Quantum correlations in a family of bipartite separable qubit states

  • Chuanmei Xie
  • Yimin Liu
  • Jianlan Chen
  • Zhanjun Zhang


Quantum correlations (QCs) in some separable states have been proposed as a key resource for certain quantum communication tasks and quantum computational models without entanglement. In this paper, a family of nine-parameter separable states, obtained from arbitrary mixture of two sets of bi-qubit product pure states, is considered. QCs in these separable states are studied analytically or numerically using four QC quantifiers, i.e., measurement-induced disturbance (Luo in Phys Rev A77:022301, 2008), ameliorated MID (Girolami et al. in J Phys A Math Theor 44:352002, 2011),quantum dissonance (DN) (Modi et al. in Phys Rev Lett 104:080501, 2010), and new quantum dissonance (Rulli in Phys Rev A 84:042109, 2011), respectively. First, an inherent symmetry in the concerned separable states is revealed, that is, any nine-parameter separable states concerned in this paper can be transformed to a three-parameter kernel state via some certain local unitary operation. Then, four different QC expressions are concretely derived with the four QC quantifiers. Furthermore, some comparative studies of the QCs are presented, discussed and analyzed, and some distinct features about them are exposed. We find that, in the framework of all the four QC quantifiers, the more mixed the original two pure product states, the bigger QCs the separable states own. Our results reveal some intrinsic features of QCs in separable systems in quantum information.


Separable qubit states Quantum correlations Measurement-induced disturbance Ameliorated MID Quantum dissonance 



Authors are very grateful to the anonymous referees and the associated editor Dr. Michael Frey for their constructive and detailed suggestions. This work is supported by the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11375011 and 11372122, the Natural Science Foundation of Anhui province under Grant No. 1408085MA12, and the 211 Project of Anhui University.


