Quantum Information & Computation
Volume 24, Issue 1
RESEARCH ARTICLE Open access

A Triality Pattern in Entanglement Theory

Daniel Cariello

Author affiliations
  • Daniel Cariello1
  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, Brazil

pp. 40-57 · 10.2478/qic-2024-0003

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Abstract

In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.
Open accessCC BY-NC-SA 4.0

Keywords

  • filter normal form
  • separability
  • complete reducibility property
  • spectral radius

Suggested citation

Daniel Cariello (2024). A Triality Pattern in Entanglement Theory. Quantum Information & Computation, 24(1), 40-57. https://doi.org/10.2478/qic-2024-0003

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References (24)

  1. O. Gühne and G. Tóth (2009). “Entanglement detection”. Physics Reports, 474: 1–6, 1–75.DOI: 10.1016/j.physrep.2009.02.004Google Scholar
  2. A. Peres (1996). “Separability criterion for density matrices”. Physical Review Letters, 77: 8, 1413.DOI: 10.1103/PhysRevLett.77.1413Google Scholar
  3. M. Horodecki, P. Horodecki and R. Horodecki (1996). “Separability of mixed states: Necessary and sufficient conditions”. Physics Letters A, 223, 1–8.DOI: 10.1016/S0375-9601(96)00706-2Google Scholar
  4. L. Gurvits (2004). “Classical complexity and quantum entanglement”. Journal of Computer and System Sciences, 69: 3, 448–484.DOI: 10.1016/j.jcss.2004.06.003Google Scholar
  5. D. Cariello (2016). “Completely reducible maps in quantum information theory”. IEEE Transactions on Information Theory, 62: 4, 1721–1732.DOI: 10.1109/TIT.2016.2535176Google Scholar
  6. G. Tóth and O. Gühne (2010). “Separability criteria and entanglement witnesses for symmetric quantum states”. Applied Physics B, 98: 4, 617–622.DOI: 10.1007/s00340-009-3839-7Google Scholar
  7. M. Horodecki, P. Horodecki and R. Horodecki (2006). “Separability of mixed quantum states: Linear contractions approach”. Open Systems & Information Dynamics, 13, 103.DOI: 10.1007/s11080-006-7271-8Google Scholar
  8. O. Rudolph (2005). “Computable cross-norm criterion for separability”. Letters in Mathematical Physics, 70, 57–64.DOI: 10.1007/s11005-004-0767-7Google Scholar
  9. O. Rudolph (2005). “Further results on the cross norm criterion for separability”. Quantum Information Processing, 4, 219–239.DOI: 10.1007/s11128-005-5664-1Google Scholar
  10. J.M. Leinaas, J. Myrheim and E. Ovrum (2006). “Geometrical aspects of entanglement”. Physical Review A, 74: 3, 012313.DOI: 10.1103/PhysRevA.74.012313Google Scholar
  11. W. Dür, G. Vidal and J.I. Cyrac (2002). “Optimal conversion of nonlocal unitary operations”. Physical Review Letters, 89: 5, 057901.DOI: 10.1103/PhysRevLett.89.057901Google Scholar
  12. B. Kraus and J.I. Cyrac (2001). “Optimal creation of entanglement using a two-qubit gate”. Physical Review A, 63: 6, 062309.DOI: 10.1103/PhysRevA.63.062309Google Scholar
  13. O. Gittsovich, O. Gühne, P. Hyllus and J. Eisert (2008). “Unifying several separability conditions using the covariance matrix criterion”. Physical Review A, 78, 052319.DOI: 10.1103/PhysRevA.78.052319Google Scholar
  14. P. Horodecki, M. Lewenstein, G. Vidal and I. Cirac (2000). “Operational criterion and constructive checks for the separability of low-rank density matrices”. Physical Review A, 62: 3, 032310.DOI: 10.1103/PhysRevA.62.032310Google Scholar
  15. P. Horodecki, J.A. Smolin, B.M. Terhal and A.V. Thapliyal (2003). “Rank two bipartite bound entangled states do not exist”. Theoretical Computer Science, 292: 3, 589–596.DOI: 10.1016/S0304-3975(01)00376-0Google Scholar
  16. C.-K. Li, Y.-T. Poon and X. Wang (2014). “Ranks and eigenvalues of states with prescribed reduced states”. Electronic Journal of Linear Algebra, 27, 935–950.DOI: 10.13001/1081-3810.2882Google Scholar
  17. H. Chen (2003). “Schmidt numbers of low-rank bipartite mixed states”. Physical Review A, 67: 6, 062301.DOI: 10.1103/PhysRevA.67.062301Google Scholar
  18. E.P. Hanson, C. Rouzé and D. Stilck França (2020). “Eventually entanglement breaking markovian dynamics: Structure and characteristic times”. Annales Henri Poincaré, 21, 1517–1571.DOI: 10.1007/s00023-020-00906-4Google Scholar
  19. M. Weiner (2013). “A gap for the maximum number of mutually unbiased bases”. Proceedings of the American Mathematical Society, 141: 6, 1963–1969.DOI: 10.1090/S0002-9939-2013-11487-5Google Scholar
  20. D. Cariello (2014). “Separability for weakly irreducible matrices”. Quantum Information & Computation, 14: 15–16, 1308–1337.DOI: 10.26421/QIC14.15-16-4Google Scholar
  21. D. Cariello (2019). “Sinkhorn-Knopp theorem for rectangular positive maps”. Linear and Multilinear Algebra, 67, 2345–2365.DOI: 10.1080/03081087.2018.1491524Google Scholar
  22. M. Marcus and H. Minc (1992). A Survey of Matrix Theory and Matrix Inequalities, volume 14, Courier Corporation.Google Scholar
  23. R. Sinkhorn and P. Knopp (1967). “Concerning nonnegative matrices and doubly stochastic matrices”. Pacific Journal of Mathematics, 21: 2, 343–348.DOI: 10.2140/pjm.1967.21.343Google Scholar
  24. R. Bhatia (2009). Positive Definite Matrices, Princeton University Press.DOI: 10.1515/9781400827787Google Scholar