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

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Abstract

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.

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References

  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)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  3. Bennett, C.H., Bessette, F., Brassard, G., Salvail, L., Smolin, J.: Experimental quantum cryptography. J. Cryptol. 5(1), 3–28 (1992)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.70.1895

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89(4), 041003 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  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. Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963). https://doi.org/10.1103/PhysRev.131.2766

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963). https://doi.org/10.1103/PhysRevLett.10.277

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  11. Lloyd, S.: Quantum coherence in biological systems. J. Phys. Conf. Ser. 302, 012037 (2011)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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). https://doi.org/10.1103/PhysRevA.91.052115

    Article  ADS  Google Scholar 

  14. Yadin, B., Vedral, V.: General framework for quantum macroscopicity in terms of coherence. Phys. Rev. A 93(2), 022122 (2016)

    Article  ADS  Google Scholar 

  15. Alok, A.K., Banerjee, S., Sankar, S.U.: Quantum correlations in terms of neutrino oscillation probabilities. Nucl. Phys. B 909, 65–72 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  17. Mani, A., Karimipour, V.: Cohering and decohering power of quantum channels. Phys. Rev. A 92, 032331 (2015). https://doi.org/10.1103/PhysRevA.92.032331

    Article  ADS  Google Scholar 

  18. Maniscalco, S., Olivares, S., Paris, M.G.A.: Entanglement oscillations in non-Markovian quantum channels. Phys. Rev. A 75, 062119 (2007). https://doi.org/10.1103/PhysRevA.75.062119

    Article  ADS  Google Scholar 

  19. Banerjee, S.: Open Quantum Systems: Dynamics of Nonclassical Evolution. Springer, Berlin (2018)

    Book  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  22. Horodecki, R., Horodecki, M., Horodecki, P.: Teleportation, Bell’s inequalities and inseparability. Phys. Lett. A 222(1–2), 21–25 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  23. Horodecki, M., Horodecki, P., Horodecki, R.: General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 60(3), 1888 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  24. Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41(12), 2315–2323 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  25. Saulo, V.M., Cunha, M.T.: Quantifying quantum invasiveness. Phys. Rev. A 99, 022124 (2019). https://doi.org/10.1103/PhysRevA.99.022124

    Article  ADS  Google Scholar 

  26. Shahbeigi, F., Akhtarshenas, S.J.: Quantumness of quantum channels. Phys. Rev. A 98, 042313 (2018). https://doi.org/10.1103/PhysRevA.98.042313

    Article  ADS  Google Scholar 

  27. Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  28. Holevo, A.S.: Quantum Systems, Channels, Information: A Mathematical Introduction, vol. 16. Walter de Gruyter, Berlin (2012)

    Book  Google Scholar 

  29. Zhao, M.-J., Ma, T., Quan, Q., Fan, H., Pereira, R.: \({l}_{1}\)-norm coherence of assistance. Phys. Rev. A 100, 012315 (2019). https://doi.org/10.1103/PhysRevA.100.012315

    Article  ADS  MathSciNet  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.116.150502

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  32. Hillery, M.: Coherence as a resource in decision problems: the Deutsch–Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016). https://doi.org/10.1103/PhysRevA.93.012111

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevA.95.032307

    Article  ADS  MathSciNet  Google Scholar 

  34. Anand, N., Pati, A.K.: Coherence and entanglement monogamy in the discrete analogue of analog Grover search (2016). arXiv:1611.04542

  35. Bu, K., Kumar, A., Zhang, L., Wu, J.: Cohering power of quantum operations. Phys. Lett. A. 381(19), 1670–1676 (2017). https://doi.org/10.1016/j.physleta.2017.03.022

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1103/PhysRevA.97.022342

    Article  ADS  MathSciNet  Google Scholar 

  37. John, W.: The Theory of Quantum Information. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

  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). https://doi.org/10.1103/PhysRevA.70.010304

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1142/S1230161218500142

    Article  MATH  Google Scholar 

  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. Shrikant, U., Srikanth, R., Banerjee, S.: Non-Markovian dephasing and depolarizing channels. Phys. Rev. A 98(3), 032328 (2018)

    Article  ADS  Google Scholar 

  42. Banerjee, S., Ghosh, R.: Dynamics of decoherence without dissipation in a squeezed thermal bath. J. Phys. A Math. Theor. 40(45), 13735 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  43. Srikanth, R., Banerjee, S.: Squeezed generalized amplitude damping channel. Phys. Rev. A 77(1), 012318 (2008)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  46. Srikanth, R., Banerjee, S.: An environment-mediated quantum deleter. Phys. Lett. A 367(4–5), 295–299 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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Acknowledgements

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

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  1. Indian Institute of Technology Jodhpur, Jodhpur, 342011, India

    Javid Naikoo & Subhashish Banerjee

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  1. Javid Naikoo
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Naikoo, J., Banerjee, S. Coherence-based measure of quantumness in (non-) Markovian channels. Quantum Inf Process 19, 29 (2020). https://doi.org/10.1007/s11128-019-2533-x

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