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Recurrence and Transience of Continuous-Time Open Quantum Walks

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Séminaire de Probabilités L

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2252))

Abstract

This paper is devoted to the study of continuous-time processes known as continuous-time open quantum walks (CTOQW). A CTOQW represents the evolution of a quantum particle constrained to move on a discrete graph, but which also has internal degrees of freedom modeled by a state (in the quantum mechanical sense). CTOQW contain as a special case continuous-time Markov chains on graphs. Recurrence and transience of a vertex are an important notion in the study of Markov chains, and it is known that all vertices must be of the same nature if the Markov chain is irreducible. In the present paper we address the corresponding result in the context of irreducible CTOQW. Because of the “quantum” internal degrees of freedom, CTOQW exhibit non standard behavior, and the classification of recurrence and transience properties obeys a “trichotomy” rather than the classical dichotomy. Essential tools in this paper are the so-called “quantum trajectories” which are jump stochastic differential equations which can be associated with CTOQW.

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References

  1. S. Attal, F. Petruccione, C. Sabot, I. Sinayskiy, Open quantum random walks. J. Stat. Phys. 147(4), 832–852 (2012)

    Article  MathSciNet  Google Scholar 

  2. S. Attal, F. Petruccione, I. Sinayskiy, Open quantum walks on graphs. Phys. Lett. A 376(18), 1545–1548 (2012)

    Article  MathSciNet  Google Scholar 

  3. S. Attal, N. Guillotin-Plantard, C. Sabot, Central limit theorems for open quantum random walks and quantum measurement records. Ann. Henri Poincaré 16(1), 15–43 (2015)

    Article  MathSciNet  Google Scholar 

  4. A. Barchielli, V.P. Belavkin, Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A 24(7), 1495–1514 (1991)

    Article  MathSciNet  Google Scholar 

  5. I. Bardet, D. Bernard, Y. Pautrat, Passage times, exit times and Dirichlet problems for open quantum walks. J. Stat. Phys. 167(2), 173–204 (2017)

    Article  MathSciNet  Google Scholar 

  6. M. Bauer, D. Bernard, A. Tilloy, The open quantum Brownian motions. J. Stat. Mech. Theory Exp. 2014(9), p09001, 48 (2014)

    Article  MathSciNet  Google Scholar 

  7. L. Bouten, M. Guţă, H. Maassen, Stochastic Schrödinger equations. J. Phys. A 37(9), 3189–3209 (2004)

    Article  MathSciNet  Google Scholar 

  8. R. Carbone, Y. Pautrat, Homogeneous open quantum random walks on a lattice. J. Stat. Phys. 160(5), 1125–1153 (2015)

    Article  MathSciNet  Google Scholar 

  9. R. Carbone, Y. Pautrat, Open quantum random walks: reducibility, period, ergodic properties. Ann. Henri Poincaré 17(1), 99–135 (2016)

    Article  MathSciNet  Google Scholar 

  10. S.L. Carvalho, L.F. Guidi, C.F. Lardizabal, Site recurrence of open and unitary quantum walks on the line. Quantum Inf. Process. 16(1), 32 (2017). Art. 17

    Google Scholar 

  11. E.B. Davies, Quantum stochastic processes. II. Commun. Math. Phys. 19, 83–105 (1970)

    Article  MathSciNet  Google Scholar 

  12. E.B. Davies, Quantum Theory of Open Systems (Academic/Harcourt Brace Jovanovich Publishers, London/New York, 1976)

    MATH  Google Scholar 

  13. J. Dereziński, W. De Roeck, C. Maes, Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. 131(2), 341–356 (2008)

    Article  MathSciNet  Google Scholar 

  14. K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

    Google Scholar 

  15. D.E. Evans, R. Høegh-Krohn, Spectral properties of positive maps on C∗-algebras. J. Lond. Math. Soc. (2) 17(2), 345–355 (1978)

    Article  MathSciNet  Google Scholar 

  16. F. Fagnola, R. Rebolledo, Subharmonic projections for a quantum Markov semigroup. J. Math. Phys. 43(2), 1074–1082 (2002)

