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Exponential Stability of Subspaces for Quantum Stochastic Master Equations

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Abstract

We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant.

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References

  1. Adler, S.L., Brody, D.C., Brun, T.A., Hughston, L.P.: Martingale models for quantum state reduction. J. Phys. A: Math. Gen. 34, 8795–8820 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Springer, Berlin (1987)

    MATH  Google Scholar 

  3. Altafini, C.: Controllability properties for finite dimensional quantum Markovian master equations. J. Math. Phys. 44, 2357–2372 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Altafini, C., Nishio, K., Ticozzi, F.: Stabilization of stochastic quantum dynamics via open and closed loop control. IEEE Trans. Automat. Contr. 58, 74–85 (2013)

    Article  MathSciNet  Google Scholar 

  5. Altafini, C., Ticozzi, F.: Modeling and control of quantum systems: an introduction. IEEE Trans. Automat. Contr. 57, 1898–1917 (2012)

    Article  MathSciNet  Google Scholar 

  6. Amini, H., Rouchon, P., Pellegrini, C.: Stability of continuous-time quantum filters with measurement imperfections. Russ. J. Math. Phys. 21(3), 297–315 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Amini, H., Somaraju, A., Dotsenko, I., Sayrin, C., Mirrahimi, M., Rouchon, P.: Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays. Automatica 49(9), 2683–2692 (2013)

    Article  MathSciNet  Google Scholar 

  8. Amini, H., Mirrahimi, M., Rouchon, P.: On stability of continuous-time quantum-filters. CDC/ECC pp. 6242–6247, (2011)

  9. Attal, S., Pellegrini, C.: Return to Equilibrium in Quantum Trajectory Theory. Nova Publisher Book stochastic differential equations ISBN: 978-1-61324-278-0 (2011)

  10. Ballesteros, M., Fraas, M., Fröhlich, J., Schubnel, B.: Indirect retrieval of information and the emergence of facts in quantum mechanics. (2015) Preprint arXiv:1506.01213

  11. Barchielli, A., Gregoratti, M.: Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case. Ser. Lect. Notes Phys. 782. Springer, Berlin (2009)

  12. Barchielli, A., Holevo, A.S.: Constructing quantum measurement processes via classical stochastic calculus. Stoch. Process. Appl. 58, 293–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bauer, M., Bernard, D.: Convergence of repeated quantum nondemolition measurements and wave-function collapse. Phys. Rev. A 84, 044103 (2011)

    Article  ADS  Google Scholar 

  14. Bauer, M., Bernard, D.: Real time imaging of quantum and thermal fluctuations: The case of a two-level system. Lett. Math. Phys. 104, 707–729 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bauer, M., Benoist, T., Bernard, D.: Repeated quantum non-demolition measurements: convergence and continuous time limit. Ann. H. Poincaré 14, 639–679 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bauer, M., Bernard, D., Benoist, T.: Iterated stochastic measurements. J. Phys. A: Math. Theor. 45, 494020 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bauer, M., Bernard, D., Tilloy, A.: Open quantum random walks: bistability on pure states and ballistically induced diffusion. Phys. Rev. A 88, 062340 (2013)

    Article  ADS  Google Scholar 

  18. Bauer, M., Bernard, D., Tilloy, A.: The open quantum brownian motions. J. Stat. Mech.: Theor. Exp 2014, P09001 (2014)

    Article  MathSciNet  Google Scholar 

  19. Bauer, M., Bernard, D., Tilloy, A.: Computing the rates of measurement-induced quantum jumps. J. Phys. A: Math. Theor 48, 25FT02 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Baumgartner, B., Narnhofer, H.: Analysis of quantum semigroups with GKS-Lindblad generators: II. General. J. Phys. A: Math. Theor. 41, 395303 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Belavkin, V.P.: Nondemolition measurements and control in quantum dynamical systems. In: Blaquiere, A., Diner, S., Lochak, G. (eds.) Proceedings, Information Complexity and Control in Quantum Physics, Udine, pp. 311–336. Springer, New York (1985)

    Google Scholar 

  22. Belavkin, V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42, 171–201 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Belavkin, V.P.: Measurement, filtering and control in quantum open dynamical systems. Rep. Math. Phys. 43, 405–425 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Benoist, T., Pellegrini, C.: Large time behaviour and convergence rate for non demolition quantum trajectories. Comm. Math. Phys. 331, 703–723 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Bouten, L., van Handel, R., James, M.R.: A discrete invitation to quantum filtering and feedback control. SIAM Rev. 51, 239–316 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Cirillo, G.I., Ticozzi, F.: Decompositions of hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups. J. Phys. A: Math. Theor. 48, 085302 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Dirr, G., Helmke, U., Kurniawan, I., Schulte-Herbrüggen, T.: Lie-semi group structures for reachability and control of open quantum systems: Kossakowski-Lindblad generators from Lie wedges to Markovian channels. Rep. Math. Phys. 64, 93–121 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Evans, D.E., Høegh-Krohn, R.: Spectral properties of positive maps on C*-algebras. J. London Math. Soc. 2, 345–355 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gardiner, C .W., Zoller, P.: Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd edn. Springer, New York (2004)

    MATH  Google Scholar 

  31. Cook, R.L., Martin, P.J., Geremia, J.M.: Optical coherent state discrimination using a closed-loop quantum measurement. Nature 446, 774–777 (2007)

    Article  ADS  Google Scholar 

  32. Gorini, V., Kossakowski, A., Sudarshan, E.: Completely positive dynamical semigroups of n-level systems. J. Math. Phys. 17, 821–825 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  33. Gorini, V., Frigerio, A., Verri, M., Kossakowski, A., Sudarshan, E.C.G.: Properties of quantum Markovian master equations. Rep. Math. Phys. 13, 149–173 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Haroche, S., Raimond, J.-M.: Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  35. Hopkins, A., Jacobs, K., Habib, S., Schwab, K.: Feedback cooling of a nanomechanical resonator. Phys. Rev. B 68, 235328 (2003)

