Alternatives of entanglement depth and metrological entanglement criteria

Szilárd Szalay; Géza Tóth

doi:10.22331/q-2025-04-18-1718

Alternatives of entanglement depth and metrological entanglement criteria

Szilárd Szalay^1,2,3 and Géza Tóth^2,3,4,5,1

¹HUN-REN Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary
²Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain
³EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, E-48940 Leioa, Biscay, Spain
⁴IKERBASQUE, Basque Foundation for Science, E-48009 Bilbao, Spain
⁵Donostia International Physics Center DIPC, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain

Published:	2025-04-18, volume 9, page 1718
Editor:	Aaron Goldberg
Eprint:	arXiv:2408.15350v3
Doi:	https://doi.org/10.22331/q-2025-04-18-1718
Citation:	Quantum 9, 1718 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We work out the general theory of one-parameter families of partial entanglement properties and the resulting entanglement depth-like quantities. Special cases of these are the depth of partitionability, the depth of producibility (or simply entanglement depth) and the depth of stretchability, which are based on one-parameter families of partial entanglement properties known earlier. We also construct some further physically meaningful properties, for instance the squareability, the toughness, the degree of freedom, and also several ones of entropic motivation. Metrological multipartite entanglement criteria with the quantum Fisher information fit naturally into this framework. Here we formulate these for the depth of squareability, which therefore turns out to be the natural choice, leading to stronger bounds than the usual entanglement depth. Namely, the quantum Fisher information turns out to provide a lower bound not only on the maximal size of entangled subsystems, but also on the average size of entangled subsystems for a random choice of elementary subsystems. We also formulate criteria with convex quantities for both cases, which are much stronger than the original ones. In particular, the quantum Fisher information puts a lower bound on the average size of entangled subsystems. We also argue that one-parameter partial entanglement properties, which carry entropic meaning, are more suitable for the purpose of defining metrological bounds.

Featured image: The structure of the new multipartite entanglement criteria.

Popular summary

We formulate new metrological multipartite entanglement criteria by the generalization of the entanglement depth. In particular, the quantum Fisher information puts a lower bound not only on the size of the largest entangled subsystems but also on the average size of entangled subsystems.

► BibTeX data

@article{Szalay2025alternativesof,
  doi = {10.22331/q-2025-04-18-1718},
  url = {https://doi.org/10.22331/q-2025-04-18-1718},
  title = {Alternatives of entanglement depth and metrological entanglement criteria},
  author = {Szalay, Szil{\'{a}}rd and T{\'{o}}th, G{\'{e}}za},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1718},
  month = apr,
  year = {2025}
}

► References

[1] Anders S. Sørensen and Klaus Mølmer. ``Entanglement and extreme spin squeezing''. Physical Review Letters 86, 4431–4434 (2001).
https://doi.org/10.1103/PhysRevLett.86.4431

[2] Otfried Gühne, Géza Tóth, and Hans J. Briegel. ``Multipartite entanglement in spin chains''. New Journal of Physics 7, 229 (2005).
https://doi.org/10.1088/1367-2630/7/1/229

[3] Bernd Lücke, Jan Peise, Giuseppe Vitagliano, Jan Arlt, Luis Santos, Géza Tóth, and Carsten Klempt. ``Detecting multiparticle entanglement of Dicke states''. Physical Review Letters 112, 155304 (2014).
https://doi.org/10.1103/PhysRevLett.112.155304

[4] Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, and Géza Tóth. ``Entanglement and extreme spin squeezing of unpolarized states''. New Journal of Physics 19, 013027 (2017).
https://doi.org/10.1088/1367-2630/19/1/013027

[5] Giuseppe Vitagliano, Giorgio Colangelo, Ferran Martin Ciurana, Morgan W. Mitchell, Robert J. Sewell, and Géza Tóth. ``Entanglement and extreme planar spin squeezing''. Physical Review A 97, 020301 (2018).
https://doi.org/10.1103/PhysRevA.97.020301

[6] Bastian Jungnitsch, Tobias Moroder, and Otfried Gühne. ``Taming multiparticle entanglement''. Physical Review Letters 106, 190502 (2011).
https://doi.org/10.1103/PhysRevLett.106.190502

[7] Liang-Liang Sun, Xiang Zhou, Armin Tavakoli, Zhen-Peng Xu, and Sixia Yu. ``Bounding the amount of entanglement from witness operators''. Physical Review Letters 132, 110204 (2024).
https://doi.org/10.1103/PhysRevLett.132.110204

