Entangled symmetric states and copositive matrices

Carlo Marconi1, Albert Aloy2, Jordi Tura3,4, and Anna Sanpera1,5

1Física Teòrica: Informació i Fenòmens Quàntics. Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
2ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
4Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
5ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain

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Abstract

Entanglement in symmetric quantum states and the theory of copositive matrices are intimately related concepts. For the simplest symmetric states, i.e., the diagonal symmetric (DS) states, it has been shown that there exists a correspondence between exceptional (non-exceptional) copositive matrices and non-decomposable (decomposable) Entanglement Witnesses (EWs). Here we show that EWs of symmetric, but not DS, states can also be constructed from extended copositive matrices, providing new examples of bound entangled symmetric states, together with their corresponding EWs, in arbitrary odd dimensions.

Entanglement is one of the most intriguing phenomena in quantum physics whose implications have profound consequences not only from a theoretical point of view but also in light of some computational tasks that would be otherwise unfeasible with classical systems.
For this reason, deciding whether a quantum state is entangled or not, is a problem of paramount importance whose solution, unfortunately, is known to be NP-hard in the general scenario.
In some cases, however, symmetries provide a useful framework to recast the separability problem in a simpler way, thus reducing the original complexity of this task.
In this work we focus on symmetric states, i.e., states that are invariant under permutations of the parties, showing how, in the case of the qudits, the characterization of the entanglement can be accomplished by means of a class of matrices known as copositive. In particular, we establish a connection between entanglement witnesses, i.e., hermitian operators that are able to detect entanglement, and copositive matrices, showing how only a subset of them, dubbed as exceptional, can be used to assess PPT-entanglement in any dimension, with the PPT-entangled edge states detected by the so-called extremal matrices.
Finally we illustrate our findings discussing some examples of families of PPT-entangled states in 3-level and 4-level systems, along with the entanglement witnesses that detect them.
We conjecture that any PPT-entangled state of two qudits can be detected by means of an entanglement witness of the form that we propose.

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[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of modern physics, 81 (2): 865, 2009. 10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[2] Charles H Bennett, Herbert J Bernstein, Sandu Popescu, and Benjamin Schumacher. Concentrating partial entanglement by local operations. Physical Review A, 53 (4): 2046, 1996. 10.1103/​PhysRevA.53.2046.
https:/​/​doi.org/​10.1103/​PhysRevA.53.2046

[3] Leonid Gurvits. Classical deterministic complexity of edmonds' problem and quantum entanglement. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 10–19, 2003. 10.1145/​780542.780545.
https:/​/​doi.org/​10.1145/​780542.780545

[4] Asher Peres. Separability criterion for density matrices. Physical Review Letters, 77 (8): 1413, 1996. 10.1103/​PhysRevLett.77.1413.
https:/​/​doi.org/​10.1103/​PhysRevLett.77.1413

[5] Barbara M Terhal and Karl Gerd H Vollbrecht. Entanglement of formation for isotropic states. Physical Review Letters, 85 (12): 2625, 2000. 10.1103/​PhysRevLett.85.2625.
https:/​/​doi.org/​10.1103/​PhysRevLett.85.2625

[6] Maciej Lewenstein, Barabara Kraus, J Ignacio Cirac, and P Horodecki. Optimization of entanglement witnesses. Physical Review A, 62 (5): 052310, 2000. 10.1103/​PhysRevA.62.052310.
https:/​/​doi.org/​10.1103/​PhysRevA.62.052310

[7] Dariusz Chruściński and Gniewomir Sarbicki. Entanglement witnesses: construction, analysis and classification. Journal of Physics A: Mathematical and Theoretical, 47 (48): 483001, 2014. 10.1088/​1751-8113/​47/​48/​483001.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​48/​483001

[8] Maciej Lewenstein, B Kraus, P Horodecki, and JI Cirac. Characterization of separable states and entanglement witnesses. Physical Review A, 63 (4): 044304, 2001. 10.1103/​PhysRevA.63.044304.
https:/​/​doi.org/​10.1103/​PhysRevA.63.044304

[9] Fernando GSL Brandao. Quantifying entanglement with witness operators. Physical Review A, 72 (2): 022310, 2005. 10.1103/​physreva.72.022310.
https:/​/​doi.org/​10.1103/​physreva.72.022310

[10] Karl Gerd H Vollbrecht and Reinhard F Werner. Entanglement measures under symmetry. Physical Review A, 64 (6): 062307, 2001. 10.1103/​PhysRevA.64.062307.
https:/​/​doi.org/​10.1103/​PhysRevA.64.062307

[11] Géza Tóth and Otfried Gühne. Separability criteria and entanglement witnesses for symmetric quantum states. Applied Physics B, 98 (4): 617–622, 2010. 10.1007/​s00340-009-3839-7.
https:/​/​doi.org/​10.1007/​s00340-009-3839-7

[12] Tilo Eggeling and Reinhard F Werner. Separability properties of tripartite states with u $\otimes$ u $\otimes$ u $\otimes$ symmetry. Physical Review A, 63 (4): 042111, 2001. 10.1103/​physreva.63.042111.
https:/​/​doi.org/​10.1103/​physreva.63.042111

[13] Jordi Tura, Albert Aloy, Ruben Quesada, Maciej Lewenstein, and Anna Sanpera. Separability of diagonal symmetric states: a quadratic conic optimization problem. Quantum, 2: 45, 2018. 10.22331/​q-2018-01-12-45.
https:/​/​doi.org/​10.22331/​q-2018-01-12-45

