Open quantum random walks, quantum Markov chains and recurrence

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 ρ of OQRW. This sheds new light on some properties of the measure ρ. 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 φ-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.