[8] Jerome Estève, Christian Gross, Andreas Weller, Stefano Giovanazzi, and Markus K. Oberthaler. ``Squeezing and entanglement in a Bose–Einstein condensate''. Nature 455, 1216–1219 (2008).
https://doi.org/10.1038/nature07332

[9] Christian Gross, Tilman Zibold, Eike Nicklas, Jerome Estève, and Markus K. Oberthaler. ``Nonlinear atom interferometer surpasses classical precision limit''. Nature 464, 1165–1169 (2010).
https://doi.org/10.1038/nature08919

[10] Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuletić. ``Implementation of cavity squeezing of a collective atomic spin''. Physical Review Letters 104, 073602 (2010).
https://doi.org/10.1103/PhysRevLett.104.073602

[11] Onur Hosten, Nils J. Engelsen, Rajiv Krishnakumar, and Mark A. Kasevich. ``Measurement noise 100 times lower than the quantum-projection limit using entangled atoms''. Nature 529, 505–508 (2016).
https://doi.org/10.1038/nature16176

[12] Robert McConnell, Hao Zhang, Jiazhong Hu, Senka Ć uk, and Vladan Vuletić. ``Entanglement with negative wigner function of almost 3,000 atoms heralded by one photon''. Nature 519, 439–442 (2015).
https://doi.org/10.1038/nature14293

[13] Florian Haas, Jürgen Volz, Roger Gehr, Jakob Reichel, and Jérôme Estève. ``Entangled states of more than 40 atoms in an optical fiber cavity''. Science 344, 180–183 (2014).
https://doi.org/10.1126/science.1248905

[14] Yi-Quan Zou, Ling-Na Wu, Qi Liu, Xin-Yu Luo, Shuai-Feng Guo, Jia-Hao Cao, Meng Khoon Tey, and Li You. ``Beating the classical precision limit with spin-$1$ Dicke states of more than 10,000 atoms''. Proceedings of the National Academy of Sciences 115, 6381–6385 (2018).
https://doi.org/10.1073/pnas.1715105115

[15] Lin Xin, Maryrose Barrios, Julia T. Cohen, and Michael S. Chapman. ``Long-lived squeezed ground states in a quantum spin ensemble''. Physical Review Letters 131, 133402 (2023).
https://doi.org/10.1103/PhysRevLett.131.133402

[16] Mareike Hetzel, Luca Pezzé, Cebrail Pür, Martin Quensen, Andreas Hüper, Jiao Geng, Jens Kruse, Luis Santos, Wolfgang Ertmer, Augusto Smerzi, and Carsten Klempt. ``Tomography of a number-resolving detector by reconstruction of an atomic many-body quantum state''. Physical Review Letters 131, 260601 (2023).
https://doi.org/10.1103/PhysRevLett.131.260601

[17] Martin Quensen, Mareike Hetzel, Luis Santos, Augusto Smerzi, Géza Tóth, Luca Pezzé, and Carsten Klempt. ``Hong-Ou-Mandel interference of more than $10$ indistinguishable atoms'' (2025). arXiv:2504.02691.
arXiv:2504.02691

[18] Szilárd Szalay. ``Multipartite entanglement measures''. Physical Review A 92, 042329 (2015).
https://doi.org/10.1103/PhysRevA.92.042329

[19] Szilárd Szalay, Gergely Barcza, Tibor Szilvási, Libor Veis, and Örs Legeza. ``The correlation theory of the chemical bond''. Scientific Reports 7, 2237 (2017).
https://doi.org/10.1038/s41598-017-02447-z

[20] Szilárd Szalay. ``$k$-stretchability of entanglement, and the duality of $k$-separability and $k$-producibility''. Quantum 3, 204 (2019).
https://doi.org/10.22331/q-2019-12-02-204

[21] Luca Pezzé and Augusto Smerzi. ``Entanglement, nonlinear dynamics, and the Heisenberg limit''. Physical Review Letters 102, 100401 (2009).
https://doi.org/10.1103/PhysRevLett.102.100401

[22] Luca Pezzé and Augusto Smerzi. ``Quantum theory of phase estimation''. In Guglielmo M. Tino and Mark A. Kasevich, editors, Atom Interferometry. Volume Course 188 of Proceedings of the International School of Physics ``Enrico Fermi'', page 691. IOS Press, Amsterdam (2014).
https://doi.org/10.3254/978-1-61499-448-0-691

