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International Journal of Theoretical Physics

, Volume 57, Issue 9, pp 2722–2737 | Cite as

Quantum Non-Markovianity via the Covariance Matrix

  • Yajing Fan
  • Liang Chen
  • Huaixin Cao
  • Huixian Meng
Article

Abstract

In this paper, we propose a new characterization of non-Markovian quantum evolution based on the covariance matrix. The fundamental properties of covariance matrices are elucidated. The measure captures quite directly the characteristics of non-Markovianity from the perspective of uncertainty. We consider several typical examples and compare the covariance matrix characterization of quantum non-Markovianity with Fisher-information matrix, divisibility and the Breuer-Laine-Piilo characterization of quantum non-Markovianity.

Keywords

Markovianity Covariance matrix Quantum evolution 

Notes

Acknowledgements

This subject was supported by the NNSF of China (Nos. 11701011, 11601300, 61462002, 11761001, 11761003, 61463001), the SRP for North Minzu University (No. 2017SXKY02),the First-Class Disciplines Foundation of Ningxia(No. NXYLXK20 17B09).

References

  1. 1.
    Breuer, H.P., Vacchini, B.: Quantum semi-Markov processes. Phys. Rev. Lett. 101, 140402 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Piilo, J., Maniscalco, S., Härkönen, K., Suominen, K.A.: Non-Markovian quantum jumps. Phys. Rev. Lett. 100, 180402 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, W.M., Lo, P.Y., Xiong, H.N., Tu, M.W.Y., Nori, F.: General non-Markovian dynamics of open quantum systems. Phys. Rev. Lett. 109, 170402 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Chruściński, D., Kossakowski, A.: Non-Markovian quantum dynamics: Local versus nonlocal. Phys. Rev. Lett. 104, 070406 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Wolf, M.M., Cirac, J.I.: Dividing quantum channels. Comm. Math. Phys. 279, 147–168 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hou, S.C., Yi, X.X., Yu, S.X., Oh, C.H.: Alternative non-Markovianity measure by divisibility of dynamical maps. Phys. Rev. A 83, 062115 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Breuer, H.P., Laine, E.M., Piilo, J.: Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lu, X.M., Wang, X., Sun, C.P.: Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A 82, 042103 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Rajagopal, A.K., Usha Devi, A.R., Rendell, R.W.: Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms. Phys. Rev. A 82, 042107 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Luo, S., Fu, S., Song, H.: Quantifying non-Markovianity via correlation. Phys. Rev. A 86, 044101 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Jiang, M., Luo, S.: Comparing quantum Markovianities: Distinguishability versus correlations. Phys. Rev. A 88, 034101 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Lorenzo, S., Plastina, F., Paternostro, M.: Geometrical characterization of non-Markovianity. Phys. Rev. A 88(2), 8323–8331 (2013)CrossRefGoogle Scholar
  13. 13.
    Benatti, F., Floreanini, R., Scholeseds, G.: Special issue on loss of coherence and memory effects in quantum dynamics. J. Phys. B 45, 15 (2012)CrossRefGoogle Scholar
  14. 14.
    Chruściński, D., Wudarski, F.: Non-Markovian random unitary qubit dynamics. Phys. Lett. A 377, 1425 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chruściński, D., Maniscalco, S.: Degree of non-Markovianity of quantum evolution. Phys. Rev. Lett. 112, 120404 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    Chruściński, D., Kossakowski, A.: Witnessing non-Markovianity of quantum evolution. Eur. Phys. J. D 68, 7 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Chruściński, D., Wudarski, F.: Non-Markovianity degree for random unitary evolution. Phys. Rev. A 91, 012104 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Song, H., Luo, S., Hong, Y.: Quantum non-Markovianity based on the Fisher-information matrix. Phys. Rev. A 91, 042110 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Paula, F.M., Obando, P.C.: Non-Markovianity through multipartite correlation measures. Phys. Rev. A 93, 042337 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Karlsson, A., Lyyra, H., Laine, E.M., Maniscalco, S., Piilo, J.: Non-Markovian dynamics in two-qubit dephasing channels with an application to superdense coding. Phys. Rev. A 93, 032135 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    Mahmoudi, M., Mahdavifar, S., Mohammad Ali Zadeh, T., Soltani, M.R.: Non-Markovian dynamics in the extended cluster spin-1/2 XX chain. Phys. Rev. A 95, 012336 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    Werner, R.F., Wolf, M.M.: Bound entangled gaussian states. Phys. Rev. Lett. 86, 3658 (2001)ADSCrossRefGoogle Scholar
  23. 23.
    Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)ADSCrossRefGoogle Scholar
  24. 24.
    Gühne, O., Hyllus, P., Gittsovich, O., Eisert, J.: Covariance matrices and the separability problem. Phys. Rev. Lett. 99, 130504 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Robertson, H.P.: An indeterminacy relation for several observables and its classical interpretation. Phys. Rev. A 46, 794 (1934)ADSCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceNorth Minzu UniversityYinchuanChina
  2. 2.Department of MathematicsChangji CollegeChangjiChina
  3. 3.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  4. 4.Theoretical Physics Division, Chern Institute of MathematicsNankai UniversityTianjinChina

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