Russomanno, Alessandro Silva, and Rosario Fazio. ``Multipartite entanglement after a quantum quench''. J. Stat. Mech.: Theory Exp. 2017, 053104 (2017). arXiv:1701.05883. https://doi.org/10.1088/1742-5468/aa6809 arXiv:1701.05883 [44] Alessio Lerose and Silvia Pappalardi. ``Bridging entanglement dynamics and chaos in semiclassical systems''. Phys. Rev. A 102 (2020). arXiv:2005.03670. https://doi.org/10.1103/physreva.102.032404 arXiv:2005.03670 [45] Ferenc Iglói and Géza Tóth. ``Entanglement witnesses in the $xy$ chain: Thermal equilibrium and postquench nonequilibrium states''. Phys. Rev. Res. 5, 013158 (2023). arXiv:2207.04842. https://doi.org/10.1103/PhysRevResearch.5.013158 arXiv:2207.04842 [46] Giacomo Mazza and Costantino Budroni. ``Entanglement detection in quantum materials with competing orders''. Phys. Rev. B 111 (2025). arXiv:2404.12931. https://doi.org/10.1103/physrevb.111.l100302 arXiv:2404.12931 [47] Mark R. Dowling, Andrew C. Doherty, and Stephen D. Bartlett. ``Energy as an entanglement witness for quantum many-body systems''. Phys. Rev. A 70, 062113 (2004). arXiv:quant-ph/0408086. https://doi.org/10.1103/PhysRevA.70.062113 arXiv:quant-ph/0408086 [48] Janet Anders, Dagomir Kaszlikowski, Christian Lunkes, Toshio Ohshima, and Vlatko Vedral. ``Detecting entanglement with a thermometer''. New J. Phys. 8, 140 (2006). arXiv:quant-ph/0512181. https://doi.org/10.1088/1367-2630/8/8/140 arXiv:quant-ph/0512181 [49] Marcin Wieśniak, Vlatko Vedral, and Časlav Brukner. ``Heat capacity as an indicator of entanglement''. Phys. Rev. B 78, 064108 (2008). arXiv:quant-ph/0508193. https://doi.org/10.1103/PhysRevB.78.064108 arXiv:quant-ph/0508193 [50] Marcin Wieśniak, Vlatko Vedral, and Časlav Brukner. ``Magnetic susceptibility as a macroscopic entanglement witness''. New J. Phys. 7, 258 (2005). arXiv:quant-ph/0503037. https://doi.org/10.1088/1367-2630/7/1/258 arXiv:quant-ph/0503037 [51] Časlav Brukner, Vlatko Vedral, and Anton Zeilinger. ``Crucial role of quantum entanglement in bulk properties of solids''. Phys. Rev. A 73, 012110 (2006). arXiv:quant-ph/0410138. https://doi.org/10.1103/PhysRevA.73.012110 arXiv:quant-ph/0410138 [52] Philipp Krammer, Hermann Kampermann, Dagmar Bruß, Reinhold A. Bertlmann, Leong Chuang Kwek, and Chiara Macchiavello. ``Multipartite entanglement detection via structure factors''. Phys. Rev. Lett. 103, 100502 (2009). arXiv:0904.3860. https://doi.org/10.1103/PhysRevLett.103.100502 arXiv:0904.3860 [53] Philipp Hauke, Markus Heyl, Luca Tagliacozzo, and Peter Zoller. ``Measuring multipartite entanglement through dynamic susceptibilities''. Nat. Phys. 12, 778 – 782 (2016). arXiv:1509.01739. https://doi.org/10.1038/nphys3700 arXiv:1509.01739 [54] Otfried Gühne and Géza Tóth. ``Energy and multipartite entanglement in multidimensional and frustrated spin models''. Phys. Rev. A 73, 052319 (2006). arXiv:quant-ph/0510186. https://doi.org/10.1103/PhysRevA.73.052319 arXiv:quant-ph/0510186 [55] Pedro Ribeiro, Julien Vidal, and Rémy Mosseri. ``Thermodynamical Limit of the Lipkin-Meshkov-Glick Model''. Phys. Rev. Lett. 99 (2007). arXiv:cond-mat/0703490. https://doi.org/10.1103/physrevlett.99.050402 arXiv:cond-mat/0703490 [56] Jian Ma, Xiaoguang Wang, C. P. Sun, and Franco Nori. ``Quantum spin squeezing''. Phys. Rep. 509, 89–165 (2011). arXiv:1011.2978. https://doi.org/10.1016/j.physrep.2011.08.003 arXiv:1011.2978 [57] Nicolò Defenu, Alessio Lerose, and Silvia Pappalardi. ``Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions''. Phys. Rep. 1074, 1–92 (2024). arXiv:2307.04802. https://doi.org/10.1016/j.physrep.2024.04.005 arXiv:2307.04802 [58] Julien Vidal, Rémy Mosseri, and Jorge Dukelsky. ``Entanglement in a first-order quantum phase transition''. Phys. Rev. A 69, 054101 (2004). arXiv:cond-mat/0312130. https://doi.org/10.1103/PhysRevA.69.054101 arXiv:cond-mat/0312130 [59] Charlie-Ray Mann, Mark A. Oehlgrien, Błażej Jaworowski, Giuseppe Calajó, Jamir Marino, Kyung S. Choi, and Darrick E. Chang. ``Squeezing Classical Antiferromagnets into Quantum Spin Liquids via Global Cavity Fluctuations'' (2025). arXiv:2512.05630. arXiv:2512.05630 [60] Masahiro Kitagawa and Masahito Ueda. ``Squeezed spin states''. Phys. Rev. A 47, 5138–5143 (1993). https://doi.org/10.1103/PhysRevA.47.5138 [61] Anders Sørensen, Lu-Ming Duan, J. Ignacio Cirac, and Peter Zoller. ``Many-particle entanglement with Bose-Einstein condensates''. Nature 409, 63–66 (2001). arXiv:quant-ph/0006111. https://doi.org/10.1038/35051038 arXiv:quant-ph/0006111 [62] Anders Sørensen and Klaus Mølmer. ``Entanglement and Extreme Spin Squeezing''. Phys. Rev. Lett. 86 (2001). arXiv:quant-ph/0011035. https://doi.org/10.1103/PhysRevLett.86.4431 arXiv:quant-ph/0011035 [63] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Optimal Spin Squeezing Inequalities Detect Bound Entanglement in Spin Models''. Phys. Rev. Lett. 99, 250405 (2007). arXiv:quant-ph/0702219. https://doi.org/10.1103/PhysRevLett.99.250405 arXiv:quant-ph/0702219 [64] Giuseppe Vitagliano, Philipp Hyllus, Inigo L. Egusquiza, and Géza Tóth. ``Spin Squeezing Inequalities for Arbitrary Spin''. Phys. Rev. Lett. 107, 240502 (2011). arXiv:1104.3147. https://doi.org/10.1103/PhysRevLett.107.240502 arXiv:1104.3147 [65] Giuseppe Vitagliano, Iagoba Apellaniz, Inigo L. Egusquiza, and Géza Tóth. ``Spin squeezing and entanglement for an arbitrary spin''. Phys. Rev. A 89, 032307 (2014). arXiv:1310.2269. https://doi.org/10.1103/PhysRevA.89.032307 arXiv:1310.2269 [66] Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, and Géza Tóth. ``Entanglement and extreme spin squeezing of unpolarized states''. New J. Phys. 19, 013027 (2017). arXiv:1605.07202. https://doi.org/10.1088/1367-2630/19/1/013027 arXiv:1605.07202 [67] Oliver Marty, Marcus Cramer, Giuseppe Vitagliano, Géza Tóth, and Martin B. Plenio. ``Multiparticle entanglement criteria for nonsymmetric collective variances'' (2017). arXiv:1708.06986. arXiv:1708.06986 [68] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018). arXiv:1609.01609. https://doi.org/10.1103/RevModPhys.90.035005 arXiv:1609.01609 [69] Tilo Eggeling and Reinhard F. Werner. ``Separability properties of tripartite states with $U{\bigotimes}U{\bigotimes}U$ symmetry''. Phys. Rev. A 63, 042111 (2001). arXiv:quant-ph/0010096. https://doi.org/10.1103/PhysRevA.63.042111 arXiv:quant-ph/0010096 [70] Vlatko Vedral, Martin B. Plenio, Marcus A. Rippin, and Peter L. Knight. ``Quantifying entanglement''. Phys. Rev. Lett. 78, 2275–2279 (1997). arXiv:quant-ph/9702027. https://doi.org/10.1103/physrevlett.78.2275 arXiv:quant-ph/9702027 [71] Alexander Streltsov, Hermann Kampermann, and Dagmar Bruß. ``Linking a distance measure of entanglement to its convex roof''. New J. Phys. 12, 123004 (2010). arXiv:1006.3077. https://doi.org/10.1088/1367-2630/12/12/123004 arXiv:1006.3077 [72] Maciej Lewenstein and Anna Sanpera. ``Separability and Entanglement of Composite Quantum Systems''. Phys. Rev. Lett. 80, 2261–2264 (1998). arXiv:quant-ph/9707043. https://doi.org/10.1103/PhysRevLett.80.2261 arXiv:quant-ph/9707043 [73] Sinisa Karnas and Maciej Lewenstein. ``Separable approximations of density matrices of composite quantum systems''. J. Phys. A: Math. Gen. 34, 6919–6937 (2001). arXiv:quant-ph/0011066. https://doi.org/10.1088/0305-4470/34/35/318 arXiv:quant-ph/0011066 [74] Michael Steiner. ``Generalized robustness of entanglement''. Phys. Rev. A 67, 054305 (2003). arXiv:quant-ph/0304009. https://doi.org/10.1103/PhysRevA.67.054305 arXiv:quant-ph/0304009 [75] Ayatola Gabdulin and Aikaterini Mandilara. ``Investigating bound entangled two-qutrit states via the best separable approximation''. Phys. Rev. A 100, 062322 (2019). arXiv:1906.08963. https://doi.org/10.1103/PhysRevA.100.062322 arXiv:1906.08963 [76] Antoine Girardin, Nicolas Brunner, and Tamás Kriváchy. ``Building separable approximations for quantum states via neural networks''. Phys. Rev. Res. 4, 023238 (2022). arXiv:2112.08055. https://doi.org/10.1103/PhysRevResearch.4.023238 arXiv:2112.08055 [77] Marcus Cramer, Alain Bernard, Nicole Fabbri, Leonardo Fallani, Chiara Fort, Sara Rosi, Filippo Caruso, Massimo Inguscio, and Martin B. Plenio. ``Spatial entanglement of bosons in optical lattices''. Nat. Commun. 