[23] Géza Tóth and Iagoba Apellaniz. ``Quantum metrology from a quantum information science perspective''. Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006

[24] Luca Pezzé, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Review of Modern Physics 90, 035005 (2018).
https://doi.org/10.1103/RevModPhys.90.035005

[25] Bernd Lücke, Manuel Scherer, Jens Kruse, Luca Pezzé, Frank Deuretzbacher, Philipp Hyllus, Oliver Topic, Jan Peise, Wolfgang Ertmer, Jan Arlt, Luis Santos, Augusto Smerzi, and Carsten Klempt. ``Twin matter waves for interferometry beyond the classical limit''. Science 334, 773–776 (2011).
https://doi.org/10.1126/science.1208798

[26] Roland Krischek, Christian Schwemmer, Witlef Wieczorek, Harald Weinfurter, Philipp Hyllus, Luca Pezzé, and Augusto Smerzi. ``Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation''. Physical Review Letters 107, 080504 (2011).
https://doi.org/10.1103/PhysRevLett.107.080504

[27] Caspar F. Ockeloen, Roman Schmied, Max F. Riedel, and Philipp Treutlein. ``Quantum metrology with a scanning probe atom interferometer''. Physical Review Letters 111, 143001 (2013).
https://doi.org/10.1103/PhysRevLett.111.143001

[28] Helmut Strobel, Wolfgang Muessel, Daniel Linnemann, Tilman Zibold, David B. Hume, Luca Pezzé, Augusto Smerzi, and Markus K. Oberthaler. ``Fisher information and entanglement of non-Gaussian spin states''. Science 345, 424–427 (2014).
https://doi.org/10.1126/science.1250147

[29] Wolfgang Muessel, Helmut Strobel, Daniel Linnemann, David B. Hume, and Markus K. Oberthaler. ``Scalable spin squeezing for quantum-enhanced magnetometry with Bose-Einstein condensates''. Physical Review Letters 113, 103004 (2014).
https://doi.org/10.1103/PhysRevLett.113.103004

[30] Wolfgang Muessel, Helmut Strobel, Daniel Linnemann, Tilman Zibold, Bruno Juliá-Díaz, and Markus K. Oberthaler. ``Twist-and-turn spin squeezing in Bose-Einstein condensates''. Physical Review A 92, 023603 (2015).
https://doi.org/10.1103/PhysRevA.92.023603

[31] Luca Pezzé, Yan Li, Weidong Li, and Augusto Smerzi. ``Witnessing entanglement without entanglement witness operators''. Proceedings of the National Academy of Sciences 113, 11459–11464 (2016).
https://doi.org/10.1073/pnas.1603346113

[32] Ilka Kruse, Karsten Lange, Jan Peise, Bernd Lücke, Luca Pezzé, Jan Arlt, Wolfgang Ertmer, Christian Lisdat, Luis Santos, Augusto Smerzi, and Carsten Klempt. ``Improvement of an atomic clock using squeezed vacuum''. Physical Review Letters 117, 143004 (2016).
https://doi.org/10.1103/PhysRevLett.117.143004

[33] Giovanni Barontini, Leander Hohmann, Florian Haas, Jérôme Estève, and Jakob Reichel. ``Deterministic generation of multiparticle entanglement by quantum Zeno dynamics''. Science 349, 1317–1321 (2015).
https://doi.org/10.1126/science.aaa0754

[34] Justin G. Bohnet, Brian C. Sawyer, Joseph W. Britton, Michael L. Wall, Ana Maria Rey, Michael Foss-Feig, and John J. Bollinger. ``Quantum spin dynamics and entanglement generation with hundreds of trapped ions''. Science 352, 1297–1301 (2016).
https://doi.org/10.1126/science.aad9958

[35] Philipp Hyllus, Wiesław Laskowski, Roland Krischek, Christian Schwemmer, Witlef Wieczorek, Harald Weinfurter, Luca Pezzé, and Augusto Smerzi. ``Fisher information and multiparticle entanglement''. Physical Review A 85, 022321 (2012).
https://doi.org/10.1103/PhysRevA.85.022321