[14] Anna Sanpera, Dagmar Bruß, and Maciej Lewenstein. Schmidt-number witnesses and bound entanglement. Physical Review A, 63 (5): 050301, 2001. 10.1103/​PhysRevA.63.050301.
https:/​/​doi.org/​10.1103/​PhysRevA.63.050301

[15] Lieven Clarisse. Construction of bound entangled edge states with special ranks. Physics Letters A, 359 (6): 603–607, 2006. 10.1016/​j.physleta.2006.07.045.
https:/​/​doi.org/​10.1016/​j.physleta.2006.07.045

[16] Seung-Hyeok Kye and Hiroyuki Osaka. Classification of bi-qutrit positive partial transpose entangled edge states by their ranks. Journal of mathematical physics, 53 (5): 052201, 2012. 10.1063/​1.4712302.
https:/​/​doi.org/​10.1063/​1.4712302

[17] Lin Chen and Dragomir Ž Ðoković. Description of rank four entangled states of two qutrits having positive partial transpose. Journal of mathematical physics, 52 (12): 122203, 2011. 10.1063/​1.3663837.
https:/​/​doi.org/​10.1063/​1.3663837

[18] Jon Magne Leinaas, Jan Myrheim, and Per Øyvind Sollid. Low-rank extremal positive-partial-transpose states and unextendible product bases. Phys. Rev. A, 81: 062330, Jun 2010. 10.1103/​PhysRevA.81.062330.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062330

[19] Nengkun Yu. Separability of a mixture of dicke states. Physical Review A, 94 (6): 060101, 2016. 10.1103/​PhysRevA.94.060101.
https:/​/​doi.org/​10.1103/​PhysRevA.94.060101

[20] Katta G. Murty and Santosh N. Kabadi. Some np-complete problems in quadratic and nonlinear programming. Mathematical Programming, 39: 117–129, 1987. 10.1007/​BF02592948.
https:/​/​doi.org/​10.1007/​BF02592948

[21] Li Ping and Feng Yu Yu. Criteria for copositive matrices of order four. Linear algebra and its applications, 194: 109–124, 1993. 10.1016/​0024-3795(93)90116-6.
https:/​/​doi.org/​10.1016/​0024-3795(93)90116-6

[22] J-B Hiriart-Urruty and Alberto Seeger. A variational approach to copositive matrices. SIAM review, 52 (4): 593–629, 2010. 10.1137/​090750391.
https:/​/​doi.org/​10.1137/​090750391

[23] Palahenedi Hewage Diananda. On non-negative forms in real variables some or all of which are non-negative. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 58, pages 17–25. Cambridge University Press, 1962. 10.1017/​s0305004100036185.
https:/​/​doi.org/​10.1017/​s0305004100036185

[24] Marshall Hall and Morris Newman. Copositive and completely positive quadratic forms. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 59, pages 329–339. Cambridge University Press, 1963. 10.1017/​s0305004100036951.
https:/​/​doi.org/​10.1017/​s0305004100036951

[25] Charles Johnson and Robert Reams. Constructing copositive matrices from interior matrices. The Electronic Journal of Linear Algebra, 17: 9–20, 2008. 10.13001/​1081-3810.1245.
https:/​/​doi.org/​10.13001/​1081-3810.1245

[26] Alan J Hoffman and Francisco Pereira. On copositive matrices with- 1, 0, 1 entries. Journal of Combinatorial Theory, Series A, 14 (3): 302–309, 1973. 10.1016/​0097-3165(73)90006-x.
https:/​/​doi.org/​10.1016/​0097-3165(73)90006-x

[27] Dariusz Chruściński and Andrzej Kossakowski. Circulant states with positive partial transpose. Phys. Rev. A, 76: 032308, Sep 2007. 10.1103/​PhysRevA.76.032308.
https:/​/​doi.org/​10.1103/​PhysRevA.76.032308

[28] Andrew C Doherty, Pablo A Parrilo, and Federico M Spedalieri. Complete family of separability criteria. Physical Review A, 69 (2): 022308, 2004. 10.1103/​PhysRevA.69.022308.
https:/​/​doi.org/​10.1103/​PhysRevA.69.022308

[29] Andrew C Doherty, Pablo A Parrilo, and Federico M Spedalieri. Distinguishing separable and entangled states. Physical Review Letters, 88 (18): 187904, 2002. 10.1103/​physrevlett.88.187904.
https:/​/​doi.org/​10.1103/​physrevlett.88.187904

Cited by

[1] Adam Burchardt, Jakub Czartowski, and Karol Życzkowski, "Entanglement in highly symmetric multipartite quantum states", Physical Review A 104 2, 022426 (2021).

[2] Joonwoo Bae, Anindita Bera, Dariusz Chruściński, Beatrix C. Hiesmayr, and Daniel McNulty, "How many measurements are needed to detect bound entangled states?", arXiv:2108.01109.

[3] Beatrix C. Hiesmayr, "Free versus bound entanglement, a NP-hard problem tackled by machine learning", Scientific Reports 11, 19739 (2021).

[4] Beatrix C. Hiesmayr, "Free versus Bound Entanglement: Machine learning tackling a NP-hard problem", arXiv:2106.03977.

[5] Hari krishnan S V, Ashish Ranjan, and Manik Banik, "State space structure of tripartite quantum systems", Physical Review A 104 2, 022437 (2021).

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