4, 2161 (2013). arXiv:1302.4897. https://doi.org/10.1038/ncomms3161 arXiv:1302.4897 [78] Robert H. Dicke. ``Coherence in spontaneous radiation processes''. Phys. Rev. 93, 99–110 (1954). https://doi.org/10.1103/PhysRev.93.99 [79] Peter Kirton, Mor M. Roses, Jonathan Keeling, and Emanuele G. Dalla Torre. ``Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa''. Adv. Quantum Technol. 2 (2018). arXiv:1805.09828. https://doi.org/10.1002/qute.201800043 arXiv:1805.09828 [80] Harry J. Lipkin, N. Meshkov, and Arnold J. Glick. ``Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory''. Nucl. Phys. 62, 188–198 (1965). https://doi.org/10.1016/0029-5582(65)90862-X [81] N. Meshkov, Arnold J. Glick, and Harry J. Lipkin. ``Validity of many-body approximation methods for a solvable model: (II). Linearization procedures''. Nucl. Phys. 62, 199–210 (1965). https://doi.org/10.1016/0029-5582(65)90863-1 [82] Arnold J. Glick, Harry J. Lipkin, and N. Meshkov. ``Validity of many-body approximation methods for a solvable model: (III). Diagram summations''. Nucl. Phys. 62, 211–224 (1965). https://doi.org/10.1016/0029-5582(65)90864-3 [83] David J. Wineland, John J. Bollinger, Wayne M. Itano, F. L. Moore, and Daniel J. Heinzen. ``Spin squeezing and reduced quantum noise in spectroscopy''. Phys. Rev. A 46, R6797 (1992). https://doi.org/10.1103/PhysRevA.46.R6797 [84] David J. Wineland, John J. Bollinger, Wayne M. Itano, and Daniel J. Heinzen. ``Squeezed atomic states and projection noise in spectroscopy''. Phys. Rev. A 50, 67–88 (1994). https://doi.org/10.1103/PhysRevA.50.67 [85] Julien Vidal. ``Concurrence in collective models''. Phys. Rev. A 73, 062318 (2006). arXiv:quant-ph/0603108. https://doi.org/10.1103/PhysRevA.73.062318 arXiv:quant-ph/0603108 [86] Erling Størmer. ``Symmetric states of infinite tensor products of $C*$-algebras''. J. Funct. Anal. 3, 48–68 (1969). https://doi.org/10.1016/0022-1236(69)90050-0 [87] Robin L. Hudson and Graham R. Moody. ``Locally normal symmetric states and an analogue of de Finetti's theorem''. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33, 343–351 (1976). https://doi.org/10.1007/BF00534784 [88] Mark Fannes, John T. Lewis, and André Verbeure. ``Symmetric states of composite systems''. Lett. Math. Phys. 15, 255–260 (1988). https://doi.org/10.1007/BF00398595 [89] Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack. ``Unknown quantum states: The quantum de Finetti representation''. J. Math. Phys. 43, 4537–4559 (2002). arXiv:quant-ph/0104088. https://doi.org/10.1063/1.1494475 arXiv:quant-ph/0104088 [90] Robert König and Renato Renner. ``A de Finetti representation for finite symmetric quantum states''. J. Math. Phys. 46 (2005). arXiv:quant-ph/0410229. https://doi.org/10.1063/1.2146188 arXiv:quant-ph/0410229 [91] Matthias Christandl, Robert König, Graeme Mitchison, and Renato Renner. ``One-and-a-Half Quantum de Finetti Theorems''. Commun. Math. Phys. 273 (2007). arXiv:quant-ph/0602130. https://doi.org/10.1007/s00220-007-0189-3 arXiv:quant-ph/0602130 [92] Robert König and Graeme Mitchison. ``A most compendious and facile quantum de Finetti theorem''. J. Math. Phys. 50 (2009). arXiv:quant-ph/0703210. https://doi.org/10.1063/1.3049751 arXiv:quant-ph/0703210 [93] Friederike Trimborn, Reinhard F. Werner, and Dirk Witthaut. ``Quantum de Finetti theorems and mean-field theory from quantum phase space representations''. J. Phys. A: Math Theo 49, 135302 (2016). https://doi.org/10.1088/1751-8113/49/13/135302 [94] Jarosław. K. Korbicz, Ignacio J. Cirac, and Maciej Lewenstein. ``Spin Squeezing Inequalities and Entanglement of $N$ Qubit States''. Phys. Rev. Lett. 95, 120502 (2005). arXiv:quant-ph/0504005. https://doi.org/10.1103/PhysRevLett.95.120502 arXiv:quant-ph/0504005 [95] Jarosław. K. Korbicz, Otfried Gühne, Maciej Lewenstein, H. Häffner, Christian F. Roos, and Rainer Blatt. ``Generalized spin-squeezing inequalities in $n$-qubit systems: Theory and experiment''. Phys. Rev. A 74, 052319 (2006). arXiv:quant-ph/0601038. https://doi.org/10.1103/PhysRevA.74.052319 arXiv:quant-ph/0601038 [96] Miguel Navascués, Masaki Owari, and Martin B. Plenio. ``Complete criterion for separability detection''. Phys. Rev. Lett. 103, 160404 (2009). arXiv:0906.2735. https://doi.org/10.1103/PhysRevLett.103.160404 arXiv:0906.2735 [97] Julio T Barreiro, Philipp Schindler, Otfried Gühne, Thomas Monz, Michael Chwalla, Christian F Roos, Markus Hennrich, and Rainer Blatt. ``Experimental multiparticle entanglement dynamics induced by decoherence''. Nat. Phys. 6, 943–946 (2010). arXiv:1005.1965. https://doi.org/10.1038/nphys1781 arXiv:1005.1965 [98] Alastair Kay. ``Optimal detection of entanglement in Greenberger-Horne-Zeilinger states''. Phys. Rev. A 83, 020303 (2011). arXiv:1006.5197. https://doi.org/10.1103/PhysRevA.83.020303 arXiv:1006.5197 [99] Otfried Gühne. ``Entanglement criteria and full separability of multi-qubit quantum states''. Phys. Lett. A 375, 406–410 (2011). arXiv:1009.3782. https://doi.org/10.1016/j.physleta.2010.11.032 arXiv:1009.3782 [100] Hermann Kampermann, Otfried Gühne, Colin Wilmott, and Dagmar Bruß. ``Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle entanglement''. Phys. Rev. A 86, 032307 (2012). arXiv:1203.5872. https://doi.org/10.1103/PhysRevA.86.032307 arXiv:1203.5872 [101] Stephen Brierley, Miguel Navascues, and Tamas Vertesi. ``Convex separation from convex optimization for large-scale problems'' (2017). arXiv:1609.05011. arXiv:1609.05011 [102] Jiangwei Shang and Otfried Gühne. ``Convex Optimization over Classes of Multiparticle Entanglement''. Phys. Rev. Lett. 120, 050506 (2018). arXiv:1707.02958. https://doi.org/10.1103/PhysRevLett.120.050506 arXiv:1707.02958 [103] Ties-A. Ohst, Xiao-Dong Yu, Otfried Gühne, and H. Chau Nguyen. ``Certifying quantum separability with adaptive polytopes''. SciPost Phys. 16, 063 (2024). arXiv:2210.10054. https://doi.org/10.21468/SciPostPhys.16.3.063 arXiv:2210.10054 [104] Leonid Gurvits and Howard Barnum. ``Largest separable balls around the maximally mixed bipartite quantum state''. Phys. Rev. A 66, 062311 (2002). arXiv:quant-ph/0204159. https://doi.org/10.1103/PhysRevA.66.062311 arXiv:quant-ph/0204159 [105] S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack. ``Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing''. Phys. Rev. Lett. 