Skip to main content
SpringerLink
Log in

Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A new type of quantum random walks, called Open Quantum Random Walks, has been developed and studied in Attal et al. (Open quantum random walks, preprint) and (Central limit theorems for open quantum random walks, preprint). In this article we present a natural continuous time extension of these Open Quantum Random Walks. This continuous time version is obtained by taking a continuous time limit of the discrete time Open Quantum Random Walks. This approximation procedure is based on some adaptation of Repeated Quantum Interactions Theory (Attal and Pautrat in Annales Henri Poincaré Physique Théorique 7:59–104, 2006) coupled with the use of correlated projectors (Breuer in Phys Rev A 75:022103, 2007). The limit evolutions obtained this way give rise to a particular type of quantum master equations. These equations appeared originally in the non-Markovian generalization of the Lindblad theory (Breuer in Phys Rev A 75:022103, 2007). We also investigate the continuous time limits of the quantum trajectories associated with Open Quantum Random Walks. We show that the limit evolutions in this context are described by jump stochastic differential equations. Finally we present a physical example which can be described in terms of Open Quantum Random Walks and their associated continuous time limits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The operators \(R^{\alpha j}_{\, k}\) correspond to operators \(R^{jk}_{\alpha }\) in [7]. Our choice of notations is imposed by our definition of OQRW.

  2. In the sequel we denote \(C^2_c(\mathcal {B}(\mathcal {H}_S\otimes \mathbb {C}^V))\) the set of \(C^2\) functions with compact support.

  3. We refer to [10] for a complete reference on problems of martingale.

  4. The Skorohod topology is the usual topology of weak convergence (convergence of distribution) for càdlàg processes (see [6] for a complete reference on this theory).

  5. The author applies namely a correlated projector technique of type (18) to derive the appropriate master equation.

References

  1. Attal, S., Guillotin, N., Sabot, C.: Central Limit Theorems for Open Quantum Random Walks, preprint

  2. Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Annales Henri Poincaré Physique Théorique 7, 59–104 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Attal, S., Petruccione, F., Sabot, C., Sinayskiy, I.: Open Quantum Random Walks, preprint

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Barchielli, A., Pellegrini, C.: Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation. J. Math. Phys. 51, 112104 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  6. Billingsley, P.: Convergence of Probability Measure. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication, 2nd edn. Wiley, New York (1999)

  7. Breuer, H.P.: Non-Markovian generalization of the Lindblad theory of open quantum systems. Phys. Rev. A 75, 022103 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  8. Breuer, H.P.: Non-Markovian Quantum Dynamics and the Method of Correlated Projection Super-Operators. Lecture Notes in Physics, vol. 787/2010, pp. 125–139 (2010)

  9. Budini, A.A.: Lindblad rate equations. Phys. Rev. A 74, 053815 (2006)

    Article  ADS  Google Scholar 

  10. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)

  11. Kempe, J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  12. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119130 (1976)

    Article  MathSciNet  Google Scholar 

  13. Pellegrini, C.: Markov chains approximation of jumpdiffusion stochastic master equations. Ann. Inst. H. Poincar Probab. Statist. 46(4), 924–948 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Moodley, M., Petruccione, F.: Stochastic wave-function unraveling of the generalized Lindblad master equation Phys. Rev. A 79, 042103 (2009)

    Article  Google Scholar 

  15. Mora, C.M., Rebolledo, R.: Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions. Ann. Appl. Probab. 18(2), 591–619 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Statistique et de Probabilité, Institut de Mathématiques de Toulouse, Université Paul Sabatier (Toulouse III), 31062 , Toulouse Cedex 9, France

    Clément Pellegrini

Authors
  1. Clément Pellegrini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clément Pellegrini.

Additional information

Work supported by ANR project “HAM-MARK”, No ANR-09-BLAN-0098-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pellegrini, C. Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations. J Stat Phys 154, 838–865 (2014). https://doi.org/10.1007/s10955-013-0910-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-013-0910-x

Keywords

Search

Navigation