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Robustness of \(\varLambda \)-entanglement of multipartite states

  • Ying Yang
  • Huai-Xin CaoEmail author
  • Hui-Xian Meng
Article

Abstract

By introducing the relative robustness of \(\varLambda \)-entanglement of a multipartite state related to a \(\varLambda \)-separable state, the robustness of \(\varLambda \)-entanglement of a multipartite state is defined as the minimum of the relative robustness of \(\varLambda \)-entanglement of a state related to all \(\varLambda \)-separable states. It is proved that, as a function on the set of all quantum states of an n-partite system, robustness of \(\varLambda \)-entanglement is nonnegative, lower semi-continuous, and convex, and it is zero if and only if the state is \(\varLambda \)-separable. Thus, robustness of \(\varLambda \)-entanglement not only quantifies the endurance of \(\varLambda \)-entanglement of a state against linear noise, but also can be used to distinguish \(\varLambda \)-separable states from \(\varLambda \)-entangled states. Furthermore, influences of a quantum channel on robustness of \(\varLambda \)-entanglement are discussed.

Keywords

\(\varLambda \)-separability \(\varLambda \)-entanglement Robustness Quantum channel 

Notes

Acknowledgements

This subject was supported by the National Natural Science Foundation of China (Nos. 11871318, 11771009, 11571213, 11601300) and the Fundamental Research Funds for the Central Universities (GK20181020, GK201801011), China Post-doctoral Science Foundation (No. 2018M631726), Shaanxi Province Innovation Ability Support Program (2018KJXX-054), and Subject Research Project of Yuncheng University (XK-2018032).