  1. 1.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696 (1935)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Ekert, A.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bennett, C.H., Brassard, G., Mermin, N.D.: Quantum cryptography without bell’s theorem. Phys. Rev. Lett. 68, 557 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Long, G.L., Liu, X.S.: Theoretically efficient high-capacity quantum-key-distribution scheme. Phys. Rev. A 65, 032302 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    Bennett, C.H., Brassard, G., Crepeau, C., et al.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zhang, Z.J., Liu, Y.M.: Perfect teleportation of arbitrary n-qudit states using different quantum channels. Phys. Lett. A 372, 28 (2007)ADSCrossRefMATHGoogle Scholar
  8. 8.
    Cheung, C.Y., Zhang, Z.J.: Criterion for faithful teleportation with an arbitrary multiparticle channel. Phys. Rev. A 80, 022327 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    Bouwmeester, D., et al.: Experimental quantum teleportation. Nature 390, 575 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    Furusawa, A., et al.: Unconditional quantum teleportation. Science 282, 706 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    Boschi, D., et al.: Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 80, 1121 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Xiao, L., Long, G.L., Deng, F.G., Pan, J.W.: Efficient multiparty quantum-secret-sharing schemes. Phys. Rev. A 69, 052307 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Zhang, Z.J., Man, Z.X.: Multiparty quantum secret sharing of classical messages based on entanglement swapping. Phys. Rev. A 72, 022303 (2005)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Li, T., Ren, B.C., Wei, H.R., Hua, M., Deng, F.G.: High-efficiency multipartite entanglement purification of electron-spin states with charge detection. Quantum Inf. Process. 12, 855 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bennett, C.H., DiVincenzo, D.P., Shor, P.W., Smolin, J.A.: Remote state preparation. Phys. Rev. Lett. 87, 077902 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Lo, H.K.: Classical-communication cost in distributed quantum-information processing: a generalization of quantum-communication complexity. Phys. Rev. A 62, 012313 (2000)ADSCrossRefGoogle Scholar
  18. 18.
    Yu, C.S., Song, H.S., Wang, Y.H.: Remote preparation of a qudit using maximally entangled states of qubits. Phys. Rev. A 73, 022340 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    Deng, F.G., Long, G.L., Liu, X.S.: Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block. Phys. Rev. A 68, 042317 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Zhu, A.D., Xia, Y., Fan, Q.B., Zhang, S.: Secure direct communication based on secret transmitting order of particles. Phys. Rev. A 73, 022338 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    Li, Q.T., Cui, J.L., Wang, S.H., Long, G.L.: Study of a monogamous entanglement measure for three-qubit quantum systems. Quantum Inf. Process. 15, 2405 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Munro, W.J., Van, M.R., et al.: High-bandwidth hybrid quantum repeater. Phys. Rev. Lett. 101, 040502 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    Zukowski, M., Zeilinger, A., et al.: “Event-Ready-Detectors” bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287 (1993)ADSCrossRefGoogle Scholar
  24. 24.
    Goebel, A.M., Wagenknecht, C., Zhang, Q., et al.: Multistage entanglement swapping. Phys. Rev. Lett. 101, 080403 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    Branciard, C., Gisin, N., Pironio, S.: Characterizing the nonlocal correlations created via entanglement swapping. Phys. Rev. Lett. 104, 170401 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefMATHGoogle Scholar
  27. 27.
    Luo, S.: Using measurement-induced disturbance to characterize correlations as classical or quantum. Phys. Rev. A 77, 022301 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    Modi, K., Paterek, T., Son, W., Vedral, V., Williamson, M.: Unified view of quantum and classical correlations. Phys. Rev. Lett. 104, 080501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Dakic, B., Vedral, V., Brukner, C.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefMATHGoogle Scholar
  30. 30.
    Luo, S., Fu, S.: Geometric measure of quantum discord. Phys. Rev. A 82, 034302 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zhou, T., Cui, J., Long, G.L.: Measure of nonclassical correlation in coherence-vector representation. Phys. Rev. A 84, 062105 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Girolami, D., Paternostro, M., Adesso, G.: Faithful nonclassicality indicators and extremal quantum correlations in two-qubit states. J. Phys. A Math. Theor. 44, 352002 (2011)CrossRefMATHGoogle Scholar
  33. 33.
    Rulli, C.C., Sarandy, M.S.: Global quantum discord in multipartite systems. Phys. Rev. A 84, 042109 (2011)ADSCrossRefGoogle Scholar
  34. 34.
    Zhang, F.L., Chen, J.L.: Irreducible multiqutrit correlations in Greenberger-Horne-Zeilinger-type states. Phys. Rev. A. 84, 062328 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Ye, B.L., Liu, Y.M., Chen, J.L., Liu, X.S., Zhang, Z.J.: Analytic expressions of quantum correlations in qutrit Werner states. Quantum Inf. Process. 12, 2355 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Tang, H.J., Liu, Y.M., Chen, J.L., Ye, B.L., Zhang, Z.J.: Analytic expressions of discord and geometric discord in Werner derivatives. Quantum Inf. Process. 13, 1331 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008)ADSCrossRefGoogle Scholar
  39. 39.
    Madhok, V., Datta, A.: Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011)ADSCrossRefGoogle Scholar
