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Bayes cost of parameter estimation for a quantum system interacting with an environment

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Abstract

The Bayes cost of parameter estimation is studied for a quantum system which is influenced by an external environment, where the cost function is assumed to be a quadratic function of a difference between true and estimated values. When the reduced time evolution of a quantum system is determined by the time-dependent Lindblad equation, it is found how the Bayes cost changes with time. The Bayes cost increases monotonously with time for the Markovian environment, while it shows an oscillatory behavior for the non-Markovian environment due to the memory effect. Furthermore, in order to investigate how initial correlation between quantum system and environment, an analytic expression of the Bayes cost is derived for a qubit-oscillator system. It is found for both Markovian and non-Markovian environments that the Bayes cost can take a value smaller than the initial one in the presence of the initial correlation. The decrease in the Bayes cost is due to the backflow of information that is included in the initially correlated part.

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References

  1. Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)

    MATH  Google Scholar 

  2. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  3. Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Paris, M.G.A., Řeháček, J. (eds.): Quantum Estimation Theory. Lecture Notes in Physics. Springer, Berlin (2010)

    Google Scholar 

  5. Pezzè, L., Smerzi, A.: Quantum theory of phase estimation. LANL arXiv:1411.5164 (2014)

  6. Micadei, K., Rowlands, D.A., Pollock, F.A., Céleri, L.C., Serra, R.M., Modi, K.: Coherent measurements in quantum metrology. New J. Phys. 17, 023057 (2015)

    Article  ADS  Google Scholar 

  7. Sasaki, M., Ban, M., Barnett, S.M.: Optimal parameter estimation of a depolarizing channel. Phys. Rev. A 66, 022308 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  8. Weiss, U.: Quantum Dissipative Systems. World Scientific, Singapore (1993)

    Book  MATH  Google Scholar 

  9. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  10. Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Jaeger, G.: Quantum Information. Springer, Berlin (2007)

    MATH  Google Scholar 

  13. Jaeger, G.: Entanglement, Information, and the Interpretation of Quantum Mechanics. Springer, Berlin (2009)

    Book  Google Scholar 

  14. 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 

  15. Vacchini, B., Smirne, A., Laine, E.-M., Piilo, J., Breuer, H.-P.: Markovianity and non-Markovianity in quantum and classical systems. New J. Phys. 13, 093004 (2011)

    Article  ADS  Google Scholar 

  16. Chruściński, D., Kossakowski, A., Rivas, A.: Measures of non-Markovianity: divisibility versus backflow of information Phys. Rev. A 83, 052128 (2011)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  18. 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  Google Scholar 

  19. Bellomo, B., Lo Franco, R., Compagno, G.: Non-Markovian effects on the dynamics of entanglement. Phys. Rev. Lett. 99, 160502 (2007)

    Article  ADS  Google Scholar 

  20. Bellomo, B., Lo Franco, R., Compagno, G.: Entanglement dynamics of two independent qubits in environments with and without memory. Phys. Rev. A 77, 032342 (2008)

    Article  ADS  Google Scholar 

  21. Bellomo, B., Lo Franco, R., Maniscalco, S., Compagno, G.: Entanglement trapping in structured environments. Phys. Rev. A 78, 060302 (2008)

    Article  ADS  Google Scholar 

  22. Mazzola, L., Bellomo, B., Lo Franco, R., Compagno, G.: Connection among entanglement, mixedness, and nonlocality in a dynamical context. Phys. Rev. A 81, 052116 (2010)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  24. Vasile, R., Maniscalco, S., Paris, M.G.A., Breuer, H., Piilo, J.: Quantifying non-Markovianity of continuous-variable Gaussian dynamical maps. Phys. Rev. 84, 052118 (2011)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Bylicka, B., Chrusścński, D., Maniscalco, S.: Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective. Sci. Rep. 4, 5720 (2014)

    Article  ADS  Google Scholar 

  27. Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 73, 1060–1062 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Jordan, T.F., Shaji, A., Sudarshan, E.C.G.: Dynamics of initially entangled open quantum systems. Phys. Rev. A 70, 052110 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Rodoriguez-Rosario, C.A., Modi, K., Kuah, A.-M., Shaji, A., Sudarshan, E.C.G.: Completely positive maps and classical correlations. J. Phys. A 41, 205301 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Devi, Usha, Rajagopal, A.R., Sudha, A.K.: Open-system quantum dynamics with correlated initial states, not completely positive maps, and non-Markovianity. Phys. Rev. A 83, 022109 (2011)

    Article  ADS  Google Scholar 

  31. Laine, E.-M., Piilo, J., Breuer, H.-P.: Witness for initial system-environment correlations in open-system dynamics. Europhys. Lett. 92, 60010 (2010)

    Article  ADS  Google Scholar 

  32. Smirne, A., Breuer, H.-P., Pillo, J., Vacchini, B.: Initial correlations in open-systems dynamics: the Jaynes-Cummings model. Phys. Rev. A 82, 062114 (2010)

    Article  ADS  Google Scholar 

  33. Dajka, J., Luczka, J.: Distance growth of quantum states due to initial system-environment correlations. Phys. Rev. A 82, 012341 (2010)

    Article  ADS  Google Scholar 

  34. Dajka, J., Luczka, J., Hänggi, P.: Distance between quantum states in the presence of initial qubit-environment correlations: a comparative study. Phys. Rev. A 84, 032120 (2011)

