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Separability of multi-qubit states in terms of diagonal and anti-diagonal entries

  • Kil-Chan Ha
  • Kyung Hoon Han
  • Seung-Hyeok Kye
Article
  • 39 Downloads

Abstract

We give separability criteria for general multi-qubit states in terms of diagonal and anti-diagonal entries. We define two numbers which are obtained from diagonal and anti-diagonal entries, respectively, and compare them to get criteria. They give rise to characterizations of separability when all the entries are zero except for diagonal and anti-diagonal, like Greenberger–Horne–Zeilinger diagonal states. The criteria are strong enough to detect nonzero volume of entanglement with positive partial transposes.

Keywords

Multi-qubit states X-states Separability criterion Irreducible balanced multisets Phase identities Phase difference 

Mathematics Subject Classification

81P15 15A30 

References

  1. 1.
    Choi, M.-D.: Operator algebras and applications. In: Kadison, R.V. (ed.) Part 2: Proceedings of the 28th Summer Institute of the American Mathematical Society, Queen’s University, Kingston, ON, July 14–August 2, 1980; Proceedings of Symposia in Pure Mathematics, vol. 38. American Mathematical Society, Providence, RI (1982)Google Scholar
  2. 2.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Størmer, E.: Positive linear maps of operator algebras. Acta Math. 110, 233–278 (1963)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Woronowicz, S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10, 165–183 (1976)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Dür, W., Cirac, J.I., Tarrach, R.: Separability and distillability of multiparticle quantum systems. Phys. Rev Lett. 83, 3562–3565 (1999)ADSCrossRefGoogle Scholar
  7. 7.
    Mendonca, P.E.M.F., Rafsanjani, S.M.H., Galetti, D., Marchiolli, M.A.: Maximally genuine multipartite entangled mixed X-states of N-qubits. J. Phys. A Math. Theor. 48, 215304 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Rau, A.R.P.: Algebraic characterization of X-states in quantum information. J. Phys. A Math. Theor. 42, 412002 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Vinjanampathy, S., Rau, A.R.P.: Generalized \(X\) states of \(N\) qubits and their symmetries. Phys. Rev. A 82, 032336 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Weinstein, Y.S.: Entanglement dynamics in three-qubit \(X\) states. Phys. Rev. A 82, 032326 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Yu, T., Eberly, J.H.: Quantum open system theory: bipartite aspects. Phys. Rev. Lett. 97, 140403 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Bouwmeester, D., Pan, J.W., Daniell, M., Weifurter, H., Zeilinger, A.: Observation of three-photon Greenberger–Horne–Zeilinger entanglement. Phys. Rev. Lett. 82, 1345 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequality. Am. J. Phys. 58, 1131–1143 (1990)ADSCrossRefGoogle Scholar
  14. 14.
    Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Kluwer Academic Publishers. pp. 69–72 (1989)Google Scholar
  15. 15.
    Gühne, O.: Entanglement criteria and full separability of multi-qubit quantum states. Phys. Lett. A 375, 406–410 (2011)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gühne, O., Seevinck, M.: Separability criteria for genuine multiparticle entanglement. New J. Phys. 2, 053002 (2010)CrossRefGoogle Scholar
  17. 17.
    Kay, A.: Optimal detection of entanglement in Greenberger–Horne–Zeilinger states. Phys. Rev. A 83, 020303(R) (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Han, K.H., Kye, S.-H.: Separability of three qubit Greenberger–Horne–Zeilinger diagonal states. Phys. A Math. Theor. 50, 145303 (2017)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Han, K.H., Kye, S.-H.: The role of phases in detecting three qubit entanglement. J. Math. Phys. 58, 102201 (2017)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chen, L., Han, K.H., Kye, S.-H.: Separability criterion for three-qubit states with a four dimensional norm. J. Phys. A Math. Theor. 50, 345303 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kye, S.-H.: Three-qubit entanglement witnesses with the full spanning properties. J. Phys. A Math. Theor. 48, 235303 (2015)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Han, K.H., Kye, S.-H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A Math. Theor. 49, 175303 (2016)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Han, K.H., Kye, S.-H.: Various notions of positivity for bi-linear maps and applications to tri-partite entanglement. J. Math. Phys. 57, 015205 (2016)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Kye, S.-H.: Facial structures for various notions of positivity and applications to the theory of entanglement. Rev. Math. Phys. 25, 1330002 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Choi, M.-D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Chen, L., Djoković, D.Ž.: Boundary of the set of separable states. Proc. Roral Soc. A 471, 20150102 (2015)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Ha, K.-C., Kye, S.-H.: Separable states with unique decompositions. Commun. Math. Phys. 328, 131–153 (2014)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Ha, K.-C., Kye, S.-H.: Construction of exposed indecomposable positive linear maps between matrix algebras. Linear Multilinear Algorithms 64, 2188–2198 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kye, S.-H.: Indecomposable exposed positive bi-linear maps between two by two matrices. Acta Math. Vietnam. 43(4), 619–627 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gao, T., Hong, Y.: Separability criteria for several classes of \(n\)-partite quantum states. Eur. Phys. J. D 61, 765–771 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Rafsanjani, S.M.H., Huber, M., Broadbent, C.J., Eberly, J.H.: Genuinely multipartite concurrence of N-qubit X matrices. Phys. Rev. A 86, 062303 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Seevinck, M., Uffink, J.: Partial separability and entanglement criteria for multiqubit quantum states. Phys. Rev. A 78, 032101 (2007)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and StatisticsSejong UniversitySeoulKorea
  2. 2.Department of Data ScienceThe University of SuwonHwaseongKorea
  3. 3.Department of Mathematics, Institute of MathematicsSeoul National UniversitySeoulKorea

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