83, 1054–1057 (1999). arXiv:quant-ph/9811018. https://doi.org/10.1103/PhysRevLett.83.1054 arXiv:quant-ph/9811018 [106] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. ``Volume of the set of separable states''. Phys. Rev. A 58, 883–892 (1998). arXiv:quant-ph/9804024. https://doi.org/10.1103/PhysRevA.58.883 arXiv:quant-ph/9804024 [107] José I. Latorre, Román Orús, Enrique Rico, and Julien Vidal. ``Entanglement entropy in the Lipkin-Meshkov-Glick model''. Phys. Rev. A 71 (2005). arXiv:cond-mat/0409611. https://doi.org/10.1103/PhysRevA.71.064101 arXiv:cond-mat/0409611 [108] Julien Vidal, Guillaume Palacios, and Claude Aslangul. ``Entanglement dynamics in the Lipkin-Meshkov-Glick model''. Phys. Rev. A 70, 062304 (2004). arXiv:cond-mat/0406481. https://doi.org/10.1103/PhysRevA.70.062304 arXiv:cond-mat/0406481 [109] Román Orús. ``Geometric entanglement in a one-dimensional valence-bond solid state''. Phys. Rev. A 78, 062332 (2008). arXiv:0808.0938. https://doi.org/10.1103/PhysRevA.78.062332 arXiv:0808.0938 [110] Xiaoguang Wang and Klaus Mølmer. ``Pairwise entanglement in symmetric multi-qubit systems''. Eur. Phys. J. D 18, 385–391 (2002). arXiv:quant-ph/0106145. https://doi.org/10.1140/epjd/e20020045 arXiv:quant-ph/0106145 [111] Xiaoguang Wang and Barry C. Sanders. ``Spin squeezing and pairwise entanglement for symmetric multiqubit states''. Phys. Rev. A 68, 012101 (2003). arXiv:quant-ph/0302014. https://doi.org/10.1103/PhysRevA.68.012101 arXiv:quant-ph/0302014 [112] John K. Stockton, JM Geremia, Andrew C. Doherty, and Hideo Mabuchi. ``Characterizing the entanglement of symmetric many-particle spin-1/2 systems''. Phys. Rev. A 67, 022112 (2003). arXiv:quant-ph/0210117. https://doi.org/10.1103/PhysRevA.67.022112 arXiv:quant-ph/0210117 [113] Neill Lambert, Clive Emary, and Tobias Brandes. ``Entanglement and the Phase Transition in Single-Mode Superradiance''. Phys. Rev. Lett. 92, 073602 (2004). arXiv:quant-ph/0309027. https://doi.org/10.1103/PhysRevLett.92.073602 arXiv:quant-ph/0309027 [114] Julien Vidal, Guillaume Palacios, and Rémy Mosseri. ``Entanglement in a second-order quantum phase transition''. Phys. Rev. A 69, 022107 (2004). arXiv:cond-mat/0305573. https://doi.org/10.1103/PhysRevA.69.022107 arXiv:cond-mat/0305573 [115] N. Lambert, C. Emary, and T. Brandes. ``Entanglement and entropy in a spin-boson quantum phase transition''. Phys. Rev. A 71, 053804 (2005). arXiv:quant-ph/0405109. https://doi.org/10.1103/PhysRevA.71.053804 arXiv:quant-ph/0405109 [116] Thomas Barthel, Sébastien Dusuel, and Julien Vidal. ``Entanglement entropy beyond the free case''. Phys. Rev. Lett. 97, 220402 (2006). arXiv:cond-mat/0606436. https://doi.org/10.1103/PhysRevLett.97.220402 arXiv:cond-mat/0606436 [117] Julien Vidal, Sébastien Dusuel, and Thomas Barthel. ``Entanglement entropy in collective models''. J. Stat. Mech.: Theo. Exp. (2007). arXiv:cond-mat/0610833. https://doi.org/10.1088/1742-5468/2007/01/p01015 arXiv:cond-mat/0610833 [118] Román Orús. ``Universal Geometric Entanglement Close to Quantum Phase Transitions''. Phys. Rev. Lett. 100, 130502 (2008). arXiv:0711.2556. https://doi.org/10.1103/PhysRevLett.100.130502 arXiv:0711.2556 [119] Román Orús, Sébastien Dusuel, and Julien Vidal. ``Equivalence of Critical Scaling Laws for Many-Body Entanglement in the Lipkin-Meshkov-Glick Model''. Phys. Rev. Lett. 101, 025701 (2008). arXiv:0803.3151. https://doi.org/10.1103/PhysRevLett.101.025701 arXiv:0803.3151 [120] Román Orús and Tzu-Chieh Wei. ``Visualizing elusive phase transitions with geometric entanglement''. Phys. Rev. B 82 (2010). arXiv:0910.2488. https://doi.org/10.1103/PhysRevB.82.155120 arXiv:0910.2488 [121] Xiaoqian Wang, Jian Ma, Lijun Song, Xihe Zhang, and Xiaoguang Wang. ``Spin squeezing, negative correlations, and concurrence in the quantum kicked top model''. Phys. Rev. E 82, 056205 (2010). arXiv:1010.0109. https://doi.org/10.1103/PhysRevE.82.056205 arXiv:1010.0109 [122] Xiaolei Yin, Xiaoqian Wang, Jian Ma, and Xiaoguang Wang. ``Spin squeezing and concurrence''. J. Phys. B: At. Mol. Opt. Phys. 44, 015501 (2011). arXiv:0912.1752. https://doi.org/10.1088/0953-4075/44/1/015501 arXiv:0912.1752 [123] Philipp Krammer, Hermann Kampermann, Dagmar Bruß, Reinhold A. Bertlmann, Leong Chuang Kwek, and Chiara Macchiavello. ``Multipartite Entanglement Detection via Structure Factors''. Phys. Rev. Lett. 103, 100502 (2009). arXiv:0904.3860. https://doi.org/10.1103/PhysRevLett.103.100502 arXiv:0904.3860 [124] Marcus Cramer, Martin B. Plenio, and Harald Wunderlich. ``Measuring Entanglement in Condensed Matter Systems''. Phys. Rev. Lett. 106, 020401 (2011). arXiv:1009.2956. https://doi.org/10.1103/PhysRevLett.106.020401 arXiv:1009.2956 [125] Yumang Jing, Matteo Fadel, Valentin Ivannikov, and Tim Byrnes. ``Split spin-squeezed Bose–Einstein condensates''. New J. Phys. 21, 093038 (2019). arXiv:1808.10679. https://doi.org/10.1088/1367-2630/ab3fcf arXiv:1808.10679 [126] N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, Géza Tóth, R. J. Sewell, and M. W. Mitchell. ``Generation of Macroscopic Singlet States in a Cold Atomic Ensemble''. Phys. Rev. Lett. 113, 093601 (2014). arXiv:https://arxiv.org/abs/1403.1964. https://doi.org/10.1103/PhysRevLett.113.093601 arXiv:1403.1964 [127] Lars Dammeier, René Schwonnek, and Reinhard F Werner. ``Uncertainty relations for angular momentum''. New J. Phys. 17, 093046 (2015). arXiv:1505.00049. https://doi.org/10.1088/1367-2630/17/9/093046 arXiv:1505.00049 [128] William M. Kirby and Frederick W. Strauch. ``A practical quantum algorithm for the schur transform''. Quantum Inf. Comput. 18, 721–742 (2018). arXiv:1709.07119. https://doi.org/10.26421/qic18.9-10-1 arXiv:1709.07119 [129] Vladimir M. Akulin, Grigory A. Kabatiansky, and Aikaterini Mandilara. ``Essentially entangled component of multipartite mixed quantum states, its properties, and an efficient algorithm for its extraction''. Phys. Rev. A 92 (2015). arXiv:1504.06440. https://doi.org/10.1103/PhysRevA.92.042322 arXiv:1504.06440 [130] Thomas L. Curtright, Thomas S. Van Kortryk, and Cosmas K. Zachos. ``Spin multiplicities''. Phys. Lett. A 381, 422–427 (2017). arXiv:1607.05849. https://doi.org/10.1016/j.physleta.2016.12.006 arXiv:1607.05849 [131] Mark A. Caprio, Pavel Cejnar, and Francesco Iachello. ``Excited state quantum phase transitions in many-body systems''. Ann. Phys. 323, 1106–1135 (2008). arXiv:0707.0325. https://doi.org/10.1016/j.aop.2007.06.011 arXiv:0707.0325 [132] Robert J Lewis-Swan, Arghavan Safavi-Naini, John J Bollinger, and Ana M Rey. ``Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the dicke model''. Nat. Commun. 