Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory

Abstract

In this work we study certain aspects of Open Quantum Random Walks (OQRWs), a class of quantum channels described by S. Attal et al. \cite{attal}. As a first objective we consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga \cite{saloff}, we define a notion of ergodicity for finite nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a second objective, and based on a quantum trajectory approach, we study a notion of hitting time for OQRWs and we see that many constructions are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. In this way we are able to examine open quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.

Revised version. arXiv admin note: substantial text overlap with arXiv:1504.05398, arXiv:1506.08325

References in corpus (2)

• The Generalized Lyapunov Theorem and its Application to Quantum Channels
• Perturbation Bounds for Quantum Markov Processes and their Fixed Points

Cited by in corpus (7)

• Quantum Markov chains associated with open quantum random walks
• Passage times, exit times and Dirichlet problems for open quantum walks
• Site recurrence of open and unitary quantum walks on the line
• Lazy Open Quantum Walks
• Open quantum random walks on the half-line: the Karlin-McGregor formula, path counting and Foster's Theorem
• Mean hitting times of quantum Markov chains in terms of generalized inverses
• Hitting statistics from quantum jumps