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Communications in Theoretical Physics

Return Probability of the Open Quantum Random Walk with Time-Dependence

Clement Boateng Ampadu

Published May 1, 2013• 4 citations
Publisher

Abstract

We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"
DOI
10.1088/0253-6102/59/5/08

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A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random walker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we take care of a discrete-time three-state quantum walk on a hexagonal lattice from the view point of mathematics. The mathematical result shows a limit of the return probability when the walker starts off at the origin. The result of the limit tells us about a possibility of localization at the position and a dependence of localization on the initial state.
In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.