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


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.

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  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)

  2. 2.

    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Størmer, E.: Positive linear maps of operator algebras. Acta Math. 110, 233–278 (1963)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Woronowicz, S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10, 165–183 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Dür, W., Cirac, J.I., Tarrach, R.: Separability and distillability of multiparticle quantum systems. Phys. Rev Lett. 83, 3562–3565 (1999)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  8. 8.

    Rau, A.R.P.: Algebraic characterization of X-states in quantum information. J. Phys. A Math. Theor. 42, 412002 (2009)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Vinjanampathy, S., Rau, A.R.P.: Generalized \(X\) states of \(N\) qubits and their symmetries. Phys. Rev. A 82, 032336 (2010)

    ADS  Article  Google Scholar 

  10. 10.

    Weinstein, Y.S.: Entanglement dynamics in three-qubit \(X\) states. Phys. Rev. A 82, 032326 (2010)

    ADS  Article  Google Scholar 

  11. 11.

    Yu, T., Eberly, J.H.: Quantum open system theory: bipartite aspects. Phys. Rev. Lett. 97, 140403 (2006)

    ADS  Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    ADS  Article  Google 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)

  15. 15.

    Gühne, O.: Entanglement criteria and full separability of multi-qubit quantum states. Phys. Lett. A 375, 406–410 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Gühne, O., Seevinck, M.: Separability criteria for genuine multiparticle entanglement. New J. Phys. 2, 053002 (2010)

    Article  Google Scholar 

  17. 17.

    Kay, A.: Optimal detection of entanglement in Greenberger–Horne–Zeilinger states. Phys. Rev. A 83, 020303(R) (2011)

    ADS  Article  Google 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)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Han, K.H., Kye, S.-H.: The role of phases in detecting three qubit entanglement. J. Math. Phys. 58, 102201 (2017)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kye, S.-H.: Three-qubit entanglement witnesses with the full spanning properties. J. Phys. A Math. Theor. 48, 235303 (2015)

    ADS  MathSciNet  Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Choi, M.-D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Chen, L., Djoković, D.Ž.: Boundary of the set of separable states. Proc. Roral Soc. A 471, 20150102 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Ha, K.-C., Kye, S.-H.: Separable states with unique decompositions. Commun. Math. Phys. 328, 131–153 (2014)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  32. 32.

    Seevinck, M., Uffink, J.: Partial separability and entanglement criteria for multiqubit quantum states. Phys. Rev. A 78, 032101 (2007)

    ADS  MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Kyung Hoon Han.

Additional information

Both K.H.H. and S.H.K. were partially supported by NRF-2017R1A2B4006655, Korea. K.C.H. was partially supported by NRF-2016R1D1A1A09916730, Korea.

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Ha, K., Han, K.H. & Kye, S. Separability of multi-qubit states in terms of diagonal and anti-diagonal entries. Quantum Inf Process 18, 34 (2019).

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

Mathematics Subject Classification

  • 81P15
  • 15A30