[36] Géza Tóth. ``Multipartite entanglement and high-precision metrology''. Physical Review A 85, 022322 (2012).
https://doi.org/10.1103/PhysRevA.85.022322

[37] Zhihong Ren, Weidong Li, Augusto Smerzi, and Manuel Gessner. ``Metrological detection of multipartite entanglement from Young diagrams''. Physical Review Letters 126, 080502 (2021).
https://doi.org/10.1103/PhysRevLett.126.080502

[38] Kevin C. Cox, Graham P. Greve, Joshua M. Weiner, and James K. Thompson. ``Deterministic squeezed states with collective measurements and feedback''. Physical Review Letters 116, 093602 (2016).
https://doi.org/10.1103/PhysRevLett.116.093602

[39] Justin G. Bohnet, Kevin C. Cox, Matthew A. Norcia, Joshua M. Weiner, Zilong Chen, and James K. Thompson. ``Reduced spin measurement back-action for a phase sensitivity ten times beyond the standard quantum limit''. Nature Photonics 8, 731–736 (2014).
https://doi.org/10.1038/nphoton.2014.151

[40] Dietrich Leibfried, Brian DeMarco, Volker Meyer, David Lucas, Murray Barrett, Joseph W. Britton, Wayne M. Itano, Branislav Jelenković, Christopher Langer, Till Rosenband, and David J. Wineland. ``Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate''. Nature 422, 412–415 (2003).
https://doi.org/10.1038/nature01492

[41] Cass A. Sackett, David Kielpinski, Brian E. King, Christopher Langer, Volker Meyer, Chris J. Myatt, Mary Rowe, Quentin A. Turchette, Wayne M. Itano, David J. Wineland, and Christopher Monroe. ``Experimental entanglement of four particles''. Nature 404, 256–259 (2000).
https://doi.org/10.1038/35005011

[42] Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael Chwalla, Daniel Nigg, William A. Coish, Maximilian Harlander, Wolfgang Hänsel, Markus Hennrich, and Rainer Blatt. ``14-qubit entanglement: Creation and coherence''. Physical Review Letters 106, 130506 (2011).
https://doi.org/10.1103/PhysRevLett.106.130506

[43] Dietrich Leibfried, Murray D. Barrett, Tobias Schaetz, Joseph W. Britton, John Chiaverini, Wayne M. Itano, John D. Jost, Christopher Langer, and David J. Wineland. ``Toward Heisenberg-limited spectroscopy with multiparticle entangled states''. Science 304, 1476–1478 (2004).
https://doi.org/10.1126/science.1097576

[44] Dietrich Leibfried, Emanuel Knill, Seigne Seidelin, Joseph W. Britton, R. Brad Blakestad, John Chiaverini, David B. Hume, Wayne M. Itano, John D. Jost, Christopher Langer, Roee Ozeri, Rainer Reichle, and David J. Wineland. ``Creation of a six-atom `Schrödinger cat' state''. Nature 438, 639–642 (2005).
https://doi.org/10.1038/nature04251

[45] Géza Tóth and Dénes Petz. ``Extremal properties of the variance and the quantum Fisher information''. Physical Review A 87, 032324 (2013).
https://doi.org/10.1103/PhysRevA.87.032324

[46] Sixia Yu. ``Quantum Fisher information as the convex roof of variance'' (2013). arXiv:1302.5311.
arXiv:1302.5311

[47] George E. Andrews. ``The theory of partitions''. Addison-Wesley Publishing Company. (1984).
https://doi.org/10.1017/CBO9780511608650

[48] Richard P. Stanley. ``Enumerative combinatorics, volume 1''. Cambridge University Press. (2012). 2nd edition.
https://doi.org/10.1017/CBO9781139058520

[49] Wolfgang Dür, J. Ignacio Cirac, and Rolf Tarrach. ``Separability and distillability of multiparticle quantum systems''. Physical Review Letters 83, 3562–3565 (1999).
https://doi.org/10.1103/PhysRevLett.83.3562

[50] Wolfgang Dür and J. Ignacio Cirac. ``Classification of multiqubit mixed states: Separability and distillability properties''. Physical Review A 61, 042314 (2000).
https://doi.org/10.1103/PhysRevA.61.042314

[51] Antonio Acín, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera. ``Classification of mixed three-qubit states''. Physical Review Letters 87, 040401 (2001).
https://doi.org/10.1103/PhysRevLett.87.040401