10, 1581 (2019). arXiv:1808.07134. https://doi.org/10.1038/s41467-019-09436-y arXiv:1808.07134 [133] Zeyang Li, Simone Colombo, Chi Shu, Gustavo Velez, Saúl Pilatowsky-Cameo, Roman Schmied, Soonwon Choi, Mikhail Lukin, Edwin Pedrozo-Peñafiel, and Vladan Vuletić. ``Improving metrology with quantum scrambling''. Science 380, 1381–1384 (2023). arXiv:2212.13880. https://doi.org/10.1126/science.adg9500 arXiv:2212.13880 [134] Daniel Cavalcanti. ``Connecting the generalized robustness and the geometric measure of entanglement''. Phys. Rev. A 73, 044302 (2006). arXiv:quant-ph/0602031. https://doi.org/10.1103/PhysRevA.73.044302 arXiv:quant-ph/0602031Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-21 11:41:45: Could not fetch cited-by data for 10.22331/q-2026-04-21-2078 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-21 11:41:46: 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. AbstractWe discuss the estimation of the distance of a given mixed many-body quantum state to the set of fully separable states, applied to the concrete scenario of collective spin states. Concretely, we discuss lower bounds to distances from the set of fully separable states based on entanglement criteria and upper bounds to those distances using an iterative algorithm to find the optimal separable state closest to the target. Focusing on collective states of $N$ spin-$1/2$ particles, we consider spin-squeezing inequalities (SSIs), which provide a complete set of nonlinear entanglement criteria based on collective spin variances. First, we find a lower bound to distance-based entanglement monotones, specifically the so-called best separable approximation (BSA) from the complete set of SSIs, thereby bypassing entirely a numerical optimization over a (potentially very large) set of linear entanglement witnesses. Then, we improve current algorithms to iteratively find the closest separable state to a given target state, exploiting the symmetry of the system. These results allow us to study entanglement quantitatively on thermal states of spin systems on fully-connected graphs at nonzero temperature, as well as potentially similar states arising in out-of-equilibrium situations. We thus apply our methods to investigate entanglement across different phases of a fully-connected XXZ model. We observe that our lower bound becomes often tight for zero temperature as well as for the temperature at which entanglement disappears, both of which are thus precisely captured by the SSIs. We further observe, among other things, that entanglement can arise at nonzero temperature even in the ordered phase, where the ground state is separable, revealing the potential usefulness of entanglement quantification also beyond the ground state paradigm.Featured image: We estimate the best separable approximation for an $N$-qubit state (top right), by computing lower and upper bounds (left). Lower bounds are computed from the negative expectation value of an entanglement witness W that is illustrated as blue lines not crossing the set of separable states (shaded ellipse). Upper bounds can be computed from the distance to any given separable state (red/purple lines), minimized over as many separable states as possible. Optimal bounds are illustrated as straight or dotted lines, non-optimal bounds as dashed lines. For states featuring symmetries, the problem can be formulated in a symmetric subspace (bottom right: example matrix for $N = 4$, blocks of different color are a result of permutational invariance, while diagonal matrices have an additional rotation symmetry). Popular summaryEntanglement in many-body systems is usually analyzed for pure ground states, but realistic systems are often mixed because of temperature, noise, or nonequilibrium dynamics. In such cases, even deciding whether a state is entangled can be difficult, let alone quantifying how much entanglement it contains. In this work, we study this problem for collective spin states by asking how far a given mixed state is from the set of fully separable states. This distance is quantified by the best separable approximation, which tells us how well the state can still be described by a classical-like mixture of unentangled particles. We derive a lower bound on this quantity from spin-squeezing inequalities built from standard collective observables, and an upper bound from an iterative algorithm that searches for the closest separable state while exploiting the symmetries of the system. Applying these tools to thermal states of fully connected spin models, we show that this method can provide insightful quantitative information about mixed-state entanglement across different phases. In particular, we find that entanglement may appear at nonzero temperature even in regimes where the ground state itself is separable, highlighting that relevant quantum correlations can emerge beyond the usual ground-state picture. Our results also suggest that entanglement quantification may refine the usual phase diagram by revealing a finer structure within conventional phases, where states with different amounts of entanglement define distinct regimes.► BibTeX data@article{Mathe2026estimatingbest, doi = {10.22331/q-2026-04-21-2078}, url = {https://doi.org/10.22331/q-2026-04-21-2078}, title = {Estimating the best separable approximation of non-pure spin-squeezed states}, author = {Math{\'{e}}, Julia and Usui, Ayaka and G{\"{u}}hne, Otfried and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2078}, month = apr, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). arXiv:quant-ph/0702225. https://doi.org/10.1103/RevModPhys.81.865 arXiv:quant-ph/0702225 [2] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. ``Entanglement in many-body systems''. Rev. Mod. Phys. 80, 517–576 (2008). arXiv:quant-ph/0703044. https://doi.org/10.1103/RevModPhys.80.517 arXiv:quant-ph/0703044 [3] Nicolas Laflorencie. ``Quantum entanglement in condensed matter systems''. Phys. Rep. 646, 1–59 (2016). arXiv:1512.03388. https://doi.org/10.1016/j.physrep.2016.06.008 arXiv:1512.03388 [4] Subir Sachdev. ``Quantum Phase Transitions''.
Cambridge University Press. (2011). https://doi.org/10.1017/CBO9780511973765 [5] Steven M. Girvin and Kun Yang. ``Modern Condensed Matter Physics''.
Cambridge University Press. (2019). https://doi.org/10.1017/9781316480649 [6] Ulrich Schollwöck. ``The density-matrix renormalization group''. Rev. Mod. Phys. 77, 259–315 (2005). arXiv:cond-mat/0409292. https://doi.org/10.1103/RevModPhys.77.259 arXiv:cond-mat/0409292 [7] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Ann. Phys. 326, 96–192 (2011). arXiv:1008.3477. https://doi.org/10.1016/j.aop.2010.09.012 arXiv:1008.3477 [8] Román Orús. ``Tensor networks for complex quantum systems''. Nat. Rev. Phys. 1, 538–550 (2019). arXiv:1812.04011. https://doi.org/10.1038/s42254-019-0086-7 arXiv:1812.04011 [9] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. Rep. 474, 1–75 (2009). arXiv:0811.2803. https://doi.org/10.1016/j.physrep.2009.02.004 arXiv:0811.2803 [10] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber. ``Entanglement certification from theory to experiment''. Nat. Rev. Phys. 1, 72–87 (2018). arXiv:1906.10929. https://doi.org/10.1038/s42254-018-0003-5 arXiv:1906.10929 [11] V. Vedral and M. B. Plenio. ``Entanglement measures and purification procedures''. Phys. Rev. A 57, 1619–1633 (1998). arXiv:quant-ph/9707035. https://doi.org/10.1103/physreva.57.1619 arXiv:quant-ph/9707035 [12] Martin B. Plenio and Shashank Virmani. ``An introduction to entanglement measures''. Quant. Inf. Comput. 