k-stretchability of entanglement, and the duality of k-separability and k-producibility

Szilárd Szalay

Strongly Correlated Systems ``Lendület'' Research Group, Wigner Research Centre for Physics, 29-33, Konkoly-Thege Miklós str., H-1121 Budapest, Hungary

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The notions of $k$-separability and $k$-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call $k$-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated with (or separable from) one another.

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[1] Ervin Schrödinger. Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23: 807, 1935a. 10.1007/​BF01491987.

[2] Ervin Schrödinger. Discussion of probability relations between separated systems. Math. Proc. Camb. Phil. Soc., 31: 555, 1935b. 10.1017/​S0305004100013554.

[3] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81 (2): 865–942, Jun 2009. 10.1103/​RevModPhys.81.865.

[4] Kavan Modi, Tomasz Paterek, Wonmin Son, Vlatko Vedral, and Mark Williamson. Unified view of quantum and classical correlations. Phys. Rev. Lett., 104: 080501, Feb 2010. 10.1103/​PhysRevLett.104.080501.

[5] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 1 edition, October 2000. ISBN 0521635039. 10.1017/​CBO9780511976667.

[6] Dénes Petz. Quantum Information Theory and Quantum Statistics. Springer, 2008. 10.1007/​978-3-540-74636-2.

[7] Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. 10.1017/​CBO9781139525343.

[8] Valerie Coffman, Joydip Kundu, and William K. Wootters. Distributed entanglement. Phys. Rev. A, 61: 052306, Apr 2000. 10.1103/​PhysRevA.61.052306.

[9] Masato Koashi and Andreas Winter. Monogamy of quantum entanglement and other correlations. Phys. Rev. A, 69: 022309, Feb 2004. 10.1103/​PhysRevA.69.022309.

[10] Jens Eisert, Marcus Cramer, and Martin B. Plenio. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys., 82: 277–306, Feb 2010. 10.1103/​RevModPhys.82.277.

[11] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 517–576, May 2008. 10.1103/​RevModPhys.80.517.

[12] Örs Legeza and Jenő Sólyom. Quantum data compression, quantum information generation, and the density-matrix renormalization-group method. Phys. Rev. B, 70: 205118, Nov 2004. 10.1103/​PhysRevB.70.205118.

[13] Szilárd Szalay, Max Pfeffer, Valentin Murg, Gergely Barcza, Frank Verstraete, Reinhold Schneider, and Örs Legeza. Tensor product methods and entanglement optimization for ab initio quantum chemistry. Int. J. Quantum Chem., 115 (19): 1342–1391, 2015. ISSN 1097-461X. 10.1002/​qua.24898.

[14] Szilárd Szalay, Gergely Barcza, Tibor Szilvási, Libor Veis, and Örs Legeza. The correlation theory of the chemical bond. Scientific Reports, 7: 2237, May 2017. 10.1038/​s41598-017-02447-z.

[15] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40 (8): 4277–4281, Oct 1989. 10.1103/​PhysRevA.40.4277.

[16] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher. Concentrating partial entanglement by local operations. Phys. Rev. A, 53: 2046–2052, Apr 1996a. 10.1103/​PhysRevA.53.2046.

[17] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54: 3824–3851, Nov 1996b. 10.1103/​PhysRevA.54.3824.

[18] Eric Chitambar, Debbie Leung, Laura Mancinska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys., 328 (1): 303–326, 2014. ISSN 0010-3616. 10.1007/​s00220-014-1953-9.

[19] Wolfgang Dür, Guifre Vidal, and J. Ignacio Cirac. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A, 62: 062314, Nov 2000. 10.1103/​PhysRevA.62.062314.

[20] Michael A. Nielsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83: 436–439, Jul 1999. 10.1103/​PhysRevLett.83.436.

[21] Guifré Vidal. Entanglement of pure states for a single copy. Phys. Rev. Lett., 83: 1046–1049, Aug 1999. 10.1103/​PhysRevLett.83.1046.

[22] Guifre Vidal. Entanglement monotones. J. Mod. Opt, 47 (2): 355–376, Feb 2000. 10.1080/​09500340008244048.

[23] Michał Horodecki. Entanglement measures. Quant. Inf. Comp., 1: 3, May 2001. 10.26421/​QIC1.1.

