Coherence-based measure of quantumness in (non-) Markovian channels

  • Javid NaikooEmail author
  • Subhashish Banerjee


We make a detailed analysis of quantumness for various quantum noise channels, both Markovian and non-Markovian. The noise channels considered include dephasing channels like random telegraph noise, non-Markovian dephasing and phase damping, as well as the non-dephasing channels such as generalized amplitude damping and Unruh channels. We make use of a recently introduced witness for quantumness based on the square \(l_1\) norm of coherence. It is found that the increase in the degree of non-Markovianity increases the quantumness of the channel. This may be attributed to the fact that the non-Markovian dynamics involves the generation of entanglement between the system and environment degrees of freedom.


Channels Quantumness Non-Markovian dynamics 



We thank Prof. R. Srikanth of PPISR, Bangalore, India, for useful discussions during the preparation of this manuscript.


  1. 1.
    Hu, M.-L., Hu, X., Wang, J., Yi, P., Zhang, Y.-R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762, 1–100 (2018)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., Bessette, F., Brassard, G., Salvail, L., Smolin, J.: Experimental quantum cryptography. J. Cryptol. 5(1), 3–28 (1992)CrossRefGoogle Scholar
  4. 4.
    Grosshans, F., Van Assche, G., Wenger, J., Brouri, R., Cerf, N.J., Grangier, P.: Quantum key distribution using Gaussian-modulated coherent states. Nature 421(6920), 238 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89(4), 041003 (2017)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Radhakrishnan, C., Ding, Z., Shi, F., Du, J., Byrnes, T.: Basis-independent quantum coherence and its distribution (2018). arXiv preprint arXiv:1805.09263
  8. 8.
    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Almeida, J., De Groot, P.C., Huelga, S.F., Liguori, A.M., Plenio, M.B.: Probing quantum coherence in qubit arrays. J. Phys. B At. Mol. Opt. Phys. 46(10), 104002 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    Lloyd, S.: Quantum coherence in biological systems. J. Phys. Conf. Ser. 302, 012037 (2011)CrossRefGoogle Scholar
  12. 12.
    Bhattacharya, S., Banerjee, S., Pati, A.K.: Evolution of coherence and non-classicality under global environmental interaction. Quantum Inf. Process. 17(9), 236 (2018)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Singh, U., Bera, M.N., Dhar, H.S., Pati, A.K.: Maximally coherent mixed states: complementarity between maximal coherence and mixedness. Phys. Rev. A 91, 052115 (2015). ADSCrossRefGoogle Scholar
  14. 14.
    Yadin, B., Vedral, V.: General framework for quantum macroscopicity in terms of coherence. Phys. Rev. A 93(2), 022122 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Alok, A.K., Banerjee, S., Sankar, S.U.: Quantum correlations in terms of neutrino oscillation probabilities. Nucl. Phys. B 909, 65–72 (2016)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Dixit, K., Naikoo, J., Banerjee, S., Alok, A.K.: Study of coherence and mixedness in meson and neutrino systems. Eur. Phys. J. C 79(2), 96 (2019)ADSCrossRefGoogle Scholar
  17. 17.
    Mani, A., Karimipour, V.: Cohering and decohering power of quantum channels. Phys. Rev. A 92, 032331 (2015). ADSCrossRefGoogle Scholar
  18. 18.
    Maniscalco, S., Olivares, S., Paris, M.G.A.: Entanglement oscillations in non-Markovian quantum channels. Phys. Rev. A 75, 062119 (2007). ADSCrossRefGoogle Scholar
  19. 19.
    Banerjee, S.: Open Quantum Systems: Dynamics of Nonclassical Evolution. Springer, Berlin (2018)CrossRefGoogle Scholar
  20. 20.
    Braunstein, S.L., Fuchs, C.A., Kimble, H.J.: Criteria for continuous-variable quantum teleportation. J. Mod. Opt. 47(2–3), 267–278 (2000)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Horodecki, R., Horodecki, P., Horodecki, M.: Violating bell inequality by mixed spin-12 states: necessary and sufficient condition. Phys. Lett. A 200(5), 340–344 (1995)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Horodecki, R., Horodecki, M., Horodecki, P.: Teleportation, Bell’s inequalities and inseparability. Phys. Lett. A 222(1–2), 21–25 (1996)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Horodecki, M., Horodecki, P., Horodecki, R.: General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 60(3), 1888 (1999)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41(12), 2315–2323 (1994)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Saulo, V.M., Cunha, M.T.: Quantifying quantum invasiveness. Phys. Rev. A 99, 022124 (2019). ADSCrossRefGoogle Scholar
