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Theoretical approach to characterize the non-Markovianity and diffusion through the influx of the information

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Abstract

In this paper, we study the Fisher information for a quantum system consisting of two identical qubits, each of them locally interacting with a bosonic reservoir in the same environment for non-Markovian open, dissipative quantum system. Based on the influx of the information, we propose an information-theoretical approach for characterizing the time-dependent memory effect of environment and diffusion function under the effect of the physical parameters. More precisely, an interesting monotonic relation between the time derivative of quantum Fisher information (QFI) and diffusion function behavior is observed during the time evolution. The phenomenon is that the QFI, namely the precision of estimation, changes dramatically with the environment structure. The dependence of the physical parameters shows that the increasing in the temperature will damage the amount of the QFI with respect of the ratio between the reservoir cutoff frequency and the system oscillation frequency.

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References

  1. Dorner, U., Demkowicz-Dobrzanski, R., Smith, B.J., Lundeen, J.S., Wasilewski, W., Banaszek, K., Walmsley, I.A.: Optimal quantum phase estimation. Phys. Rev. Lett. 102, 040403 (2009)

    Article  ADS  Google Scholar 

  2. Demkowicz-Dobrzanski, R., Dorner, U., Smith, B.J., Lundeen, J.S., Wasilewski, W., Banaszek, K., Walmsley, I.A.: Quantum phase estimation with lossy interferometers. Phys. Rev. A 80, 013825 (2009)

    Article  ADS  Google Scholar 

  3. Leibfried, D., Barrett, M.D., Schaetz, T., Britton, J., Chiaverini, J., Itano, W.M., Jost, J.D., Langer, C., Wineland, D.J.: Toward Heisenberg-limited spectroscopy with multiparticle entangled states. Science 304, 1476 (2004)

    Article  ADS  Google Scholar 

  4. Petz, D.: Monotone metrics on matrix spaces. Linear Algebra Appl. 244, 81 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rivas, A., Luis, A.: Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes. Phys. Rev. Lett. 105, 010403 (2010)

    Article  ADS  Google Scholar 

  6. Sun, Z., Ma, J., Lu, X.M., Wang, X.G.: Fisher information in a quantum-critical environment. Phys. Rev. A 82, 022306 (2010)

    Article  ADS  Google Scholar 

  7. Ma, J., Wang, X.G., Sun, C.P., Nori, F.: Quantum spin squeezing. Phys. Rep. 509, 89 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  8. Roos, C.F., Chwalla, M., Kim, K., Riebe, M., Blatt, R.: Designer atoms for quantum metrology. Nature 443, 316 (2006)

    Article  ADS  Google Scholar 

  9. Huelga, S.F., Macchiavello, C., Pellizzari, T., Ekert, A.K., Plenio, M.B., Cirac, J.I.: Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79, 3865 (1997)

    Article  ADS  Google Scholar 

  10. Peters, A., Chung, K.Y., Chu, S.: Measurement of gravitational acceleration by dropping atoms. Nature (London) 400, 849 (1999)

    Article  ADS  Google Scholar 

  11. Jozsa, R., Abrams, D.S., Dowling, J.P., Williams, C.P.: Quantum clock synchronization based on shared prior entanglement. Phys. Rev. Lett. 85, 2010 (2000)

    Article  ADS  Google Scholar 

  12. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330 (2004)

    Article  ADS  Google Scholar 

  13. Dowling, J.: Quantum optical metrology-the lowdown on high-NOON states. Contemp. Phys. 49, 125 (2008)

    Article  ADS  Google Scholar 

  14. Jones, J.A., Karlen, S.D., Fitzsimons, J., Ardavan, A., Benjamin, S.C., Briggs, G.A.D., Morton, J.J.L.: Magnetic field sensing beyond the standard quantum limit using 10-spin NOON states. Science 324, 1166 (2009)

    Article  ADS  Google Scholar 

  15. Simmons, S., Jones, J.A., Karlen, S.D., Ardavan, A., Morton, J.J.L.: Magnetic field sensors using 13-spin cat states. Phys. Rev. A 82, 022330 (2010)

    Article  ADS  Google Scholar 

  16. Higgins, B.L., Berry, D.W., Bartlett, S.D., Wiseman, H.M., Pryde, G.J.: Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393 (2007)

    Article  ADS  Google Scholar 

  17. Dorner, U., Smith, B.J., Lundeen, J.S., Wasilewski, W., Banaszek, K., Walmsley, I.A.: Quantum phase estimation with lossy interferometers. Phys. Rev. A 80, 013825 (2009)

    Article  ADS  Google Scholar 

  18. Jha, Pankaj K., Eleuch, Hichem, Rostovtsev, Yuri V.: Coherent control of atomic excitation using off-resonant strong few-cycle pulses. Phys. Rev. A 82, 045805 (2010)

    Article  ADS  Google Scholar 

  19. Eleuch, H., Rachid, N.: Autocorrelation function of microcavity-emitting field in the non-linear regime. Eur. Phys. J. D 57, 259 (2010)

    Article  ADS  Google Scholar 

  20. Berrada, K.: Quantum metrology with SU(1,1) coherent states in the presence of nonlinear phase shifts. Phys. Rev. A 88, 013817 (2013)

    Article  ADS  Google Scholar 

  21. Fisher, R.A.: Theory of statistical estimation. Proc. Camb. Philos. Soc. 22, 700 (1925) [reprinted in Collected Papers of R. A. Fisher, edited by J. H. Bennett (Univ. of Adelaide Press, South Australia, 1972), pp. 15–40]

