Quantum Markov chains associated with open quantum random walks

TL;DR

This paper introduces quantum Markov chains linked to open quantum random walks, providing tools to analyze their fundamental properties and establishing their relation to classical Markov chains.

Contribution

It constructs quantum Markov chains for open quantum random walks and demonstrates their equivalence to previous concepts, with examples and connections to classical chains.

Findings

Quantum Markov chains can analyze properties like reducibility and irreducibility.

The approach is equivalent to previous methods by Carbone and Pautrat.

Classical Markov chains can be reconstructed as quantum Markov chains.

Abstract

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.