7, 1–51 (2007). arXiv:quant-ph/0504163. https://doi.org/10.26421/QIC7.1-2-1 arXiv:quant-ph/0504163 [13] William K. Wootters. ``Entanglement of Formation of an Arbitrary State of Two Qubits''. Phys. Rev. Lett. 80, 2245–2248 (1998). arXiv:quant-ph/9709029. https://doi.org/10.1103/PhysRevLett.80.2245 arXiv:quant-ph/9709029 [14] Karl Gerd H. Vollbrecht and Reinhard F. Werner. ``Entanglement measures under symmetry''. Phys. Rev. A 64, 062307 (2001). arXiv:quant-ph/0010095. https://doi.org/10.1103/PhysRevA.64.062307 arXiv:quant-ph/0010095 [15] Armin Uhlmann. ``Fidelity and concurrence of conjugated states''. Phys. Rev. A 62, 032307 (2000). arXiv:quant-ph/9909060. https://doi.org/10.1103/PhysRevA.62.032307 arXiv:quant-ph/9909060 [16] Robert Lohmayer, Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Entangled Three-Qubit States without Concurrence and Three-Tangle''. Phys. Rev. Lett. 97, 260502 (2006). arXiv:quant-ph/0606071. https://doi.org/10.1103/PhysRevLett.97.260502 arXiv:quant-ph/0606071 [17] Christopher Eltschka, Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Three-tangle for mixtures of generalized GHZ and generalized W states''. New J. Phys. 10, 043014 (2008). arXiv:0711.4477. https://doi.org/10.1088/1367-2630/10/4/043014 arXiv:0711.4477 [18] Andreas Osterloh, Jens Siewert, and Armin Uhlmann. ``Tangles of superpositions and the convex-roof extension''. Phys. Rev. A 77, 032310 (2008). arXiv:0710.5909. https://doi.org/10.1103/PhysRevA.77.032310 arXiv:0710.5909 [19] Jens Siewert and Christopher Eltschka. ``Quantifying Tripartite Entanglement of Three-Qubit Generalized Werner States''. Phys. Rev. Lett. 108, 230502 (2012). https://doi.org/10.1103/PhysRevLett.108.230502 [20] Koenraad Audenaert, Jens Eisert, Martin B. Plenio, and Reinhard F. Werner. ``Entanglement properties of the harmonic chain''. Phys. Rev. A 66, 042327 (2002). arXiv:quant-ph/0205025. https://doi.org/10.1103/PhysRevA.66.042327 arXiv:quant-ph/0205025 [21] Jens Eisert, Marcus Cramer, and Martin B. Plenio. ``Colloquium: Area laws for the entanglement entropy''. Rev. Mod. Phys. 82, 277 (2010). arXiv:0808.3773. https://doi.org/10.1103/RevModPhys.82.277 arXiv:0808.3773 [22] Pasquale Calabrese, John Cardy, and Erik Tonni. ``Entanglement negativity in extended systems: A field theoretical approach''. J. Stat. Mech.: Theory Exp. 2013, P02008 (2013). arXiv:1210.5359. https://doi.org/10.1088/1742-5468/2013/02/P02008 arXiv:1210.5359 [23] Viktor Eisler and Zoltán Zimborás. ``Entanglement negativity in the harmonic chain out of equilibrium''. New J. Phys. 16, 123020 (2014). arXiv:1406.5474. https://doi.org/10.1088/1367-2630/16/12/123020 arXiv:1406.5474 [24] Pasquale Calabrese, John Cardy, and Erik Tonni. ``Finite temperature entanglement negativity in conformal field theory''. J. Phys. A: Math. Theor. 48, 015006 (2015). arXiv:1408.3043. https://doi.org/10.1088/1751-8113/48/1/015006 arXiv:1408.3043 [25] Viktor Eisler and Zoltán Zimborás. ``On the partial transpose of fermionic Gaussian states''. New J. Phys. 17, 053048 (2015). arXiv:1502.01369. https://doi.org/10.1088/1367-2630/17/5/053048 arXiv:1502.01369 [26] Viktor Eisler and Zoltán Zimborás. ``Entanglement negativity in two-dimensional free lattice models''. Phys. Rev. B 93, 115148 (2016). arXiv:1511.08819. https://doi.org/10.1103/PhysRevB.93.115148 arXiv:1511.08819 [27] Jens Eisert, Viktor Eisler, and Zoltán Zimborás. ``Entanglement negativity bounds for fermionic gaussian states''. Phys. Rev. B 97 (2018). arXiv:1611.08007. https://doi.org/10.1103/physrevb.97.165123 arXiv:1611.08007 [28] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera. ``Symmetric quantum states: a review of recent progress''. Rep. Prog. Phys. 89, 024001 (2025). arXiv:2506.10185. https://doi.org/10.1088/1361-6633/ae440a arXiv:2506.10185 [29] Alessandro Ferraro, Daniel Cavalcanti, Artur García-Saez, and Antonio Acín. ``Thermal Bound Entanglement in Macroscopic Systems and Area Law''. Phys. Rev. Lett. 100 (2008). arXiv:0804.4867. https://doi.org/10.1103/physrevlett.100.080502 arXiv:0804.4867 [30] Daniel Cavalcanti, Alessandro Ferraro, Artur García-Saez, and Antonio Acín. ``Distillable entanglement and area laws in spin and harmonic-oscillator systems''. Phys. Rev. A 78 (2008). arXiv:0705.3762. https://doi.org/10.1103/physreva.78.012335 arXiv:0705.3762 [31] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Spin squeezing and entanglement''. Phys. Rev. A 79, 042334 (2009). arXiv:0806.1048. https://doi.org/10.1103/PhysRevA.79.042334 arXiv:0806.1048 [32] Giuseppe Vitagliano, Otfried Gühne, and Géza Tóth. ``$su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems''. Quantum 9, 1844 (2025). arXiv:2406.13338. https://doi.org/10.22331/q-2025-09-03-1844 arXiv:2406.13338 [33] Fernando G. S. L. Brandão. ``Quantifying entanglement with witness operators''. Phys. Rev. A 72, 022310 (2005). arXiv:quant-ph/0503152. https://doi.org/10.1103/PhysRevA.72.022310 arXiv:quant-ph/0503152 [34] Koenraad M. R. Audenaert and Martin B. Plenio. ``When are correlations quantum?—verification and quantification of entanglement by simple measurements''. New J. Phys. 8, 266–266 (2006). arXiv:quant-ph/0608067. https://doi.org/10.1088/1367-2630/8/11/266 arXiv:quant-ph/0608067 [35] Jens Eisert, Fernando G. S. L. Brandão, and Koenraad M. R. Audenaert. ``Quantitative entanglement witnesses''. New J. Phys. 9, 46 (2007). arXiv:quant-ph/0607167. https://doi.org/10.1088/1367-2630/9/3/046 arXiv:quant-ph/0607167 [36] Otfried Gühne, Michael Reimpell, and Reinhard F. Werner. ``Estimating Entanglement Measures in Experiments''. Phys. Rev. Lett. 98, 110502 (2007). arXiv:quant-ph/0607163. https://doi.org/10.1103/PhysRevLett.98.110502 arXiv:quant-ph/0607163 [37] Otfried Gühne, Michael Reimpell, and Reinhard F. Werner. ``Lower bounds on entanglement measures from incomplete information''. Phys. Rev. A 77, 052317 (2008). arXiv:0802.1734. https://doi.org/10.1103/PhysRevA.77.052317 arXiv:0802.1734 [38] Matteo Fadel, Ayaka Usui, Marcus Huber, Nicolai Friis, and Giuseppe Vitagliano. ``Entanglement Quantification in Atomic Ensembles''. Phys. Rev. Lett. 127, 010401 (2021). arXiv:2103.15730. https://doi.org/10.1103/PhysRevLett.127.010401 arXiv:2103.15730 [39] Géza Tóth. ``Entanglement witnesses in spin models''. Phys. Rev. A 71, 010301 (2005). arXiv:quant-ph/0406061. https://doi.org/10.1103/PhysRevA.71.010301 arXiv:quant-ph/0406061 [40] Janet Anders. ``Thermal state entanglement in harmonic lattices''. Phys. Rev. A 77, 062102 (2008). arXiv:0803.1102. https://doi.org/10.1103/PhysRevA.77.062102 arXiv:0803.1102 [41] Oliver Marty, Michael Epping, Hermann Kampermann, Dagmar Bruß, Martin B. Plenio, and M. Cramer. ``Quantifying entanglement with scattering experiments''. Phys. Rev. B 89, 125117 (2014). arXiv:1310.0929. https://doi.org/10.1103/PhysRevB.89.125117 arXiv:1310.0929 [42] Oliver Marty, Marcus Cramer, and Martin B. Plenio. ``Practical Entanglement Estimation for Spin-System Quantum Simulators''. Phys. Rev. Lett. 116, 105301 (2016). arXiv:1504.03572. https://doi.org/10.1103/PhysRevLett.116.105301 arXiv:1504.03572 [43] Silvia Pappalardi, Angelo Russomanno, Alessandro Silva, and Rosario Fazio. ``Multipartite entanglement after a quantum quench''. J. Stat. Mech.: Theory Exp. 2017, 053104 (2017). arXiv:1701.05883. https://doi.org/10.1088/1742-5468/aa6809 arXiv:1701.05883 [44] Alessio Lerose and Silvia Pappalardi. ``Bridging entanglement dynamics and chaos in semiclassical