    Article  ADS  Google Scholar 

  36. Jakšić, V., Pillet, C.-A., Westrich, M.: Entropic fluctuations of quantum dynamical semigroups. J. Statist. Phys. 154, 153–187 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Mabuchi, H., Doherty, A.C.: Cavity Quantum Electrodynamics: Coherence in Context. Science 298, 1372–1377 (2002)

    Article  ADS  Google Scholar 

  39. Mancini, S., Vitali, D., Tombesi, P.: Optomechanical cooling of a macroscopic oscillator by homodyne feedback. Phys. Rev. Lett. 80, 688–691 (1998)

    Article  ADS  Google Scholar 

  40. Pellegrini, C.: Existence, uniqueness and approximation of a stochastic Schrödinger equation: the diffusive case. Ann. Probab. 36, 2332–2353 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pellegrini, C.: Poisson and Diffusion Approximation of Stochastic Schrödinger Equations with Control. Annales Henri Poincaré: Physique Théorique 10, 995–1025 (2009)

    Article  ADS  MATH  Google Scholar 

  42. Pellegrini, C.: Existence, uniqueness and approximation of the jump-type stochastic Schrödinger equation for two-level systems. Stoch. Process. Appl. 120, 1722–1747 (2010)

    Article  MATH  Google Scholar 

  43. Pellegrini, C.: Markov chains approximation of jump-diffusion stochastic master equations. Ann. Inst. H. Poincaré: Prob. Stat. 46, 924–948 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Poyatos, J.F., Cirac, J.I., Zoller, P.: Quantum reservoir engineering with laser cooled trapped ions. Phys. Rev. Lett. 77, 4728 (1996)

    Article  ADS  Google Scholar 

  45. Protter, P.E.: Stochastic Integration and Differential Equations. Springer, Berlin (2013)

    Google Scholar 

  46. Rouchon, P., Ralph, J.: Efficient quantum filtering for quantum feedback control. Phys. Rev. A 91, 012118 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  47. Sayrin, C., Dotsenko, I., Zhou, X., Peaudecerf, B., Rybarczyk, Th, Gleyzes, S., Rouchon, P., Mirrahimi, M., Amini, H., Brune, M., Raimond, J.M., Haroche, S.: Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011)

    Article  ADS  Google Scholar 

  48. Shabani, A., Lidar, D .A.: Theory of initialization-free decoherence-free subspaces and subsystems. Phys. Rev. A 72(4), 042303 (2005)

    Article  ADS  Google Scholar 

  49. Scaramuzza, P., Ticozzi, F.: Switching quantum dynamics for fast stabilization. Phys. Rev. A 91, 062314 (2015)

    Article  ADS  Google Scholar 

  50. Smith, W.P., Reiner, J.E., Orozco, L.A., Kuhr, S., Wiseman, H.M.: Capture and release of a conditional state of a cavity QED system by quantum feedback. Phys. Rev. Lett. 89, 133601 (2002)

    Article  ADS  Google Scholar 

  51. Steck, D.A., Jacobs, K., Mabuchi, H., Bhattacharya, T., Habib, S.: Quantum feedback control of atomic motion in an optical cavity. Phys. Rev. Lett. 92, 223004 (2004)

    Article  ADS  Google Scholar 

  52. Ticozzi, F., Lucchese, R., Cappellaro, P., Viola, L.: Hamiltonian control of quantum dynamical semigroups: stabilization and convergence speed. IEEE Trans. Automat. Contr. 57, 1931–1944 (2012)

    Article  MathSciNet  Google Scholar 

  53. Ticozzi, F., Viola, L.: Analysis and synthesis of attractive quantum Markovian dynamics. Automatica 45, 2002–2009 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  54. Ticozzi, F., Viola, L.: Quantum markovian subsystems: invariance, attractivity and control. IEEE Trans. Automat. Contr. 53, 2048–2063 (2008)

    Article  MathSciNet  Google Scholar 

  55. Ticozzi, F., Viola, L.: Steady-state entanglement by engineered quasi-local Markovian dissipation. Quant. Inf. Comput. 14, 0265–0294 (2014)

    Google Scholar 

  56. Ticozzi, F., Viola, L.: Quantum information encooding, protection and correction via trace-norm isometries. Phys. Rev. A 81(3), 032313 (2010)

    Article  ADS  Google Scholar 

  57. Wiseman, H.M.: Adaptive phase measurements of optical modes: Going beyond the marginal \(q\) distribution. Phys. Rev. Lett. 75, 4587–4590 (1995)

    Article  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse UPS, 31062, Toulouse, France

    Tristan Benoist

  2. Institut de Mathématiques de Toulouse Equipe de Statistique et de Probabilité, Université Paul Sabatier, 31062, Toulouse Cedex 9, France

    Clément Pellegrini

  3. Department of Information Engineering, Universitá degli Studi di Padova, via gradenigo 6/b, 35131, Padova, Italy

    Francesco Ticozzi

  4. Department of Physics and Astronomy, Dartmouth College, 6127 Wilder, Hanover, NH, USA

    Francesco Ticozzi

Authors
  1. Tristan Benoist
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  2. Clément Pellegrini
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  3. Francesco Ticozzi
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Correspondence to Tristan Benoist.

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Communicated by Claude Alain Pillet.

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Benoist, T., Pellegrini, C. & Ticozzi, F. Exponential Stability of Subspaces for Quantum Stochastic Master Equations. Ann. Henri Poincaré 18, 2045–2074 (2017). https://doi.org/10.1007/s00023-017-0556-3

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