Open quantum random walks on the half-line: the Karlin-McGregor formula, path counting and Foster's Theorem

Abstract

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.

arXiv commentRevised version, includes a general version of the gambler's ruin problem and other applications to Foster's Theorem

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Cited by in corpus (3)

• Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators
• Quantum Markov chains on the line: matrix orthogonal polynomials, spectral measures and their statistics
• One-dimensional Continuous-Time Quantum Markov Chains: qubit probabilities and measures