[24] Martin B. Plenio and Shashank Virmani. An introduction to entanglement measures. Quant. Inf. Comp., 7: 1, Jan 2007. 10.26421/​QIC7.1-2.

[25] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde. Four qubits can be entangled in nine different ways. Phys. Rev. A, 65: 052112, Apr 2002. 10.1103/​PhysRevA.65.052112.

[26] Jean-Gabriel Luque and Jean-Yves Thibon. Polynomial invariants of four qubits. Phys. Rev. A, 67: 042303, Apr 2003. 10.1103/​PhysRevA.67.042303.

[27] Christopher Eltschka and Jens Siewert. Quantifying entanglement resources. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424005, 2014. 10.1088/​1751-8113/​47/​42/​424005.

[28] Hayata Yamasaki, Alexander Pirker, Mio Murao, Wolfgang Dür, and Barbara Kraus. Multipartite entanglement outperforming bipartite entanglement under limited quantum system sizes. Phys. Rev. A, 98: 052313, Nov 2018. 10.1103/​PhysRevA.98.052313.

[29] Martin. Hebenstreit, Cornelia. Spee, and Barbara Kraus. Maximally entangled set of tripartite qutrit states and pure state separable transformations which are not possible via local operations and classical communication. Phys. Rev. A, 93: 012339, Jan 2016. 10.1103/​PhysRevA.93.012339.

[30] Cornelia Spee, Julio I. de Vicente, and Barbara Kraus. The maximally entangled set of 4-qubit states. Journal of Mathematical Physics, 57 (5): 052201, 2016. 10.1063/​1.4946895.

[31] Gilad Gour, Barbara Kraus, and Nolan R. Wallach. Almost all multipartite qubit quantum states have trivial stabilizer. Journal of Mathematical Physics, 58 (9): 092204, 2017. 10.1063/​1.5003015.

[32] David Sauerwein, Nolan R. Wallach, Gilad Gour, and Barbara Kraus. Transformations among pure multipartite entangled states via local operations are almost never possible. Phys. Rev. X, 8: 031020, Jul 2018. 10.1103/​PhysRevX.8.031020.

[33] Wolfgang Dür, J. Ignacio Cirac, and Rolf Tarrach. Separability and distillability of multiparticle quantum systems. Phys. Rev. Lett., 83: 3562–3565, Oct 1999. 10.1103/​PhysRevLett.83.3562.

[34] Wolfgang Dür and J. Ignacio Cirac. Classification of multiqubit mixed states: Separability and distillability properties. Phys. Rev. A, 61: 042314, Mar 2000. 10.1103/​PhysRevA.61.042314.

[35] Antonio Acín, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera. Classification of mixed three-qubit states. Phys. Rev. Lett., 87: 040401, Jul 2001. 10.1103/​PhysRevLett.87.040401.

[36] Koji Nagata, Masato Koashi, and Nobuyuki Imoto. Configuration of separability and tests for multipartite entanglement in Bell-type experiments. Phys. Rev. Lett., 89: 260401, Dec 2002. 10.1103/​PhysRevLett.89.260401.

[37] Michael Seevinck and Jos Uffink. Partial separability and entanglement criteria for multiqubit quantum states. Phys. Rev. A, 78 (3): 032101, Sep 2008. 10.1103/​PhysRevA.78.032101.

[38] Szilárd Szalay. Separability criteria for mixed three-qubit states. Phys. Rev. A, 83: 062337, Jun 2011. 10.1103/​PhysRevA.83.062337.

[39] Szilárd Szalay and Zoltán Kökényesi. Partial separability revisited: Necessary and sufficient criteria. Phys. Rev. A, 86: 032341, Sep 2012. 10.1103/​PhysRevA.86.032341.

[40] Szilárd Szalay. Multipartite entanglement measures. Phys. Rev. A, 92: 042329, Oct 2015. 10.1103/​PhysRevA.92.042329.

[41] Szilárd Szalay. The classification of multipartite quantum correlation. Journal of Physics A: Mathematical and Theoretical, 51 (48): 485302, 2018. dx.doi.org/​10.1088/​1751-8121/​aae971.