[52] Michael Seevinck and Jos Uffink. ``Sufficient conditions for three-particle entanglement and their tests in recent experiments''. Physical Review A 65, 012107 (2001).
https://doi.org/10.1103/PhysRevA.65.012107

[53] Michael Seevinck and Jos Uffink. ``Partial separability and entanglement criteria for multiqubit quantum states''. Physical Review A 78, 032101 (2008).
https://doi.org/10.1103/PhysRevA.78.032101

[54] Szilárd Szalay and Zoltán Kökényesi. ``Partial separability revisited: Necessary and sufficient criteria''. Physical Review A 86, 032341 (2012).
https://doi.org/10.1103/PhysRevA.86.032341

[55] Brian A. Davey and Hilary A. Priestley. ``Introduction to lattices and order''. Cambridge University Press. (2002). 2nd edition.
https://doi.org/10.1017/CBO9780511809088

[56] Szilárd Szalay. ``The classification of multipartite quantum correlation''. Journal of Physics A: Mathematical and Theoretical 51, 485302 (2018).
https://doi.org/10.1088/1751-8121/aae971

[57] Otfried Gühne and Géza Tóth. ``Energy and multipartite entanglement in multidimensional and frustrated spin models''. Physical Review A 73, 052319 (2006).
https://doi.org/10.1103/PhysRevA.73.052319

[58] Géza Tóth and Otfried Gühne. ``Separability criteria and entanglement witnesses for symmetric quantum states''. Applied Physics B 98, 617–622 (2010).
https://doi.org/10.1007/s00340-009-3839-7

[59] Ji-Yao Chen, Zhengfeng Ji, Nengkun Yu, and Bei Zeng. ``Entanglement depth for symmetric states''. Physical Review A 94, 042333 (2016).
https://doi.org/10.1103/PhysRevA.94.042333

[60] Guillermo García-Pérez, Oskari Kerppo, Matteo A. C. Rossi, and Sabrina Maniscalco. ``Experimentally accessible nonseparability criteria for multipartite-entanglement-structure detection''. Physical Review Research 5, 013226 (2023).
https://doi.org/10.1103/PhysRevResearch.5.013226

[61] The On-Line Encyclopedia of Integer Sequences. ``A000041: Number of partitions of $n$: ways of placing $n$ unlabelled balls into $n$ indistinguishable boxes''. http://oeis.org/A000041.
http://oeis.org/A000041

[62] Kyung Hoon Han and Seung-Hyeok Kye. ``Construction of multi-qubit optimal genuine entanglement witnesses''. Journal of Physics A: Mathematical and Theoretical 49, 175303 (2016).
https://doi.org/10.1088/1751-8113/49/17/175303

[63] Kil-Chan Ha and Seung-Hyeok Kye. ``Construction of three-qubit genuine entanglement with bipartite positive partial transposes''. Physical Review A 93, 032315 (2016).
https://doi.org/10.1103/PhysRevA.93.032315

[64] Kyung Hoon Han and Seung-Hyeok Kye. ``Separability of three qubit Greenberger-Horne-Zeilinger diagonal states''. Journal of Physics A: Mathematical and Theoretical 50, 145303 (2017).
https://doi.org/10.1088/1751-8121/aa616b

[65] Kyung Hoon Han and Seung-Hyeok Kye. ``The role of phases in detecting three-qubit entanglement''. Journal of Mathematical Physics 58, 102201 (2017).
https://doi.org/10.1063/1.5004977

[66] Lin Chen, Kyung Hoon Han, and Seung-Hyeok Kye. ``Separability criterion for three-qubit states with a four dimensional norm''. Journal of Physics A: Mathematical and Theoretical 50, 345303 (2017).
https://doi.org/10.1088/1751-8121/aa7f9a

[67] Kil-Chan Ha, Kyung Hoon Han, and Seung-Hyeok Kye. ``Separability of multi-qubit states in terms of diagonal and anti-diagonal entries''. Quantum Information Processing 18, 34 (2018).
https://doi.org/10.1007/s11128-018-2145-x

[68] Kyung Hoon Han and Seung-Hyeok Kye. ``Construction of three-qubit biseparable states distinguishing kinds of entanglement in a partial separability classification''. Physical Review A 99, 032304 (2019).
https://doi.org/10.1103/PhysRevA.99.032304

