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Russian Journal of Mathematical Physics

, Volume 21, Issue 3, pp 297–315 | Cite as

Stability of continuous-time quantum filters with measurement imperfections

  • H. Amini
  • C. Pellegrini
  • P. Rouchon
Article

Abstract

The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.

Keywords

Mathematical Physic Markov Process Poisson Process Wiener Process Quantum Trajectory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. Amini, Stabilization of Discrete-Time Quantum Systems and Stability of Continuous-Time Quantum Filters (PhD thesis, Mines ParisTech, 2012).Google Scholar
  2. 2.
    H. Amini, M. Mirrahimi, and P. Rouchon, “On Stability of Continuous-Time Quantum-Filters,” Proceedings of the 50th IEEE Conference on Decision and Control, 6242–6247 (2011).Google Scholar
  3. 3.
    S. Attal and Y. Pautrat, “From Repeated to Continuous Quantum Interactions,” Annales Henri Poincaré 7, 59–104 (2006).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Barchielli and V. P. Belavkin, “Measurements Continuous in Time and a Posteriori States in Quantum Mechanics,” J. Physics A: Mathematical and General 24(7), (1991).Google Scholar
  5. 5.
    A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case (782, Springer Verlag, 2009).CrossRefGoogle Scholar
  6. 6.
    M. Bauer, T. Benoist, and D. Bernard, “Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit,” Annales Henri Poincaré 14(4), 639–679 (2013).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    M. Bauer and D. Bernard, “Convergence of Repeated Quantum Non-Demolition Measurements and Wave Function Collapse,” Physical Review A 84(4), 44–103 (2011).CrossRefMathSciNetGoogle Scholar
  8. 8.
    V. P. Belavkin, “Quantum Filtering of Markov Signals on a Background with Quantum White Quantum Noises,” Radiotekhn. i Elektron. 25(7), 1445–1453 (1980) [Radio Engrg. Electron. Phys. 25 (7), 76 (1980) (1981)].ADSMathSciNetGoogle Scholar
  9. 9.
    V. P. Belavkin, “Quantum Stochastic Calculus and Quantum Nonlinear Filtering,” J. Multivariate Analysis 42(2), 171–201 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    V. P. Belavkin, Eventum Mechanics of Quantum Trajectories: Continual Measurements, Quantum Predictions and Feedback Control (arXiv:math-ph/0702079, 2007).Google Scholar
  11. 11.
    T. Benoist and C. Pellegrini, Large Time Behavior and Convergence Rate for Quantum Filters under Standard non Demolition Conditions (Communications in Mathematical Physics, in press, 2013).Google Scholar
  12. 12.
    V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge Univ Pr, 1995).Google Scholar
  13. 13.
    H. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, 1993).zbMATHGoogle Scholar
  14. 14.
    J. Dalibard, Y. Castin, and K. Mølmer, “Wave-Function Approach to Dissipative Processes in Quantum Optics,” Phys. Rev. Lett. 68(5), 580–583 (1992).ADSCrossRefGoogle Scholar
  15. 15.
    E. B. Davies, Quantum Theory of Open Systems (Academic Press, 1976).zbMATHGoogle Scholar
  16. 16.
    J. Gough and A. I. Sobolev, “Stochastic Schrödinger Equations as Limit of Discrete Filtering,” Open Systems & Information Dynamics 11(3), 235–255 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University Press, New York, 2006).CrossRefGoogle Scholar
  18. 18.
    C. Pellegrini, Existence, Uniqueness and Approximation for Stochastic Schrödinger Equation: the Poisson Case (arXiv preprint, arXiv:0709.3713, 2007).Google Scholar
  19. 19.
    C. Pellegrini, “Existence, Uniqueness and Approximation of a Stochastic Schrödinger Equation: the Diffusive Case,” The Annals of Probability, 2332–2353 (2008).Google Scholar
  20. 20.
    C. Pellegrini, “Markov Chains Approximation of Jump-Diffusion Stochastic Master Equations” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 46(4), 924–948 (2010).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    D. Petz, “Monotone Metrics on Matrix Spaces,” Linear Algebra and its Applications 244, 81–96 (1996).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    P. Rouchon, “Fidelity is a Sub-Martingale for Discrete-Time Quantum Filters,” IEEE Transactions on Automatic Control 56(11), 2743–2747 (2011).CrossRefMathSciNetGoogle Scholar
  23. 23.
    C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.-M. Raimond, and S. Haroche, “Real-Time Quantum Feedback Prepares and Stabilizes Photon Number States,” Nature 477(7362), 73–77 (2011).ADSCrossRefGoogle Scholar
  24. 24.
    A. Somaraju, I. Dotsenko, C. Sayrin, and P. Rouchon, “Design and Stability of Discrete-Time Quantum Filters with Measurement Imperfections,” Proceedings of American Control Conference, 5084–5089 (2012).Google Scholar
  25. 25.
    R. van Handel, Filtering, Stability, and Robustness (PhD thesis, California Institute of Technology, 2006).Google Scholar
  26. 26.
    R. van Handel, “The Stability of Quantum Markov Filters,” Infinite Dimensional Analysis, Quantum Probability and Related Topics 12(1), 153–172 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2009).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Edward L. Ginzton LaboratoryStanford UniversityStanfordUSA
  2. 2.Institut de Mathématiques, IMTUniversité de Toulouse (UMR 5219)Toulouse, Cedex 9France
  3. 3.Centre Automatique et SystèmesMines ParisTechParis cedex 06France

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