systems''. Phys. Rev. A 102 (2020). arXiv:2005.03670. https://doi.org/10.1103/physreva.102.032404 arXiv:2005.03670 [45] Ferenc Iglói and Géza Tóth. ``Entanglement witnesses in the $xy$ chain: Thermal equilibrium and postquench nonequilibrium states''. Phys. Rev. Res. 5, 013158 (2023). arXiv:2207.04842. https://doi.org/10.1103/PhysRevResearch.5.013158 arXiv:2207.04842 [46] Giacomo Mazza and Costantino Budroni. ``Entanglement detection in quantum materials with competing orders''. Phys. Rev. B 111 (2025). arXiv:2404.12931. https://doi.org/10.1103/physrevb.111.l100302 arXiv:2404.12931 [47] Mark R. Dowling, Andrew C. Doherty, and Stephen D. Bartlett. ``Energy as an entanglement witness for quantum many-body systems''. Phys. Rev. A 70, 062113 (2004). arXiv:quant-ph/0408086. https://doi.org/10.1103/PhysRevA.70.062113 arXiv:quant-ph/0408086 [48] Janet Anders, Dagomir Kaszlikowski, Christian Lunkes, Toshio Ohshima, and Vlatko Vedral. ``Detecting entanglement with a thermometer''. New J. Phys. 8, 140 (2006). arXiv:quant-ph/0512181. https://doi.org/10.1088/1367-2630/8/8/140 arXiv:quant-ph/0512181 [49] Marcin Wieśniak, Vlatko Vedral, and Časlav Brukner. ``Heat capacity as an indicator of entanglement''. Phys. Rev. B 78, 064108 (2008). arXiv:quant-ph/0508193. https://doi.org/10.1103/PhysRevB.78.064108 arXiv:quant-ph/0508193 [50] Marcin Wieśniak, Vlatko Vedral, and Časlav Brukner. ``Magnetic susceptibility as a macroscopic entanglement witness''. New J. Phys. 7, 258 (2005). arXiv:quant-ph/0503037. https://doi.org/10.1088/1367-2630/7/1/258 arXiv:quant-ph/0503037 [51] Časlav Brukner, Vlatko Vedral, and Anton Zeilinger. ``Crucial role of quantum entanglement in bulk properties of solids''. Phys. Rev. A 73, 012110 (2006). arXiv:quant-ph/0410138. https://doi.org/10.1103/PhysRevA.73.012110 arXiv:quant-ph/0410138 [52] Philipp Krammer, Hermann Kampermann, Dagmar Bruß, Reinhold A. Bertlmann, Leong Chuang Kwek, and Chiara Macchiavello. ``Multipartite entanglement detection via structure factors''. Phys. Rev. Lett. 103, 100502 (2009). arXiv:0904.3860. https://doi.org/10.1103/PhysRevLett.103.100502 arXiv:0904.3860 [53] Philipp Hauke, Markus Heyl, Luca Tagliacozzo, and Peter Zoller. ``Measuring multipartite entanglement through dynamic susceptibilities''. Nat. Phys. 12, 778 – 782 (2016). arXiv:1509.01739. https://doi.org/10.1038/nphys3700 arXiv:1509.01739 [54] Otfried Gühne and Géza Tóth. ``Energy and multipartite entanglement in multidimensional and frustrated spin models''. Phys. Rev. A 73, 052319 (2006). arXiv:quant-ph/0510186. https://doi.org/10.1103/PhysRevA.73.052319 arXiv:quant-ph/0510186 [55] Pedro Ribeiro, Julien Vidal, and Rémy Mosseri. ``Thermodynamical Limit of the Lipkin-Meshkov-Glick Model''. Phys. Rev. Lett. 99 (2007). arXiv:cond-mat/0703490. https://doi.org/10.1103/physrevlett.99.050402 arXiv:cond-mat/0703490 [56] Jian Ma, Xiaoguang Wang, C. P. Sun, and Franco Nori. ``Quantum spin squeezing''. Phys. Rep. 509, 89–165 (2011). arXiv:1011.2978. https://doi.org/10.1016/j.physrep.2011.08.003 arXiv:1011.2978 [57] Nicolò Defenu, Alessio Lerose, and Silvia Pappalardi. ``Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions''. Phys. Rep. 1074, 1–92 (2024). arXiv:2307.04802. https://doi.org/10.1016/j.physrep.2024.04.005 arXiv:2307.04802 [58] Julien Vidal, Rémy Mosseri, and Jorge Dukelsky. ``Entanglement in a first-order quantum phase transition''. Phys. Rev. A 69, 054101 (2004). arXiv:cond-mat/0312130. https://doi.org/10.1103/PhysRevA.69.054101 arXiv:cond-mat/0312130 [59] Charlie-Ray Mann, Mark A. Oehlgrien, Błażej Jaworowski, Giuseppe Calajó, Jamir Marino, Kyung S. Choi, and Darrick E. Chang. ``Squeezing Classical Antiferromagnets into Quantum Spin Liquids via Global Cavity Fluctuations'' (2025). arXiv:2512.05630. arXiv:2512.05630 [60] Masahiro Kitagawa and Masahito Ueda. ``Squeezed spin states''. Phys. Rev. A 47, 5138–5143 (1993). https://doi.org/10.1103/PhysRevA.47.5138 [61] Anders Sørensen, Lu-Ming Duan, J. Ignacio Cirac, and Peter Zoller. ``Many-particle entanglement with Bose-Einstein condensates''. Nature 409, 63–66 (2001). arXiv:quant-ph/0006111. https://doi.org/10.1038/35051038 arXiv:quant-ph/0006111 [62] Anders Sørensen and Klaus Mølmer. ``Entanglement and Extreme Spin Squeezing''. Phys. Rev. Lett. 86 (2001). arXiv:quant-ph/0011035. https://doi.org/10.1103/PhysRevLett.86.4431 arXiv:quant-ph/0011035 [63] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Optimal Spin Squeezing Inequalities Detect Bound Entanglement in Spin Models''. Phys. Rev. Lett. 99, 250405 (2007). arXiv:quant-ph/0702219. https://doi.org/10.1103/PhysRevLett.99.250405 arXiv:quant-ph/0702219 [64] Giuseppe Vitagliano, Philipp Hyllus, Inigo L. Egusquiza, and Géza Tóth. ``Spin Squeezing Inequalities for Arbitrary Spin''. Phys. Rev. Lett. 107, 240502 (2011). arXiv:1104.3147. https://doi.org/10.1103/PhysRevLett.107.240502 arXiv:1104.3147 [65] Giuseppe Vitagliano, Iagoba Apellaniz, Inigo L. Egusquiza, and Géza Tóth. ``Spin squeezing and entanglement for an arbitrary spin''. Phys. Rev. A 89, 032307 (2014). arXiv:1310.2269. https://doi.org/10.1103/PhysRevA.89.032307 arXiv:1310.2269 [66] Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, and Géza Tóth. ``Entanglement and extreme spin squeezing of unpolarized states''. New J. Phys. 19, 013027 (2017). arXiv:1605.07202. https://doi.org/10.1088/1367-2630/19/1/013027 arXiv:1605.07202 [67] Oliver Marty, Marcus Cramer, Giuseppe Vitagliano, Géza Tóth, and Martin B. Plenio. ``Multiparticle entanglement criteria for nonsymmetric collective variances'' (2017). arXiv:1708.06986. arXiv:1708.06986 [68] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018). arXiv:1609.01609. https://doi.org/10.1103/RevModPhys.90.035005 arXiv:1609.01609 [69] Tilo Eggeling and Reinhard F. Werner. ``Separability properties of tripartite states with $U{\bigotimes}U{\bigotimes}U$ symmetry''. Phys. Rev. A 63, 042111 (2001). arXiv:quant-ph/0010096. https://doi.org/10.1103/PhysRevA.63.042111 arXiv:quant-ph/0010096 [70] Vlatko Vedral, Martin B. Plenio, Marcus A. Rippin, and Peter L. Knight. ``Quantifying entanglement''. Phys. Rev. Lett. 78, 2275–2279 (1997). arXiv:quant-ph/9702027. https://doi.org/10.1103/physrevlett.78.2275 arXiv:quant-ph/9702027 [71] Alexander Streltsov, Hermann Kampermann, and Dagmar Bruß. ``Linking a distance measure of entanglement to its convex roof''. New J. Phys. 12, 123004 (2010). arXiv:1006.3077. https://doi.org/10.1088/1367-2630/12/12/123004 arXiv:1006.3077 [72] Maciej Lewenstein and Anna Sanpera. ``Separability and Entanglement of Composite Quantum Systems''. Phys. Rev. Lett. 80, 2261–2264 (1998). arXiv:quant-ph/9707043. https://doi.org/10.1103/PhysRevLett.80.2261 arXiv:quant-ph/9707043 [73] Sinisa Karnas and Maciej Lewenstein. ``Separable approximations of density matrices of composite quantum systems''. J. Phys. A: Math. Gen. 34, 6919–6937 (2001). arXiv:quant-ph/0011066. https://doi.org/10.1088/0305-4470/34/35/318 arXiv:quant-ph/0011066 [74] Michael Steiner. ``Generalized robustness of entanglement''. Phys. Rev. A 67, 054305 (2003). arXiv:quant-ph/0304009. https://doi.org/10.1103/PhysRevA.67.054305 arXiv:quant-ph/0304009 [75] Ayatola Gabdulin and Aikaterini Mandilara. ``Investigating bound entangled two-qutrit states via the best separable approximation''. Phys. Rev. A 100, 062322 (2019). arXiv:1906.08963. https://doi.org/10.1103/PhysRevA.100.062322 arXiv:1906.08963 [76] Antoine Girardin, Nicolas Brunner, and Tamás Kriváchy. ``Building separable approximations for quantum states via neural networks''. Phys. Rev. Res. 4, 023238 (2022). arXiv:2112.08055. https://doi.org/10.1103/PhysRevResearch.4.023238 arXiv:2112.08055 [77] Marcus Cramer, Alain Bernard, Nicole Fabbri, Leonardo Fallani, Chiara Fort, Sara Rosi, Filippo Caruso, Massimo Inguscio, and Martin B. Plenio. ``Spatial entanglement of bosons in optical lattices''. Nat. Commun. 