[69] Kyung Hoon Han and Seung-Hyeok Kye. ``Criteria for partial entanglement of three qubit states arising from distributive rules''. Quantum Information Processing 20, 151 (2021).
https://doi.org/10.1007/s11128-021-03095-z

[70] Kyung Hoon Han and Seung-Hyeok Kye. ``On the convex cones arising from classifications of partial entanglement in the three qubit system''. Journal of Physics A: Mathematical and Theoretical 53, 015301 (2019).
https://doi.org/10.1088/1751-8121/ab5593

[71] Kyung Hoon Han and Seung-Hyeok Kye. ``Polytope structures for Greenberger-Horne-Zeilinger diagonal states''. Journal of Physics A: Mathematical and Theoretical 54, 455302 (2021).
https://doi.org/10.1088/1751-8121/ac2c5a

[72] Kyung Hoon Han, Seung-Hyeok Kye, and Szilárd Szalay. ``Partial separability/entanglement violates distributive rules''. Quantum Information Processing 19, 202 (2020).
https://doi.org/10.1007/s11128-020-02710-9

[73] Kil-Chan Ha, Kyung Hoon Han, and Seung-Hyeok Kye. ``There exist infinitely many kinds of partial separability/entanglement''. Journal of Mathematical Physics 63, 042201 (2022).
https://doi.org/10.1063/5.0084613

[74] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher. ``Concentrating partial entanglement by local operations''. Physical Review A 53, 2046–2052 (1996).
https://doi.org/10.1103/PhysRevA.53.2046

[75] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Physical Review A 54, 3824–3851 (1996).
https://doi.org/10.1103/PhysRevA.54.3824

[76] Guifre Vidal. ``Entanglement monotones''. Journal of Modern Optics 47, 355–376 (2000).
https://doi.org/10.1080/09500340008244048

[77] Michał Horodecki. ``Entanglement measures''. Quantum Information and Computation 1, 3 (2001).
https://doi.org/10.26421/QIC1.1-2

[78] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Review of Modern Physics 81, 865–942 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[79] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters. ``Quantum nonlocality without entanglement''. Physical Review A 59, 1070–1091 (1999).
https://doi.org/10.1103/PhysRevA.59.1070

[80] Ervin Schrödinger. ``Probability relations between separated systems''. Mathematical Proceedings of the Cambridge Philosophical Society 32, 446–452 (1936).
https://doi.org/10.1017/S0305004100019137

[81] Nicolas Gisin. ``Stochastic quantum dynamics and relativity''. Helvetica Physica Acta 62, 363 (1989). url: https://www.e-periodica.ch/digbib/view?pid=hpa-001:1989:62::1121#379.
https://www.e-periodica.ch/digbib/view?pid=hpa-001:1989:62::1121#379

[82] Lane P. Hughston, Richard Jozsa, and William K. Wootters. ``A complete classification of quantum ensembles having a given density matrix''. Physics Letters A 183, 14–18 (1993).
https://doi.org/10.1016/0375-9601(93)90880-9

[83] Armin Uhlmann. ``Entropy and optimal decompositions of states relative to a maximal commutative subalgebra''. Open Systems & Information Dynamics 5, 209–228 (1998).
https://doi.org/10.1023/A:1009664331611

[84] Armin Uhlmann. ``Roofs and convexity''. Entropy 12, 1799 (2010).
https://doi.org/10.3390/e12071799

[85] Barbara M. Terhal and Paweł Horodecki. ``Schmidt number for density matrices''. Physical Review A 61, 040301 (2000).
https://doi.org/10.1103/PhysRevA.61.040301

[86] Anna Sanpera, Dagmar Bruß, and Maciej Lewenstein. ``Schmidt-number witnesses and bound entanglement''. Physical Review A 63, 050301 (2001).
https://doi.org/10.1103/PhysRevA.63.050301

[87] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states: An introduction to quantum entanglement''. Cambridge University Press. (2017). 2nd edition.
https://doi.org/10.1017/9781139207010

[88] Albert W. Marshall, Ingram Olkin, and Barry C. Arnold. ``Inequalities: Theory of majorization and its applications''. Springer New York, NY. (2010). 2nd edition.
https://doi.org/10.1007/978-0-387-68276-1

[89] Takahiro Sagawa. ``Entropy, divergence, and majorization in classical and quantum thermodynamics''. Springer Singapore. (2022).
https://doi.org/10.1007/978-981-16-6644-5