  26. 26.
    Shahbeigi, F., Akhtarshenas, S.J.: Quantumness of quantum channels. Phys. Rev. A 98, 042313 (2018). ADSCrossRefGoogle Scholar
  27. 27.
    Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  28. 28.
    Holevo, A.S.: Quantum Systems, Channels, Information: A Mathematical Introduction, vol. 16. Walter de Gruyter, Berlin (2012)CrossRefGoogle Scholar
  29. 29.
    Zhao, M.-J., Ma, T., Quan, Q., Fan, H., Pereira, R.: \({l}_{1}\)-norm coherence of assistance. Phys. Rev. A 100, 012315 (2019). ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Napoli, C., Bromley, T.R., Cianciaruso, M., Piani, M., Johnston, N., Adesso, G.: Robustness of coherence: an operational and observable measure of quantum coherence. Phys. Rev. Lett. 116, 150502 (2016). ADSCrossRefGoogle Scholar
  31. 31.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 439(1907), 553–558 (1992)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Hillery, M.: Coherence as a resource in decision problems: the Deutsch–Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016). ADSCrossRefGoogle Scholar
  33. 33.
    Shi, H.-L., Liu, S.-Y., Wang, X.-H., Yang, W.-L., Yang, Z.-Y., Fan, H.: Coherence depletion in the Grover quantum search algorithm. Phys. Rev. A 95, 032307 (2017). ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Anand, N., Pati, A.K.: Coherence and entanglement monogamy in the discrete analogue of analog Grover search (2016). arXiv:1611.04542
  35. 35.
    Bu, K., Kumar, A., Zhang, L., Wu, J.: Cohering power of quantum operations. Phys. Lett. A. 381(19), 1670–1676 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhu, H., Hayashi, M., Chen, L.: Axiomatic and operational connections between the \({l}_{1}\)-norm of coherence and negativity. Phys. Rev. A 97, 022342 (2018). ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    John, W.: The Theory of Quantum Information. Cambridge University Press, Cambridge (2018)zbMATHGoogle Scholar
  38. 38.
    Daffer, S., Wódkiewicz, K., Cresser, J.D., McIver, J.K.: Depolarizing channel as a completely positive map with memory. Phys. Rev. A 70, 010304 (2004). ADSCrossRefGoogle Scholar
  39. 39.
    Kumar, N.P., Banerjee, S., Srikanth, R., Jagadish, V., Petruccione, F.: Non-Markovian evolution: a quantum walk perspective. Open Syst. Inf. Dyn. 25, 1850014 (2018). CrossRefzbMATHGoogle Scholar
  40. 40.
    Banerjee. S., Kumar, N.P., Srikanth, R., Jagadish, V., Petruccione, F.: Non-Markovian dynamics of discrete-time quantum walks (2017). arXiv:1703.08004
  41. 41.
    Shrikant, U., Srikanth, R., Banerjee, S.: Non-Markovian dephasing and depolarizing channels. Phys. Rev. A 98(3), 032328 (2018)ADSCrossRefGoogle Scholar
  42. 42.
    Banerjee, S., Ghosh, R.: Dynamics of decoherence without dissipation in a squeezed thermal bath. J. Phys. A Math. Theor. 40(45), 13735 (2007)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Srikanth, R., Banerjee, S.: Squeezed generalized amplitude damping channel. Phys. Rev. A 77(1), 012318 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    Omkar, S., Srikanth, R., Banerjee, S.: Dissipative and non-dissipative single-qubit channels: dynamics and geometry. Quantum Inf. Process. 12(12), 3725–3744 (2013)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Omkar, S., Banerjee, S., Srikanth, R., Alok, A.K.: The unruh effect interpreted as a quantum noise channel. Quantum Inf. Comput. 16, 0757 (2016)MathSciNetGoogle Scholar
  46. 46.
    Srikanth, R., Banerjee, S.: An environment-mediated quantum deleter. Phys. Lett. A 367(4–5), 295–299 (2007)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Bina, M., Mandarino, A., Olivares, S., Paris, M.G.A.: Drawbacks of the use of fidelity to assess quantum resources. Phys. Rev. A 89(1), 012305 (2014)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Indian Institute of Technology JodhpurJodhpurIndia

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