    Article  ADS  Google Scholar 

  35. Ban, M., Kitajima, S., Shibata, F.: Qubit decoherence with an initial correlation. Phys. Lett. A 375, 2283–2290 (2011)

    Article  ADS  MATH  Google Scholar 

  36. Ban, M., Kitajima, S., Shibata, F.: Distance between qubit states with initial system-environment correlation. Int. J. Theor. Phys. 51, 2419–2426 (2012)

    Article  MATH  Google Scholar 

  37. Hu, Z., Wang, J., Zhang, Y.: Dynamics of nonclassical correlations with initial correlation. J. Phys. Soc. Jpn. 83, 114004 (2014)

    Article  ADS  Google Scholar 

  38. Sarovar, M., Milburn, G.J.: Optimal estimation of one-parameter quantum channels. J. Phys. 39, 8487–8505 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  39. Monras, A., Paris, M.G.A.: Optimal quantum estimation of loss in bosonic channels. Phys. Rev. Lett. 98, 160401 (2007)

    Article  ADS  Google Scholar 

  40. Watanabe, Y., Sagawa, T., Ueda, M.: Optimal measurement on noisy quantum systems. Phys. Rev. Lett. 104, 020401 (2010)

    Article  ADS  Google Scholar 

  41. Lu, X., Wang, X., Sun, C.P.: Quantum Fisher information flow and non-Markovian process of open systems. Phys. Rev. A 82, 042103 (2010)

    Article  ADS  Google Scholar 

  42. Escher, B.M., de Matos Filho, R.L., Davidovich, L.: General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011)

    Article  Google Scholar 

  43. Ma, J., Huang, Y., Wang, X., 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 

  44. 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 

  45. Berrada, K., Abdel-Khalek, S., Obada, A.-S.F.: Quantum Fisher information for a qubit system placed inside a dissipative cavity. Phys. Lett. A 376, 1412–1416 (2012)

    Article  ADS  MATH  Google Scholar 

  46. Zhong, W., Sun, Z., Ma, J., Wang, X., Nori, F.: Fisher information under decoherence in Bloch representation. Phys. Rev. A 87, 022337 (2013)

    Article  ADS  Google Scholar 

  47. Berrada, K.: Non-Markovian effect in the precision of parameter estimation. Phys. Rev. A 88, 035806 (2013)

    Article  ADS  Google Scholar 

  48. Ozaydin, F.: Phase damping destroys quantum Fisher information of W states. Phys. Lett. 378, 3161–3164 (2014)

    Article  Google Scholar 

  49. Alipour, S., Mehboudi, M., Rezakhani, A.T.: Quantum metrology in open systems: dissipative Cram’er-Rao bound. Phys. Rev. Lett. 112, 120405 (2014)

    Article  ADS  Google Scholar 

  50. Ban, M.: Quantum Fisher information of a qubit initially correlated with a non-Markovian environment. Quant. Inf. Process. 14, 4163–4177 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Schirmer, S.G., Oi, D.K.L.: Two-qubit Hamiltonian tomography by Bayesian analysis of noisy data. Phys. Rev. A 80, 022333 (2009)

    Article  ADS  Google Scholar 

  52. Schirmer, S.G., Langbein, F.C.: Quantum system identification: Hamiltonian estimation using spectral and Bayesian analysis. LANL quant-ph 0911.5429 (2009)

  53. Schirmer, S.G., Oi, D.K.L.: Quantum system identification by Bayesian analysis of noisy data: beyond Hamiltonian tomography. Laser Phys. 20, 1203–1209 (2010)

    Article  ADS  Google Scholar 

  54. Wiebe, N., Granade, C.: Efficient Bayesian phase estimation. LANL quant-ph 1508, 00869 (2009)

  55. Macieszczak, K., Fraas, M., Demkowicz-Dobrzański, R.: Bayesian quantum frequency estimation in presence of collective dephasing. New J. Phys. 16, 113002 (2914)

    Article  Google Scholar 

  56. Barndorff-Nielsen, O.E., Gill, R.D.: Fisher information in quantum statistics. J. Phys. A 33, 4481–4490 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  58. Rivas, A., Huelga, S.F.: Open Quantum Systems. Springer, Berlin (2011)

    MATH  Google Scholar 

  59. Personik, S.D.: Application of quantum estimation theory to analog communication over quantum channels. Trans. IEEE IT–17, 240–246 (1971)

    MathSciNet  Google Scholar 

  60. Morozov, V.G., Mathey, S., Röpke, G.: Decoherence in an exactly solvable qubit model with initial qubit-environment correlations. Phys. Rev. A 85, 022101 (2012)

    Article  ADS  Google Scholar 

  61. Ignatyuk, V.V., Morozov, V.G.: Enhancement of coherence in qubits due to interaction with the environment. Phys. Rev. A 91, 052102 (2015)

    Article  ADS  Google Scholar 

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  1. Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo, 112-8610, Japan

    Masashi Ban

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Ban, M. Bayes cost of parameter estimation for a quantum system interacting with an environment. Quantum Inf Process 15, 2213–2230 (2016). https://doi.org/10.1007/s11128-016-1267-2

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  • DOI: https://doi.org/10.1007/s11128-016-1267-2

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