[90] Thomas Brylawski. ``The lattice of integer partitions''. Discrete Mathematics 6, 201–219 (1973).
https://doi.org/10.1016/0012-365X(73)90094-0

[91] Carl W. Helstrom. ``Quantum detection and estimation theory''. Journal of Statistical Physics 1, 231–252 (1969).
https://doi.org/10.1007/bf01007479

[92] Alexander Holevo. ``Probabilistic and statistical aspects of quantum theory''. Edizioni della Normale. (2011).
https://doi.org/10.1007/978-88-7642-378-9

[93] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Physical Review Letters 72, 3439–3443 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[94] Akio Fujiwara. ``Quantum channel identification problem''. Physical Review A 63, 042304 (2001).
https://doi.org/10.1103/PhysRevA.63.042304

[95] Géza Tóth and Florian Fröwis. ``Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices''. Physical Review Research 4, 013075 (2022).
https://doi.org/10.1103/PhysRevResearch.4.013075

[96] Shao-Hen Chiew and Manuel Gessner. ``Improving sum uncertainty relations with the quantum Fisher information''. Physical Review Research 4, 013076 (2022).
https://doi.org/10.1103/PhysRevResearch.4.013076

[97] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum metrology''. Physical Review Letters 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[98] Iagoba Apellaniz, Bernd Lücke, Jan Peise, Carsten Klempt, and Géza Tóth. ``Detecting metrologically useful entanglement in the vicinity of Dicke states''. New Journal of Physics 17, 083027 (2015).
https://doi.org/10.1088/1367-2630/17/8/083027

[99] Géza Tóth. ``Entanglement detection and quantum metrology in quantum optical systems''. http://real-d.mtak.hu/1230/ (2021). D.Sc. Dissertation, Hungarian Academy of Sciences, Budapest.
http://real-d.mtak.hu/1230/

[100] Anders Sørensen, Lu-Ming Duan, J. Ignacio Cirac, and Peter Zoller. ``Many-particle entanglement with Bose–Einstein condensates''. Nature 409, 63–66 (2001).
https://doi.org/10.1038/35051038

[101] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Optimal spin squeezing inequalities detect bound entanglement in spin models''. Physical Review Letters 99, 250405 (2007).
https://doi.org/10.1103/PhysRevLett.99.250405

[102] Géza Tóth and Morgan W. Mitchell. ``Generation of macroscopic singlet states in atomic ensembles''. New Journal of Physics 12, 053007 (2010).
https://doi.org/10.1088/1367-2630/12/5/053007

[103] Giuseppe Vitagliano, Philipp Hyllus, Íñigo L. Egusquiza, and Géza Tóth. ``Spin squeezing inequalities for arbitrary spin''. Physical Review Letters 107, 240502 (2011).
https://doi.org/10.1103/PhysRevLett.107.240502

[104] Giuseppe Vitagliano, Otfried Gühne, and Géza Tóth. ``$su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems'' (2024). arXiv:2406.13338.
arXiv:2406.13338

[105] Naeimeh Behbood, Ferran Martin Ciurana, Giorgio Colangelo, Mario Napolitano, Géza Tóth, Robert J. Sewell, and Morgan W. Mitchell. ``Generation of macroscopic singlet states in a cold atomic ensemble''. Physical Review Letters 113, 093601 (2014).
https://doi.org/10.1103/PhysRevLett.113.093601

[106] Jia Kong, Ricardo Jiménez-Martínez, Charikleia Troullinou, Vito Giovanni Lucivero, Géza Tóth, and Morgan W. Mitchell. ``Measurement-induced, spatially-extended entanglement in a hot, strongly-interacting atomic system''. Nature Communications 11, 2415 (2020).
https://doi.org/10.1038/s41467-020-15899-1

[107] Lu-Ming Duan. ``Entanglement detection in the vicinity of arbitrary Dicke states''. Physical Review Letters 107, 180502 (2011).
https://doi.org/10.1103/PhysRevLett.107.180502

[108] Vlatko Vedral and Martin B. Plenio. ``Entanglement measures and purification procedures''. Physical Review A 57, 1619–1633 (1998).
https://doi.org/10.1103/PhysRevA.57.1619

[109] Eric Chitambar and Gilad Gour. ``Quantum resource theories''. Review of Modern Physics 91, 025001 (2019).
https://doi.org/10.1103/RevModPhys.91.025001

