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The quantum Markovianity criterion based on correlations under random unitary qudit dynamical evolutions

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Abstract

Random unitary dynamical evolutions play an important role in characterizing quantum Markovianity. By exploiting the maximally entangled state under the representation of a set of generalized unitary Weyl operators, we present a necessary and sufficient condition for quantum Markovianity in d-level systems from the perspective of quantum mutual information. Moreover, we analyze three quantum Markovianity criteria among divisibility, BLP and LFS\(_{0}\) to establish their connections and differences. It is shown that they coincide in many special cases, but different in general, and Markovianity in the sense of LFS\(_{0}\) is less restrictive than Markovianities in the sense of divisibility and BLP by detailed examples.

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The datasets analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2007)

    Book  MATH  Google Scholar 

  2. Weiss, U.: Quantum Dissipative Systems. World Scientific, Singapore (2000)

    Google Scholar 

  3. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Springer, Berlin (1987)

    MATH  Google Scholar 

  4. Leggett, A.J., Chakravarty, S., Dorsey, A.T., et al.: Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987)

    Article  ADS  Google Scholar 

  5. Breuer, H.-P., Laine, E.-M., Piilo, J., Vacchini, B.: Colloquium: Non-Markovian dynamics in open quantum systems. Rev. Mod. Phys. 88, 021002 (2016)

    Article  ADS  Google Scholar 

  6. de Vega, I., Alonso, D.: Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 89, 015001 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  7. Shirai, T., Thingna, J., Mori, T., et al.: Effective Floquet–Gibbs states for dissipative quantum systems. New J. Phys. 18, 053008 (2016)

    Article  ADS  Google Scholar 

  8. Flakowski, J., Osmanov, M., Taj, D., Öttinger, H.C.: Biexponential decay and ultralong coherence of a qubit. Europhys. Lett. 113, 40003 (2016)

    Article  ADS  Google Scholar 

  9. Dive, B., Burgarth, D., Mintert, F.: Isotropy and control of dissipative quantum dynamics. Phys. Rev. A 94, 012119 (2016)

    Article  ADS  Google Scholar 

  10. Vu, T.V., Hasegawa, Y.: Geometrical bounds of the irreversibility in Markovian systems. Phys. Rev. Lett. 126, 010601 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  11. Rivas, Á., Huelga, S.F., Plenio, M.B.: Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 77, 094001 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  12. Wolf, M.M., Eisert, J., Cubitt, T.S., Cirac, J.I.: Assessing non-Markovian quantum dynamics. Phys. Rev. Lett. 101, 150402 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18, 97–133 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lewis, J.T.: Quantum stochastic processes I\(^{*}\). Phys. Rep. 77, 339–349 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  15. 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)

    Article  ADS  MathSciNet  Google Scholar 

  16. Laine, E.-M., Piilo, J., Breuer, H.-P.: Measure for the non-Markovianity of quantum processes. Phys. Rev. A 81, 062115 (2010)

    Article  ADS  Google Scholar 

  17. Breuer, H.-P.: Foundations and measures of quantum non-Markovianity. J. Phys. B: At. Mol. Opt. Phys. 45, 154001 (2012)

    Article  ADS  Google Scholar 

  18. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of \(N\)-level systems. J. Math. Phys. 17, 821 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Rivas, Á., Huelga, S.F., Plenio, M.B.: Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett. 105, 050403 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  21. Chruściński, D., Kossakowski, A.: Non-Markovian quantum dynamics: local versus nonlocal. Phys. Rev. Lett. 104, 070406 (2010)

    Article  ADS  Google Scholar 

  22. Chruściński, D., Kossakowski, A.: From Markovian semigroup to non-Markovian quantum evolution. Europhys. Lett. 97, 20005 (2012)

    Article  ADS  MATH  Google Scholar 

  23. 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)

    Article  ADS  Google Scholar 

  24. Chruściński, D., Wudarski, F.A.: Non-Markovian random unitary qubit dynamics. Phys. Lett. A 377, 1425 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Luo, S.L., Fu, S.S., Song, H.T.: Quantifying non-Markovianity via correlations. Phys. Rev. A 86, 044101 (2012)