  22. Cramer, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  24. Piilo, J., Maniscalco, S., Harkonen, K., Suominen, K.-A.: Non-Markovian quantum jumps. Phys. Rev. Lett. 100, 180402 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Vasile, R., Olivares, S., Paris, M.G.A., Maniscalco, S.: Continuous-variable quantum key distribution in non-Markovian channels. Phys. Rev. A 83, 042321 (2011)

    Article  ADS  Google Scholar 

  26. Chin, A.W., Huelga, S.F., Plenio, M.B.: Quantum metrology in non-Markovian environments. Phys. Rev. Lett. 109, 233601 (2012)

    Article  ADS  Google Scholar 

  27. Laine, E.-M., Breuer, H.-P., Piilo, J.: Nonlocal memory effects allow perfect teleportation with mixed states. Sci. Rep. 4, 4620 (2014)

    Article  ADS  Google Scholar 

  28. 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 

  29. Vacchini, B., et al.: Markovianity and non-Markovianity in quantum and classical systems. New J. Phys. 13, 093004 (2011)

    Article  ADS  Google Scholar 

  30. Chiuri, A., et al.: Linear optics simulation of quantum non-Markovian dynamics. Sci. Rep. 2, 968 (2012)

    Article  ADS  Google Scholar 

  31. Benatti, F., Floreanini, R., Olivares, S.: Non-divisibility and non-Markovianity in a Gaussian dissipative dynamics. Phys. Lett. A 376, 2951 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  32. Xu, J.-S., et al.: Experimental recovery of quantum correlations in absence of system-environment back-action. Nat. Commun. 4, 2851 (2013)

    ADS  Google Scholar 

  33. Fanchini, F.F., Karpat, G., Castelano, L.K., Rossatto, D.Z.: Probing the degree of non-Markovianity for independent and common environments. Phys. Rev. A 88, 012105 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  35. 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, 062112 (2011)

    Article  ADS  Google Scholar 

  36. 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 

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

    Article  ADS  Google Scholar 

  38. 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 

  39. Zhong, Wei, Sun, Zhe, Ma, Jian, Wang, Xiaoguang, Nori, Franco: Fisher information under decoherence in Bloch representation. Phys. Rev. A 87, 022337 (2013)

    Article  ADS  Google Scholar 

  40. Ma, Jian, Huang, Yi-xiao, Wang, Xiaoguang, Sun, C.P.: Quantum Fisher information of the Greenberger-Horne-Zeilinger state in decoherence channels. Phys. Rev. A 84, 022302 (2011)

    Article  ADS  Google Scholar 

  41. Heng-Na, Xiong, Xiaoguang, Wang: Dynamical quantum Fisher information in the Ising model. Phys. A. 390, 4719 (2011)

    Article  MathSciNet  Google Scholar 

  42. Li, Y.-L., Xiao, X., Yao, Y.: Classical-driving-enhanced parameter-estimation precision of a non-Markovian dissipative two-state system. Phys. Rev. A 91, 052105 (2015)

    Article  ADS  Google Scholar 

  43. Dhar, H.S., Bera, M.N., Adesso, G.: Characterizing non-Markovianity via quantum interferometric power. Phys. Rev. A 91, 032115 (2015)

    Article  ADS  Google Scholar 

  44. Song, H., Luo, S., Hong, Y.: Quantum non-Markovianity based on the Fisher-information matrix. Phys. Rev. A 91, 042110 (2015)

    Article  ADS  Google Scholar 

  45. Lu, X.M., et al.: Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A 82, 042103 (2010)

    Article  ADS  Google Scholar 

  46. Haseli, S., et al.: Non-Markovianity through flow of information between a system and an environment. Phys. Rev. A 90, 052118 (2014)

    Article  ADS  Google Scholar 

  47. Cui, W., Xi, Z.R., Pan, Y.: Optimal decoherence control in non-Markovian open dissipative quantum systems. Phys. Rev. A 77, 032117 (2008)

    Article  ADS  Google Scholar 

  48. Maniscalco, S., Olivares, S., Paris, M.G.A.: Entanglement oscillations in non-Markovian quantum channels. Phys. Rev. A 75, 062119 (2007)

    Article  ADS  Google Scholar 

  49. Maniscalco, S., Piilo, J., Intravaia, F., Petruccione, F., Messina, A.: Lindblad-and non-Lindblad-type dynamics of a quantum Brownian particle. Phys. Rev. A 70, 032113 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Zhang, J., Wu, R.B., Li, C.W., Tarn, T.J., Wu, J.W.: Asymptotically noise decoupling for Markovian open quantum systems. Phys. Rev. A 75, 022324 (2007)

    Article  ADS  Google Scholar 

  51. Weiss, U.: Quantum Dissipative System. World Scientific Publishing, Singapore (1993)

    Book  MATH  Google Scholar 

  52. Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., Zwerger, W.: Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987)

    Article  ADS  Google Scholar 

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Acknowledgments

We acknowledge the reviewers comments and suggestions very much, which are valuable in improving the quality of our manuscript. We acknowledge enlightening discussions with Fabio Benatti. We thank the International Centre for Theoretical Physics, ICTP, for kind hospitality.

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Authors and Affiliations

  1. The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Miramare-Trieste, Italy

    K. Berrada

  2. Department of Physics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    K. Berrada

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Berrada, K. Theoretical approach to characterize the non-Markovianity and diffusion through the influx of the information. Quantum Inf Process 15, 4897–4909 (2016). https://doi.org/10.1007/s11128-016-1417-6

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