[110] Ludovico Lami and Maksim Shirokov. ``Continuity of the relative entropy of resource''. International Journal of Quantum Information 22, 2440009 (2024).
https://doi.org/10.1142/S0219749924400094

[111] John von Neumann. ``Thermodynamik quantenmechanischer Gesamtheiten''. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927). url: http://eudml.org/doc/59231.
http://eudml.org/doc/59231

[112] Masanori Ohya and Dénes Petz. ``Quantum entropy and its use''. Springer Berlin Heidelberg. (1993). 1st edition. url: https://link.springer.com/book/9783540208068.
https://link.springer.com/book/9783540208068

[113] Dénes Petz. ``Quantum information theory and quantum statistics''. Springer Berlin Heidelberg. (2008).
https://doi.org/10.1007/978-3-540-74636-2

[114] Mark M. Wilde. ``Quantum information theory''. Cambridge University Press. (2013).
https://doi.org/10.1017/CBO9781139525343

[115] Hisaharu Umegaki. ``Conditional expectation in an operator algebra. IV. Entropy and information''. Kodai Mathematical Seminar Reports 14, 59–85 (1962).
https://doi.org/10.2996/kmj/1138844604

[116] Steven Roman. ``Lattices and ordered sets''. Springer New York, NY. (2008). 1st edition.
https://doi.org/10.1007/978-0-387-78901-9

[117] Rajendra Bhatia and Chandler Davis. ``A better bound on the variance''. The American Mathematical Monthly 107, 353–357 (2000).
https://doi.org/10.1080/00029890.2000.12005203

[118] Tiberiu Popoviciu. ``Sur les équations algébriques ayant toutes leurs racines réelles''. Mathematica 9, 129 (1935).

[119] Holger F. Hofmann and Shigeki Takeuchi. ``Violation of local uncertainty relations as a signature of entanglement''. Physical Review A 68, 032103 (2003).
https://doi.org/10.1103/PhysRevA.68.032103

[120] Otfried Gühne. ``Characterizing entanglement via uncertainty relations''. Physical Review Letters 92, 117903 (2004).
https://doi.org/10.1103/PhysRevLett.92.117903

[121] Géza Tóth. ``Entanglement detection in optical lattices of bosonic atoms with collective measurements''. Physical Review A 69, 052327 (2004).
https://doi.org/10.1103/PhysRevA.69.052327

Cited by

[1] Szilárd Szalay and Péter Nyári, "Dicke subsystems are entangled", Quantum 9, 1908 (2025).

[2] Yan Hong, Mengjia Zhang, Limin Gao, Huaqi Zhou, and Limei Zhang, "Detecting k -nonstretchability via a class of informationally complete symmetric measurements", Journal of Physics A: Mathematical and Theoretical 59 30, 305305 (2026).

[3] Shaowei Du, Shuheng Liu, Matteo Fadel, Giuseppe Vitagliano, and Qiongyi He, "Uncertainty Relations between Quantum Fisher Information and Entanglement Monotones", Physical Review Letters 136 11, 110806 (2026).

[4] Frederic Bippus, Juraj Krsnik, Motoharu Kitatani, Luka Akšamović, Anna Kauch, Neven Barišić, and Karsten Held, "Entanglement in the pseudogap regime of cuprate superconductors", Physical Review B 112 8, L081110 (2025).

[5] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera, "Symmetric quantum states: a review of recent progress", Reports on Progress in Physics 89 2, 024001 (2026).

[6] Yansa Lu, Zhihua Chen, and Yuanhong Tao, "Robust Certification of Entanglement Structures in Noisy Quantum Systems: Device‐Independent and Semi‐Device‐Independent Approaches", Advanced Quantum Technologies 8 11, e00347 (2025).

[7] Da-Jian Zhang and D. M. Tong, "Krylov Shadow Tomography: Efficient Estimation of Quantum Fisher Information", Physical Review Letters 134 11, 110802 (2025).

[8] Kai Wu, Zhihua Chen, Zhen-Peng Xu, Zhihao Ma, and Shao-Ming Fei, "Hybrid of Gradient Descent And Semidefinite Programming for Certifying Multipartite Entanglement Structure", arXiv:2412.16480, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 22:31:26) and SAO/NASA ADS (last updated successfully 2026-08-08 22:31:31). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.