Quantum Markov Chains Associated with Open Quantum Random Walks

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 of the underlying dynamics such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc. So, the quantum Markov chain machinery opens many new features of the dynamics. On the other hand, as will be shown in this paper, the open quantum random walks serves as a very interesting nontrivial model for which one can construct the associated quantum Markov chains. Here, after constructing the quantum Markov chain associated with the open quantum random walks, 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 see also that the classical Markov chains can be reconstructed as quantum Markov chains.