[42] Jan Brandejs, Libor Veis, Szilárd Szalay, Ji ̆rí Pittner, and Örs Legeza. Quantum information-based analysis of electron-deficient bonds. The Journal of Chemical Physics, 150 (20): 204117, 2019. 10.1063/​1.5093497.

[43] Brian A. Davey and Hilary A. Priestley. Introduction to Lattices and Order. Cambridge University Press, second edition, 2002. ISBN 9780521784511. 10.1017/​CBO9780511809088.

[44] Steven Roman. Lattices and Ordered Sets. Springer, first edition, 2008. ISBN 978-0-387-78900-2. 10.1007/​978-0-387-78901-9.

[45] Richard P. Stanley. Enumerative Combinatorics, Volume 1. Cambridge University Press, second edition, 2012. ISBN 9781107602625. 10.1017/​CBO9781139058520.

[46] Michael Seevinck and Jos Uffink. Sufficient conditions for three-particle entanglement and their tests in recent experiments. Phys. Rev. A, 65: 012107, Dec 2001. 10.1103/​PhysRevA.65.012107.

[47] Otfried Gühne, Géza Tóth, and Hans J. Briegel. Multipartite entanglement in spin chains. New J. Phys., 7 (1): 229, 2005. 10.1088/​1367-2630/​7/​1/​229.

[48] Otfried Gühne and Géza Tóth. Energy and multipartite entanglement in multidimensional and frustrated spin models. Phys. Rev. A, 73: 052319, May 2006. 10.1103/​PhysRevA.73.052319.

[49] Géza Tóth and Otfried Gühne. Separability criteria and entanglement witnesses for symmetric quantum states. Appl. Phys. B, 98 (4): 617–622, 2010. ISSN 0946-2171. 10.1007/​s00340-009-3839-7.

[50] Anders S. Sørensen and Klaus Mølmer. Entanglement and extreme spin squeezing. Phys. Rev. Lett., 86: 4431–4434, May 2001. 10.1103/​PhysRevLett.86.4431.

[51] Bernd Lücke, Jan Peise, Giuseppe Vitagliano, Jan Arlt, Luis Santos, Géza Tóth, and Carsten Klempt. Detecting multiparticle entanglement of dicke states. Phys. Rev. Lett., 112: 155304, Apr 2014. 10.1103/​PhysRevLett.112.155304.

[52] Ji-Yao Chen, Zhengfeng Ji, Nengkun Yu, and Bei Zeng. Entanglement depth for symmetric states. Phys. Rev. A, 94: 042333, Oct 2016. 10.1103/​PhysRevA.94.042333.

[53] Florian John Curchod, Nicolas Gisin, and Yeong-Cherng Liang. Quantifying multipartite nonlocality via the size of the resource. Phys. Rev. A, 91: 012121, Jan 2015. 10.1103/​PhysRevA.91.012121.

[54] Yeong-Cherng Liang, Denis Rosset, Jean-Daniel Bancal, Gilles Pütz, Tomer Jack Barnea, and Nicolas Gisin. Family of bell-like inequalities as device-independent witnesses for entanglement depth. Phys. Rev. Lett., 114: 190401, May 2015. 10.1103/​PhysRevLett.114.190401.

[55] Pei-Sheng Lin, Jui-Chen Hung, Ching-Hsu Chen, and Yeong-Cherng Liang. Exploring bell inequalities for the device-independent certification of multipartite entanglement depth. Phys. Rev. A, 99: 062338, Jun 2019. 10.1103/​PhysRevA.99.062338.

[56] He Lu, Qi Zhao, Zheng-Da Li, Xu-Fei Yin, Xiao Yuan, Jui-Chen Hung, Luo-Kan Chen, Li Li, Nai-Le Liu, Cheng-Zhi Peng, Yeong-Cherng Liang, Xiongfeng Ma, Yu-Ao Chen, and Jian-Wei Pan. Entanglement structure: Entanglement partitioning in multipartite systems and its experimental detection using optimizable witnesses. Phys. Rev. X, 8: 021072, Jun 2018. 10.1103/​PhysRevX.8.021072.

[57] Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information science perspective. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424006, oct 2014. 10.1088/​1751-8113/​47/​42/​424006.

