Recurrence and Transience of Continuous-Time Open Quantum Walks

Abstract

This paper is devoted to the study of continuous-time processes known as continuous-time open quantum walks (CTOQWs). A CTOQW represents the evolution of a quantum particle constrained to move on a discrete graph, but also has internal degrees of freedom modeled by a state (in the quantum mechanical sense), and contain as a special case continuous-time Markov 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 results in the context of irreducible CTOQWs. Because of the "quantum" internal degrees of freedom, CTOQWs 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 CTOQWs.

One-dimensional continuous-time quantum Markov chains: qubit probabilities and measures

“…This generalizes the classical condition for continuous-time Markov chains, namely, that the columns of the generator must have entries adding up to 0. Therefore, note that in G l , the indices for R j l are inverted with respect to the ones appearing in L. In this work we assume, as in [7], that R l l = 0 for all l and R l j = 0 if |j − l| > 2. The underlying graph structure can be seen in figure 1.…”

“…With these definitions we can proceed to define the concept of site recurrence. Following [7] (see also [34]), we say that a vertex |i is ρ-recurrent if…”

Monitored recurrence of a one-parameter family of three-state quantum walks

“…We point out that in the recent years the study of recurrence was considerably broadened to open quantum systems, quantum Markov chains and open quantum walks, see e.g. [18][19][20][21][22][23][24].…”