Recurrence and Transience of Continuous-Time Open Quantum Walks

This paper is devoted to the study of continuous-time processes known as continuous-time open quantum walks (CTOQW). A CTOQW represents the evolution of a quantum particle constrained to move on a discrete graph, but which also has internal degrees of freedom modeled by a state (in the quantum mechanical sense). CTOQW contain as a special case continuous-timeMarkov chains on graphs. Recurrence and transience of a vertex are an important notion in the study of Markov chains, and it is known that all vertices must be of the same nature if the Markov chain is irreducible. In the present paper we address the corresponding result in the context of irreducible CTOQW. Because of the "quantum" internal degrees of freedom, CTOQW exhibit non standard behavior, and the classification of recurrence and transience properties obeys a "trichotomy" rather than the classical dichotomy. Essential tools in this paper are the so-called "quantum trajectories" which are jump stochastic differential equations which can be associated with CTOQW.