Separable decompositions of bipartite mixed states



We present a practical scheme for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices, using the technique developed in Li and Qiao (Sci. Rep. 8:1442, 2018). In the scheme, the correlation matrix which characterizes the bipartite entanglement is first decomposed into two matrices composed of the Bloch vectors of local states. Then, we show that the symmetries of Bloch vectors are consistent with that of the correlation matrix, and the magnitudes of the local Bloch vectors are lower bounded by the correlation matrix. Concrete examples for the separable decompositions of bipartite mixed states are presented for illustration.


Entanglement of mixed states Separability criterion Bloch vector Matrix decomposition 



This work was supported in part by the Ministry of Science and Technology of the Peoples’ Republic of China (2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB23030100); and by the National Natural Science Foundation of China (NSFC) under the Grants 11375200 and 11635009.


  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  2. 2.
    Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69, 448–484 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219–239 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, K., Ling-An, W.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193–202 (2003)MathSciNetMATHGoogle Scholar
  8. 8.
    de Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 7, 624–638 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Li, M., Wang, J., Fei, S.-M., Li-Jost, X.: Quantum separability criteria for arbitrary-dimensional multipartite states. Phys. Rev. A 89, 022325 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Hofmann, H.F., Takeuchi, S.: Violation of local uncertainty relations as a signature of entanglement. Phys. Rev. A 68, 032103 (2003)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Sperling, J., Vogel, W.: Necessary and sufficient conditions for bipartite entanglement. Phys. Rev. A 79, 022318 (2009)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, C.-J., Nha, H., Zhang, Y.-S., Guo, G.-C.: Entanglement detection via tighter local uncertainty relations. Phys. Rev. A 81, 012324 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Gühne, O., Hyllus, P., Gittsovich, O., Eisert, J.: Covariance matrices and the separability problem. Phys. Rev. Lett. 99, 130504 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gittsovich, O., Gühne, O.: Quantifying entanglement with covariance matrices. Phys. Rev. A 81, 032333 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    Li, J.-L., Qiao, C.-F.: A necessary and sufficient criterion for the separability of quantum state. Sci. Rep. 8, 1442 (2018)ADSCrossRefGoogle Scholar
  16. 16.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)MATHGoogle Scholar
  17. 17.
    Verstraete, F., Dehaene, J., De Moor, B.: Normal forms and entanglement measures for multipartite quantum state. Phys. Rev. A 68, 012103 (2003)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Zangi, S.M., Li, J.-L., Qiao, C.-F.: Entanglement classification of four-partite states under the SLOCC. J. Phys. A Math. Theor. 50, 325301 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  21. 21.
    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, Berlin (2011)CrossRefMATHGoogle Scholar
  22. 22.
    Kimura, G.: The Bloch vector for \(N\)-level systems. Phys. Lett. A 314, 339–349 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rungta, P., Bužek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Jevtic, S., Pusey, M., Jennings, D., Rudolph, T.: Quantum steering ellipsoids. Phys. Rev. Lett. 113, 020402 (2014)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhysicsUniversity of the Chinese Academy of SciencesBeijingChina
  2. 2.Key Laboratory of Vacuum PhysicsUniversity of Chinese Academy of SciencesBeijingChina

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