References

  1. 1.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Vedral, V., Plenio, M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57, 1619 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    Li, Z.G., Fei, S.M., Albeverio, S., et al.: Bound of entanglement of assistance and monogamy constraints. Phys. Rev. A 80, 034301 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Popescu, S., Rohrlich, D.: Thermodynamics and the measure of entanglement. Phys. Rev. A 56, R3319 (1997)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Vidal, G., Tarrach, R.: Robustness of entanglement. Phys. Rev. A 59, 141–5 (1999)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Du, J.F., Shi, M.J., Zhou, X.Y., Han, R.D.: Geometrical interpretation for robustness of entanglement. Phys. Lett. A 267, 244–250 (2000)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Steiner, M.: Generalized robustness of entanglement. Phys. Rev. A 67, 054305 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    Zha, L., Cao, H.X., Wang, X.X.: Influences of quantum channels on robustness of entanglement. J. Jilin Univ. (Sci. Ed.) 54, 871–877 (2016) (in chinese) Google Scholar
  9. 9.
    Meng, H.X., Cao, H.X., Wang, W.H., Chen, L., Fan, Y.J.: Continuity of the robustness of contextuality of empirical models. Sci. China Phys. Mech. Astron. 59, 100311-1:8 (2016)Google Scholar
  10. 10.
    Guo, Z.H., Cao, H.X., Chen, Z.L.: Distinguishing classical correlations from quantum correlations. J. Phys. A Math. Theor. 45, 145301 (2012)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, Z.H., Cao, H.X., Qu, S.X.: Partial correlations in a multipartite quantum system. Inf. Sci. 289, 262–272 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, Z.H., Cao, H.X., Qu, S.X.: Structures of three types of local quantum channels based on quantum correlations. Found. Phys. 45, 355–369 (2015)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Guo, Z.H., Cao, H.X.: Local quantum channels preserving classical correlations. J. Phys. A Math. Theor. 46, 065303 (2013)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Guo, Z.H., Cao, H.X., Qu, S.X.: Robustness of quantum correlations against linear noise. Found. J. Phys. A Math. Theor. 49, 195301 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Quintino, M.T., Vértesi, T., Cavalcanti, D., et al.: Inequivalence of entanglement, steering, and Bell nonlocality for general measurements. Phys. Rev. A 92, 032107 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    Li, Z.W., Guo, Z.H., Cao, H.X.: Some characterizations of EPR steering. Int. J. Theor. Phys. 57, 3285–3295 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zheng, C.M., Guo, Z.H., Cao, H.X.: Generalized steering robustness of quantum states. Int. J. Theor. Phys. 57, 1787–1801 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cao, H.X., Guo, Z.H.: Characterizing Bell nonlocality and EPR steering. Sci. China Phys. Mech. Astron. 62, 030311 (2019)CrossRefGoogle Scholar
  20. 20.
    Xiao, S., Guo, Z.H., Cao, H.X.: Quantum steering in tripartite quantum systems. Sci. Sin. Phys. Mech. Astron. 49, 010301 (2019). (in Chinese) Google Scholar
  21. 21.
    Yang, Y., Cao, H.X.: Einstein–Podolsky–Rosen steering inequalities and applications. Entropy 20, 683 (2018)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gao, T., Hong, Y., Lu, Y., Yan, Fl: Efficient \(k\)-separability criteria for mixed multipartite quantum states. EPL Europhys. Lett. 104, 20007 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Tóth, G., Knapp, C., Gühne, O., Briegel, H.J.: Optimal spin squeezing inequalities detect bound entanglement in spin models. Phys. Rev. Lett. 99, 250405 (2007)ADSCrossRefGoogle Scholar
  25. 25.
    Acín, A., Brub, D., Lewenstein, M., Sanpera, A.: Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401 (2001)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gühne, O., Seevinck, M.: Separability criteria for genuine multiparticle entanglement. New J. Phys. 12, 053002 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    Huber, M., Mintert, F., Gabriel, A., Hiesmayr, B.C.: Detection of high-dimensional genuine multi-partite entanglement of mixed states. Phys. Rev. Lett. 104, 210501 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    Gabriel, A., Hiesmayr, B.C., Huber, M.: Criterion for \(k\)-separability in mixed multipartite systems. arXiv:1002.2953 (2010)
  29. 29.
    Gao, T., Hong, Y.: Separability criteria for several classes of \(n\)-partite quantum states. Eur. Phy. J. D 61, 765 (2011)ADSCrossRefGoogle Scholar
  30. 30.
    Kwiat, P.G.: Hyper-entangled states. J. Mod. Opt. 44, 2173–2184 (1997)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Yang, T., Zhang, Q., Zhang, J., Yin, J., Zhao, Z., Żukowski, M.: All-versus-nothing violation of local realism by two-photon, four-dimensional entanglement. Phys. Rev. Lett. 95, 240406 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    Barreiro, J.T., Langford, N.K., Peters, N.A., Kwiat, P.G.: Generation of hyperentangled photon pairs. Phys. Rev. Lett. 95, 260501 (2005)ADSCrossRefGoogle Scholar
  33. 33.
    Seevinck, M., Uffink, J.: Partial separability and entanglement criteria for multiqubit quantum states. Phys. Rev. A 78, 4061–4061 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Dür, W., Cirac, J.I.: Multiparticle entanglement and its experimental detection. J. Phys. A Math. Theor. 34, 6837–6850 (2001)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Wölk, S., Gühne, O.: Characterizing the width of entanglement. New J. Phys. 18, 123024 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    Sørensen, A.S., Mølmer, K.: Entanglement and extreme spin squeezing. Phys. Rev. Lett. 86, 4431 (2001)ADSCrossRefGoogle Scholar
  37. 37.
    Gühne, O., Tóth, G., Briegel, H.J.: Multipartite entanglement in spin chains. New J. Phys. 7, 229 (2005)CrossRefGoogle Scholar
  38. 38.
    Yang, Y., Cao, H.X.: Separability criterions of multipartite states. Eur. Phys. J. D 82, 143 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  2. 2.School of Mathematics and Information TechnologyYuncheng UniversityYunchengChina

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