  40. 40.
    Dakic, B., Lipp, Y.O., Ma, X., et al.: Quantum discord as resource for remote state preparation. Nat. Phys. 8, 666 (2012)CrossRefGoogle Scholar
  41. 41.
    Li, B., Fei, S.M., Wang, Z.X., Fan, H.: Assisted state discrimination without entanglement. Phys. Rev. A 85, 022328 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    Sarandy, M.S.: Classical correlation and quantum discord in critical systems. Phys. Rev. A 80, 022108 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    Lloyd, S.: Quantum search without entanglement. Phys. Rev. A 61, 010301 (1999)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Meyer, D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85, 2014 (2000)ADSCrossRefGoogle Scholar
  45. 45.
    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655 (2012)ADSCrossRefGoogle Scholar
  46. 46.
    Werlang, T., Souza, S., Fanchini, F.F., Villas, B.C.J.: Robustness of quantum discord to sudden death. Phys. Rev. A 80, 024103 (2009)ADSCrossRefGoogle Scholar
  47. 47.
    Wei, H.R., Ren, B.C., Deng, F.G.: Geometric measure of quantum discord for a two-parameter class of states in a qubitCqutrit system under various dissipative channels. Quantum Inf. Process. 12, 1109 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Ren, B.C., Wei, H.R., Deng, F.G.: Correlation dynamics of a two-qubit system in a bell-diagonal state under non-identical local noises. Quantum Inf. Process. 13, 1175 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Guo, J.L., Li, H., Long, G.L.: Decoherent dynamics of quantum correlations in qubit–qutrit systems. Quantum Inf. Process. 12, 3421 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Li, B., Wang, Z.X., Fei, S.M.: Geometry for a class of two-qubit states. Phys. Rev. A 83, 022321 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Shi, M., Sun, C., Jiang, F., Yan, X., Du, J.: Optimal measurement for quantum discord of two-qubit states. Phys. Rev. A 85, 064104 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    Wei, H.R., Ren, B.C., Deng, F.G.: Geometric measure of quantum discord for a two-parameter class of states in a qubit–qutrit system under various dissipative channels. Quantum Inf. Process. 12, 1109 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Giorgi, G.L., Bellomo, B., Galve, F., Zambrini, R.: Genuine quantum and classical correlations in multipartite systems. Phys. Rev. Lett. 107, 190501 (2011)ADSCrossRefGoogle Scholar
  54. 54.
    Giorda, P., Paris, M.G.A.: Gaussian quantum discord. Phys. Rev. Lett. 105, 020503 (2010)ADSCrossRefGoogle Scholar
  55. 55.
    Gessner, M., Laine, E.M., Breuer, H.P., Piilo, J.: Correlations in quantum states and the local creation of quantum discord. Phys. Rev. A 85, 052122 (2012)ADSCrossRefGoogle Scholar
  56. 56.
    Madsen, L.S., Berni, A., Lassen, M., Andersen, U.L.: Experimental investigation of the evolution of gaussian quantum discord in an open system. Phys. Rev. Lett. 109, 030402 (2012)ADSCrossRefGoogle Scholar
  57. 57.
    Zou, C., Chen, X., et al.: Photonic simulation of system-environment interaction: non-Markovian processes and dynamical decoupling. Phys. Rev. A 88, 063806 (2013)ADSCrossRefGoogle Scholar
  58. 58.
    Dajka, J., et al.: Swapping of correlations via teleportation with decoherence. Phys. Rev. A 87, 022301 (2013)ADSCrossRefGoogle Scholar
  59. 59.
    Lanyon, B.P., Jurcevic, P., Hempel, C., et al.: Experimental generation of quantum discord via noisy processes. Phys. Rev. Lett. 111, 100504 (2013)ADSCrossRefGoogle Scholar
  60. 60.
    Rana, S., Parashar, P.: Tight lower bound on geometric discord of bipartite states. Phys. Rev. A 85, 024102 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    Debarba, T., Maciel, T.O., Vianna, R.O.: Witnessed entanglement and the geometric measure of quantum discord. Phys. Rev. A 86, 024302 (2012)ADSCrossRefGoogle Scholar
  62. 62.
    Montealegre, J.D., Paula, F.M., Saguia, A., Sarandy, M.S.: One-norm geometric quantum discord under decoherence. Phys. Rev. A 87, 042115 (2013)ADSCrossRefGoogle Scholar
  63. 63.
    Chang, L., Luo, S.: Remedying the local ancilla problem with geometric discord. Phys. Rev. A 87, 062303 (2013)ADSCrossRefGoogle Scholar
  64. 64.
    Miranowicz, A., Horodecki, P., Chhajlany, R.W., Tuziemski, J., Sperling, J.: Analytical progress on symmetric geometric discord: measurement-based upper bounds. Phys. Rev. A 86, 042123 (2012)ADSCrossRefGoogle Scholar
  65. 65.
    Xie, C.M., Liu, Y.M., Xing, H., Chen, J.L., Zhang, Z.J.: Quantum correlation swapping. Quantum Inf. Process. 14, 653 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Xie, C.M., Liu, Y.M., Chen, J.L., Zhang, Z.J.: Study of quantum correlation swapping with relative entropy methods. Quantum Inf. Process. 15, 809 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Cen, L.X., Li, X.Q., Shao, J.S., Yan, Y.J.: Quantifying quantum discord and entanglement of formation via unified purifications. Phys. Rev. A 83, 054101 (2011)ADSCrossRefGoogle Scholar
  68. 68.
    Xie, C.M., Liu, Y.M., Li, G.F., Zhang, Z.J.: A note on quantum correlations in Werner states under two collective noises. Quantum Inf. Process. 13, 2713 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Xie, C.M., Liu, Y.M., Chen, J.L., Yin, X.F., Zhang, Z.J.: Quantum entanglement swapping of two arbitrary bi-qubit pure states. Sci. China Phys. Mech. Astron. 59, 100314 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Chuanmei Xie
    • 1
  • Yimin Liu
    • 2
  • Jianlan Chen
    • 1
  • Zhanjun Zhang
    • 1
  1. 1.School of Physics and Material ScienceAnhui UniversityHefeiChina
  2. 2.Department of PhysicsShaoguan UniversityShaoguanChina

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