Open Quantum Random Walks, Quantum Markov Chains and Recurrence

TL;DR

This paper explores the relationship between open quantum random walks and quantum Markov chains, introducing new concepts of recurrence and analyzing their properties to deepen understanding of quantum stochastic processes.

Contribution

It constructs quantum Markov chains linked to open quantum random walks and introduces a novel notion of recurrence, connecting it to existing concepts.

Findings

Established relations between new and existing recurrence notions.

Demonstrated that the measure can be viewed as a distribution of functions of a Markov process.

Provided insights into properties of the measure $P_\rho$.

Abstract

In the present paper, we construct QMCs 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 $P_\rho$ of OQRW. This sheds new light on some properties of the measure $P_\rho$. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov process. Furthermore, we study several properties of QMC and associated measure. A new notion of $\f$-recurrence of QMC is studied, and it is established relations between the defined recurrence and the existing ones.