[58] Philipp Hyllus, Wiesław Laskowski, Roland Krischek, Christian Schwemmer, Witlef Wieczorek, Harald Weinfurter, Luca Pezzé, and Augusto Smerzi. Fisher information and multiparticle entanglement. Phys. Rev. A, 85: 022321, Feb 2012. 10.1103/​PhysRevA.85.022321.

[59] Manuel Gessner, Luca Pezzè, and Augusto Smerzi. Sensitivity bounds for multiparameter quantum metrology. Phys. Rev. Lett., 121: 130503, Sep 2018. 10.1103/​PhysRevLett.121.130503.

[60] Zhongzhong Qin, Manuel Gessner, Zhihong Ren, Xiaowei Deng, Dongmei Han, Weidong Li, Xiaolong Su, Augusto Smerzi, and Kunchi Peng. Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the fisher information. npj Quantum Information, 5 (3), 2019. 10.1038/​s41534-018-0119-6.

[61] Géza Tóth. Multipartite entanglement and high-precision metrology. Phys. Rev. A, 85: 022322, Feb 2012. 10.1103/​PhysRevA.85.022322.

[62] George E. Andrews. The Theory of Partitions. Cambridge University Press, 1984. 10.1017/​CBO9780511608650.

[63] 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.

[64] Masanori Ohya and Dénes Petz. Quantum Entropy and Its Use. Springer Verlag, 1 edition, October 1993. ISBN 978-3-540-20806-8.

[65] Ingemar Bengtsson and Karol ̊Zyczkowski. Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2006. ISBN 0521814510. 10.1017/​CBO9780511535048.

[66] Hisaharu Umegaki. Conditional expectation in an operator algebra. iv. entropy and information. Kodai Math. Sem. Rep., 14 (2): 59–85, 1962. 10.2996/​kmj/​1138844604.

[67] Göran Lindblad. Entropy, information and quantum measurements. Communications in Mathematical Physics, 33 (4): 305–322, Dec 1973. ISSN 1432-0916. 10.1007/​BF01646743.

[68] Fumio Hiai and Dénes Petz. The proper formula for relative entropy and its asymptotics in quantum probability. Communications in Mathematical Physics, 143 (1): 99–114, 1991. ISSN 0010-3616. 10.1007/​BF02100287.

[69] The On-Line Encyclopedia of Integer Sequences. A000110: Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes, a. URL http:/​/​oeis.org/​A000110.

[70] Fedor Herbut. On mutual information in multipartite quantum states and equality in strong subadditivity of entropy. Journal of Physics A: Mathematical and General, 37 (10): 3535, 2004. 10.1088/​0305-4470/​37/​10/​016.

[71] Armin Uhlmann. Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Sys. Inf. Dyn., 5: 209–228, 1998. ISSN 1230-1612. 10.1023/​A:1009664331611.

[72] Armin Uhlmann. Roofs and convexity. Entropy, 12: 1799, July 2010. 10.3390/​e12071799.

[73] Kyung Hoon Han and Seung-Hyeok Kye. Construction of three-qubit biseparable states distinguishing kinds of entanglement in a partial separability classification. Phys. Rev. A, 99: 032304, Mar 2019. 10.1103/​PhysRevA.99.032304.

[74] The On-Line Encyclopedia of Integer Sequences. A000041: Number of partitions of n: ways of placing n unlabelled balls into n indistinguishable boxes, b. URL http:/​/​oeis.org/​A000041.

[75] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, Apr 2019. 10.1103/​RevModPhys.91.025001.

[76] Thomas Brylawski. The lattice of integer partitions. Discrete Mathematics, 6: 201, 1973. 10.1016/​0012-365X(73)90094-0.

[77] Garrett Birkhoff. Lattice Theory. American Mathematical Society, New York, 3rd edition, 1973.

[78] Freeman J. Dyson. Some guesses in the theory of partitions. Eureka, 8: 10–15, 1944.

Cited by

[1] Géza Tóth, "Stretching the limits of multiparticle entanglement", Quantum Views 4, 30 (2020).

[2] Kyung Hoon Han, Seung-Hyeok Kye, and Szilárd Szalay, "Partial separability/entanglement violates distributive rules", Quantum Information Processing 19 7, 202 (2020).

[3] Matteo Fadel and Manuel Gessner, "Relating spin squeezing to multipartite entanglement criteria for particles and modes", Physical Review A 102 1, 012412 (2020).

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