Potential theory for quantum Markov states and other quantum Markov chains

We introduce a potential theory for a class of Quantum Markov Chains whose forward and backward Markov transition operators satisfy a special composition rule. We study the associated recurrence, transient and irreducibility properties and we prove that an irreducible quantum Markov chain is either recurrent or transient. Moreover, we show that our theory applies in many cases such as: quantum random walks, diagonal states, entangled Quantum Markov Chains. A characterization of Entangled Quantum Markov Chains is also given.