Journal of Statistical Physics

, Volume 154, Issue 3, pp 838–865 | Cite as

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

  • Clément Pellegrini


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.


Open quantum random walks Quantum repeated interactions   Non-Markovian Lindblad master equations Quantum trajectory Open quantum systems 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

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

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