4, 2161 (2013). arXiv:1302.4897. https://doi.org/10.1038/ncomms3161 arXiv:1302.4897 [78] Robert H. Dicke. ``Coherence in spontaneous radiation processes''. Phys. Rev. 93, 99–110 (1954). https://doi.org/10.1103/PhysRev.93.99 [79] Peter Kirton, Mor M. Roses, Jonathan Keeling, and Emanuele G. Dalla Torre. ``Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa''. Adv. Quantum Technol. 2 (2018). arXiv:1805.09828. https://doi.org/10.1002/qute.201800043 arXiv:1805.09828 [80] Harry J. Lipkin, N. Meshkov, and Arnold J. Glick. ``Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory''. Nucl. Phys. 62, 188–198 (1965). https://doi.org/10.1016/0029-5582(65)90862-X [81] N. Meshkov, Arnold J. Glick, and Harry J. Lipkin. ``Validity of many-body approximation methods for a solvable model: (II). Linearization procedures''. Nucl. Phys. 62, 199–210 (1965). https://doi.org/10.1016/0029-5582(65)90863-1 [82] Arnold J. Glick, Harry J. Lipkin, and N. Meshkov. ``Validity of many-body approximation methods for a solvable model: (III). Diagram summations''. Nucl. Phys. 62, 211–224 (1965). https://doi.org/10.1016/0029-5582(65)90864-3 [83] David J. Wineland, John J. Bollinger, Wayne M. Itano, F. L. Moore, and Daniel J. Heinzen. ``Spin squeezing and reduced quantum noise in spectroscopy''. Phys. Rev. A 46, R6797 (1992). https://doi.org/10.1103/PhysRevA.46.R6797 [84] David J. Wineland, John J. Bollinger, Wayne M. Itano, and Daniel J. Heinzen. ``Squeezed atomic states and projection noise in spectroscopy''. Phys. Rev. A 50, 67–88 (1994). https://doi.org/10.1103/PhysRevA.50.67 [85] Julien Vidal. ``Concurrence in collective models''. Phys. Rev. A 73, 062318 (2006). arXiv:quant-ph/0603108. https://doi.org/10.1103/PhysRevA.73.062318 arXiv:quant-ph/0603108 [86] Erling Størmer. ``Symmetric states of infinite tensor products of $C*$-algebras''. J. Funct. Anal. 3, 48–68 (1969). https://doi.org/10.1016/0022-1236(69)90050-0 [87] Robin L. Hudson and Graham R. Moody. ``Locally normal symmetric states and an analogue of de Finetti's theorem''. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33, 343–351 (1976). https://doi.org/10.1007/BF00534784 [88] Mark Fannes, John T. Lewis, and André Verbeure. ``Symmetric states of composite systems''. Lett. Math. Phys. 15, 255–260 (1988). https://doi.org/10.1007/BF00398595 [89] Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack. ``Unknown quantum states: The quantum de Finetti representation''. J. Math. Phys. 43, 4537–4559 (2002). arXiv:quant-ph/0104088. https://doi.org/10.1063/1.1494475 arXiv:quant-ph/0104088 [90] Robert König and Renato Renner. ``A de Finetti representation for finite symmetric quantum states''. J. Math. Phys. 46 (2005). arXiv:quant-ph/0410229. https://doi.org/10.1063/1.2146188 arXiv:quant-ph/0410229 [91] Matthias Christandl, Robert König, Graeme Mitchison, and Renato Renner. ``One-and-a-Half Quantum de Finetti Theorems''. Commun. Math. Phys. 273 (2007). arXiv:quant-ph/0602130. https://doi.org/10.1007/s00220-007-0189-3 arXiv:quant-ph/0602130 [92] Robert König and Graeme Mitchison. ``A most compendious and facile quantum de Finetti theorem''. J. Math. Phys. 50 (2009). arXiv:quant-ph/0703210. https://doi.org/10.1063/1.3049751 arXiv:quant-ph/0703210 [93] Friederike Trimborn, Reinhard F. Werner, and Dirk Witthaut. ``Quantum de Finetti theorems and mean-field theory from quantum phase space representations''. J. Phys. A: Math Theo 49, 135302 (2016). https://doi.org/10.1088/1751-8113/49/13/135302 [94] Jarosław. K. Korbicz, Ignacio J. Cirac, and Maciej Lewenstein. ``Spin Squeezing Inequalities and Entanglement of $N$ Qubit States''. Phys. Rev. Lett. 95, 120502 (2005). arXiv:quant-ph/0504005. https://doi.org/10.1103/PhysRevLett.95.120502 arXiv:quant-ph/0504005 [95] Jarosław. K. Korbicz, Otfried Gühne, Maciej Lewenstein, H. Häffner, Christian F. Roos, and Rainer Blatt. ``Generalized spin-squeezing inequalities in $n$-qubit systems: Theory and experiment''. Phys. Rev. A 74, 052319 (2006). arXiv:quant-ph/0601038. https://doi.org/10.1103/PhysRevA.74.052319 arXiv:quant-ph/0601038 [96] Miguel Navascués, Masaki Owari, and Martin B. Plenio. ``Complete criterion for separability detection''. Phys. Rev. Lett. 103, 160404 (2009). arXiv:0906.2735. https://doi.org/10.1103/PhysRevLett.103.160404 arXiv:0906.2735 [97] Julio T Barreiro, Philipp Schindler, Otfried Gühne, Thomas Monz, Michael Chwalla, Christian F Roos, Markus Hennrich, and Rainer Blatt. ``Experimental multiparticle entanglement dynamics induced by decoherence''. Nat. Phys. 6, 943–946 (2010). arXiv:1005.1965. https://doi.org/10.1038/nphys1781 arXiv:1005.1965 [98] Alastair Kay. ``Optimal detection of entanglement in Greenberger-Horne-Zeilinger states''. Phys. Rev. A 83, 020303 (2011). arXiv:1006.5197. https://doi.org/10.1103/PhysRevA.83.020303 arXiv:1006.5197 [99] Otfried Gühne. ``Entanglement criteria and full separability of multi-qubit quantum states''. Phys. Lett. A 375, 406–410 (2011). arXiv:1009.3782. https://doi.org/10.1016/j.physleta.2010.11.032 arXiv:1009.3782 [100] Hermann Kampermann, Otfried Gühne, Colin Wilmott, and Dagmar Bruß. ``Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle entanglement''. Phys. Rev. A 86, 032307 (2012). arXiv:1203.5872. https://doi.org/10.1103/PhysRevA.86.032307 arXiv:1203.5872 [101] Stephen Brierley, Miguel Navascues, and Tamas Vertesi. ``Convex separation from convex optimization for large-scale problems'' (2017). arXiv:1609.05011. arXiv:1609.05011 [102] Jiangwei Shang and Otfried Gühne. ``Convex Optimization over Classes of Multiparticle Entanglement''. Phys. Rev. Lett. 120, 050506 (2018). arXiv:1707.02958. https://doi.org/10.1103/PhysRevLett.120.050506 arXiv:1707.02958 [103] Ties-A. Ohst, Xiao-Dong Yu, Otfried Gühne, and H. Chau Nguyen. ``Certifying quantum separability with adaptive polytopes''. SciPost Phys. 16, 063 (2024). arXiv:2210.10054. https://doi.org/10.21468/SciPostPhys.16.3.063 arXiv:2210.10054 [104] Leonid Gurvits and Howard Barnum. ``Largest separable balls around the maximally mixed bipartite quantum state''. Phys. Rev. A 66, 062311 (2002). arXiv:quant-ph/0204159. https://doi.org/10.1103/PhysRevA.66.062311 arXiv:quant-ph/0204159 [105] S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack. ``Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing''. Phys. Rev. Lett. 83, 1054–1057 (1999). arXiv:quant-ph/9811018. https://doi.org/10.1103/PhysRevLett.83.1054 arXiv:quant-ph/9811018 [106] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. ``Volume of the set of separable states''. Phys. Rev. A 58, 883–892 (1998). arXiv:quant-ph/9804024. https://doi.org/10.1103/PhysRevA.58.883 