    Article  MathSciNet  Google Scholar 

  17. F. Fagnola, R. Rebolledo, Transience and recurrence of quantum Markov semigroups. Probab. Theory Relat. Fields 126(2), 289–306 (2003)

    Article  MathSciNet  Google Scholar 

  18. F. Fagnola, R. Rebolledo, Notes on the qualitative behaviour of quantum Markov semigroups, in Open Quantum Systems. III. Lecture Notes in Mathematics, vol. 1882 (Springer, Berlin, 2006), pp. 161–205

    MATH  Google Scholar 

  19. V. Jakšić, C.-A. Pillet, M. Westrich, Entropic fluctuations of quantum dynamical semigroups. J. Stat. Phys. 154(1–2), 153–187 (2014)

    Article  MathSciNet  Google Scholar 

  20. C.F. Lardizabal, R.R. Souza, Open quantum random walks: ergodicity, hitting times, gambler’s ruin and potential theory. J. Stat. Phys. 164(5), 1122–1156 (2016)

    Article  MathSciNet  Google Scholar 

  21. G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976)

    Article  MathSciNet  Google Scholar 

  22. J.R. Norris, Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2 (Cambridge University Press, Cambridge, 1998). Reprint of 1997 original

    Google Scholar 

  23. C. Pellegrini, Existence, uniqueness and approximation of a stochastic Schrödinger equation: the diffusive case. Ann. Probab. 36(6), 2332–2353 (2008)

    Article  MathSciNet  Google Scholar 

  24. C. Pellegrini, Poisson and diffusion approximation of stochastic master equations with control. Ann. Henri Poincaré 10(5), 995–1025 (2009)

    Article  MathSciNet  Google Scholar 

  25. C. Pellegrini, Continuous time open quantum random walks and non-Markovian Lindblad master equations. J. Stat. Phys. 154(3), 838–865 (2014)

    Article  MathSciNet  Google Scholar 

  26. B. Russo, H.A. Dye, A note on unitary operators in C∗-algebras. Duke Math. J. 33, 413–416 (1966)

    Article  MathSciNet  Google Scholar 

  27. R. Schrader, Perron-Frobenius theory for positive maps on trace ideals (2000). Preprint. arXiv: math-ph/0007020

    Google Scholar 

  28. I. Sinayskiy, F. Petruccione, Efficiency of open quantum walk implementation of dissipative quantum computing algorithms. Quantum Inf. Process. 11(5), 1301–1309 (2012)

    Article  MathSciNet  Google Scholar 

  29. I. Sinayskiy, F. Petruccione, Quantum optical implementation of open quantum walks. Int. J. Quantum Inf. 12(2), 1461010, 8 (2014)

    Article  MathSciNet  Google Scholar 

  30. M.D. Srinivas, E.B. Davies, Photon counting probabilities in quantum optics. Opt. Acta 28(7), 981–996 (1981)

    Article  MathSciNet  Google Scholar 

  31. M.M. Wolf, Quantum channels & operations: Guided tour (2012). http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf. Lecture notes based on a course given at the Niels-Bohr Institute

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Acknowledgements

All four authors are supported by ANR grant StoQ (ANR-14-CE25-0003-01). The research of Y.P. is also supported by ANR grant NONSTOPS (ANR-17-CE40-0006-01, ANR17-CE40-0006-02, ANR-17-CE40-0006-03).

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Authors and Affiliations

  1. Institut des Hautes Études Scientifiques, Université Paris-Saclay, Bures-sur-Yvette, France

    Ivan Bardet

  2. Institut de Mathématiques de Toulouse, UMR5219, UPS IMT, Toulouse Cedex, France

    Hugo Bringuier & Clément Pellegrini

  3. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France

    Yan Pautrat

Authors
  1. Ivan Bardet
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  2. Hugo Bringuier
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  3. Yan Pautrat
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  4. Clément Pellegrini
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Corresponding author

Correspondence to Yan Pautrat .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Alain Rouault

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Bardet, I., Bringuier, H., Pautrat, Y., Pellegrini, C. (2019). Recurrence and Transience of Continuous-Time Open Quantum Walks. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités L. Lecture Notes in Mathematics(), vol 2252. Springer, Cham. https://doi.org/10.1007/978-3-030-28535-7_18

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