    Article  ADS  Google Scholar 

  26. Jiang, M., Luo, S.L.: Comparing quantum Markovianities: distinguishability versus correlations. Phys. Rev. A 88, 034101 (2013)

    Article  ADS  Google Scholar 

  27. Lorenzo, S., Plastina, F., Paternostro, M.: Geometrical characterization of non-Markovianity. Phys. Rev. A 88, 020102 (R) (2013)

    Article  ADS  Google Scholar 

  28. Ciccarello, F., Palma, G.M., Giovannetti, V.: Collision-model-based approach to non-Markovian quantum dynamics. Phys. Rev. A 87, 040103 (R) (2013)

    Article  ADS  Google Scholar 

  29. Chruściński, D., Maniscalco, S.: Degree of non-Markovianity of quantum evolution. Phys. Rev. Lett. 112, 120404 (2014)

    Article  ADS  Google Scholar 

  30. Hall, M.J.W., Cresser, J.D., Li, L., Andersson, E.: Canonical form of master equations and characterization of non-Markovianity. Phys. Rev. A 89, 042120 (2014)

    Article  ADS  Google Scholar 

  31. Fanchini, F.F., Karpat, G., Cakmak, B., et al.: Non-Markovianity through accessible information. Phys. Rev. Lett. 112, 210402 (2014)

    Article  ADS  Google Scholar 

  32. Chruściński, D., Kossakowski, A.: Witnessing non-Markovianity of quantum evolution. Eur. Phys. J. D 68, 7 (2014)

    Article  ADS  Google Scholar 

  33. Li, L., Hall, M.J.W., Wiseman, H.M.: Concepts of quantum non-Markovianity: A hierarchy. Phys. Rep. 759, 1 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Huang, Z.Q., Guo, X.-K.: Quantifying non-Markovianity via conditional mutual information. Phys. Rev. A 104, 032212 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  35. Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin (1999)

    MATH  Google Scholar 

  36. Plenioand, M.B., Knight, P.L.: The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys. 70, 101 (1998)

    Article  ADS  Google Scholar 

  37. Aharonov, D., Kitaev, A., Preskill, J.: Fault-tolerant quantum computation with Long-Range correlated noise. Phys. Rev. Lett. 96, 050504 (2006)

    Article  ADS  Google Scholar 

  38. Andersson, E., Cresser, J.D., Hall, M.J.W.: Finding the Kraus decomposition from a master equation and vice versa. J. Mod. Opt. 54, 1695 (2007)

    Article  ADS  MATH  Google Scholar 

  39. Nakajima, S.: On quantum theory of transport phenomena: Steady diffusion. Prog. Theor. Phys. 20, 948 (1958)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Zwanzig, R.: Ensemble method in the theory of irreversibility. J. Chem. Phys. 33, 1338 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  41. Chruściński, D., Kossakowski, A.: Markovianity criteria for quantum evolution. J. Phys. B: At. Mol. Opt. Phys. 45, 154002 (2012)

    Article  ADS  Google Scholar 

  42. Chruściński, D., Wudarski, F.A.: Non-Markovianity degree for random unitary evolution. Phys. Rev. A 91, 012104 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  43. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

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Acknowledgements

This work is supported by the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515310016 and Key Research and Development Project of Guangdong Province under Grant No. 2020B0303300001; Beijing Natural Science Foundation (Z190005), and Academy for Multidisciplinary Studies, Capital Normal University; National Natural Science Foundation of China (NSFC) under Grants 12075159 and 12171044; Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (No. SIQSE202001); the Academician Innovation Platform of Hainan Province. We thank Mao-Sheng Li for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Wen Xu & Zhu-Jun Zheng

  2. Laboratory of Quantum Science and Engineering, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Wen Xu & Zhu-Jun Zheng

  3. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Shao-Ming Fei

  4. Max-Planck-Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Shao-Ming Fei

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  1. Wen Xu
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  2. Zhu-Jun Zheng
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  3. Shao-Ming Fei
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Correspondence to Zhu-Jun Zheng.

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Xu, W., Zheng, ZJ. & Fei, SM. The quantum Markovianity criterion based on correlations under random unitary qudit dynamical evolutions. Quantum Inf Process 21, 221 (2022). https://doi.org/10.1007/s11128-022-03574-x

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