Recurrence and transience of continuous-time open quantum walks

TL;DR

This paper investigates the recurrence and transience properties of continuous-time open quantum walks (CTOQWs), revealing a quantum-specific trichotomy in behavior due to internal degrees of freedom, using quantum trajectories as key tools.

Contribution

It extends classical Markov chain concepts to quantum walks, establishing a quantum-specific classification of recurrence and transience for CTOQWs.

Findings

CTOQWs exhibit a quantum-specific trichotomy in recurrence and transience.

Quantum trajectories are effective tools for analyzing CTOQWs.

The paper generalizes classical Markov chain results to quantum processes.

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.