arXiv:quant-ph/9804024 [107] José I. Latorre, Román Orús, Enrique Rico, and Julien Vidal. ``Entanglement entropy in the Lipkin-Meshkov-Glick model''. Phys. Rev. A 71 (2005). arXiv:cond-mat/0409611. https://doi.org/10.1103/PhysRevA.71.064101 arXiv:cond-mat/0409611 [108] Julien Vidal, Guillaume Palacios, and Claude Aslangul. ``Entanglement dynamics in the Lipkin-Meshkov-Glick model''. Phys. Rev. A 70, 062304 (2004). arXiv:cond-mat/0406481. https://doi.org/10.1103/PhysRevA.70.062304 arXiv:cond-mat/0406481 [109] Román Orús. ``Geometric entanglement in a one-dimensional valence-bond solid state''. Phys. Rev. A 78, 062332 (2008). arXiv:0808.0938. https://doi.org/10.1103/PhysRevA.78.062332 arXiv:0808.0938 [110] Xiaoguang Wang and Klaus Mølmer. ``Pairwise entanglement in symmetric multi-qubit systems''. Eur. Phys. J. D 18, 385–391 (2002). arXiv:quant-ph/0106145. https://doi.org/10.1140/epjd/e20020045 arXiv:quant-ph/0106145 [111] Xiaoguang Wang and Barry C. Sanders. ``Spin squeezing and pairwise entanglement for symmetric multiqubit states''. Phys. Rev. A 68, 012101 (2003). arXiv:quant-ph/0302014. https://doi.org/10.1103/PhysRevA.68.012101 arXiv:quant-ph/0302014 [112] John K. Stockton, JM Geremia, Andrew C. Doherty, and Hideo Mabuchi. ``Characterizing the entanglement of symmetric many-particle spin-1/2 systems''. Phys. Rev. A 67, 022112 (2003). arXiv:quant-ph/0210117. https://doi.org/10.1103/PhysRevA.67.022112 arXiv:quant-ph/0210117 [113] Neill Lambert, Clive Emary, and Tobias Brandes. ``Entanglement and the Phase Transition in Single-Mode Superradiance''. Phys. Rev. Lett. 92, 073602 (2004). arXiv:quant-ph/0309027. https://doi.org/10.1103/PhysRevLett.92.073602 arXiv:quant-ph/0309027 [114] Julien Vidal, Guillaume Palacios, and Rémy Mosseri. ``Entanglement in a second-order quantum phase transition''. Phys. Rev. A 69, 022107 (2004). arXiv:cond-mat/0305573. https://doi.org/10.1103/PhysRevA.69.022107 arXiv:cond-mat/0305573 [115] N. Lambert, C. Emary, and T. Brandes. ``Entanglement and entropy in a spin-boson quantum phase transition''. Phys. Rev. A 71, 053804 (2005). arXiv:quant-ph/0405109. https://doi.org/10.1103/PhysRevA.71.053804 arXiv:quant-ph/0405109 [116] Thomas Barthel, Sébastien Dusuel, and Julien Vidal. ``Entanglement entropy beyond the free case''. Phys. Rev. Lett. 97, 220402 (2006). arXiv:cond-mat/0606436. https://doi.org/10.1103/PhysRevLett.97.220402 arXiv:cond-mat/0606436 [117] Julien Vidal, Sébastien Dusuel, and Thomas Barthel. ``Entanglement entropy in collective models''. J. Stat. Mech.: Theo. Exp. (2007). arXiv:cond-mat/0610833. https://doi.org/10.1088/1742-5468/2007/01/p01015 arXiv:cond-mat/0610833 [118] Román Orús. ``Universal Geometric Entanglement Close to Quantum Phase Transitions''. Phys. Rev. Lett. 100, 130502 (2008). arXiv:0711.2556. https://doi.org/10.1103/PhysRevLett.100.130502 arXiv:0711.2556 [119] Román Orús, Sébastien Dusuel, and Julien Vidal. ``Equivalence of Critical Scaling Laws for Many-Body Entanglement in the Lipkin-Meshkov-Glick Model''. Phys. Rev. Lett. 101, 025701 (2008). arXiv:0803.3151. https://doi.org/10.1103/PhysRevLett.101.025701 arXiv:0803.3151 [120] Román Orús and Tzu-Chieh Wei. ``Visualizing elusive phase transitions with geometric entanglement''. Phys. Rev. B 82 (2010). arXiv:0910.2488. https://doi.org/10.1103/PhysRevB.82.155120 arXiv:0910.2488 [121] Xiaoqian Wang, Jian Ma, Lijun Song, Xihe Zhang, and Xiaoguang Wang. ``Spin squeezing, negative correlations, and concurrence in the quantum kicked top model''. Phys. Rev. E 82, 056205 (2010). arXiv:1010.0109. https://doi.org/10.1103/PhysRevE.82.056205 arXiv:1010.0109 [122] Xiaolei Yin, Xiaoqian Wang, Jian Ma, and Xiaoguang Wang. ``Spin squeezing and concurrence''. J. Phys. B: At. Mol. Opt. Phys. 44, 015501 (2011). arXiv:0912.1752. https://doi.org/10.1088/0953-4075/44/1/015501 arXiv:0912.1752 [123] Philipp Krammer, Hermann Kampermann, Dagmar Bruß, Reinhold A. Bertlmann, Leong Chuang Kwek, and Chiara Macchiavello. ``Multipartite Entanglement Detection via Structure Factors''. Phys. Rev. Lett. 103, 100502 (2009). arXiv:0904.3860. https://doi.org/10.1103/PhysRevLett.103.100502 arXiv:0904.3860 [124] Marcus Cramer, Martin B. Plenio, and Harald Wunderlich. ``Measuring Entanglement in Condensed Matter Systems''. Phys. Rev. Lett. 106, 020401 (2011). arXiv:1009.2956. https://doi.org/10.1103/PhysRevLett.106.020401 arXiv:1009.2956 [125] Yumang Jing, Matteo Fadel, Valentin Ivannikov, and Tim Byrnes. ``Split spin-squeezed Bose–Einstein condensates''. New J. Phys. 21, 093038 (2019). arXiv:1808.10679. https://doi.org/10.1088/1367-2630/ab3fcf arXiv:1808.10679 [126] N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, Géza Tóth, R. J. Sewell, and M. W. Mitchell. ``Generation of Macroscopic Singlet States in a Cold Atomic Ensemble''. Phys. Rev. Lett. 113, 093601 (2014). arXiv:https://arxiv.org/abs/1403.1964. https://doi.org/10.1103/PhysRevLett.113.093601 arXiv:1403.1964 [127] Lars Dammeier, René Schwonnek, and Reinhard F Werner. ``Uncertainty relations for angular momentum''. New J. Phys. 17, 093046 (2015). arXiv:1505.00049. https://doi.org/10.1088/1367-2630/17/9/093046 arXiv:1505.00049 [128] William M. Kirby and Frederick W. Strauch. ``A practical quantum algorithm for the schur transform''. Quantum Inf. Comput. 18, 721–742 (2018). arXiv:1709.07119. https://doi.org/10.26421/qic18.9-10-1 arXiv:1709.07119 [129] Vladimir M. Akulin, Grigory A. Kabatiansky, and Aikaterini Mandilara. ``Essentially entangled component of multipartite mixed quantum states, its properties, and an efficient algorithm for its extraction''. Phys. Rev. A 92 (2015). arXiv:1504.06440. https://doi.org/10.1103/PhysRevA.92.042322 arXiv:1504.06440 [130] Thomas L. Curtright, Thomas S. Van Kortryk, and Cosmas K. Zachos. ``Spin multiplicities''. Phys. Lett. A 381, 422–427 (2017). arXiv:1607.05849. https://doi.org/10.1016/j.physleta.2016.12.006 arXiv:1607.05849 [131] Mark A. Caprio, Pavel Cejnar, and Francesco Iachello. ``Excited state quantum phase transitions in many-body systems''. Ann. Phys. 323, 1106–1135 (2008). arXiv:0707.0325. https://doi.org/10.1016/j.aop.2007.06.011 arXiv:0707.0325 [132] Robert J Lewis-Swan, Arghavan Safavi-Naini, John J Bollinger, and Ana M Rey. ``Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the dicke model''. Nat. Commun. 10, 1581 (2019). arXiv:1808.07134. https://doi.org/10.1038/s41467-019-09436-y arXiv:1808.07134 [133] Zeyang Li, Simone Colombo, Chi Shu, Gustavo Velez, Saúl Pilatowsky-Cameo, Roman Schmied, Soonwon Choi, Mikhail Lukin, Edwin Pedrozo-Peñafiel, and Vladan Vuletić. ``Improving metrology with quantum scrambling''. Science 380, 1381–1384 (2023). arXiv:2212.13880. https://doi.org/10.1126/science.adg9500 arXiv:2212.13880 [134] Daniel Cavalcanti. ``Connecting the generalized robustness and the geometric measure of entanglement''. Phys. Rev. A 73, 044302 (2006). arXiv:quant-ph/0602031. https://doi.org/10.1103/PhysRevA.73.044302 arXiv:quant-ph/0602031Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-21 11:41:45: Could not fetch cited-by data for 10.22331/q-2026-04-21-2078 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-21 11:41:46: 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.