Steady states of continuous-time open quantum walks

TL;DR

This paper studies the long-term behavior of continuous-time open quantum walks on graphs, showing convergence to steady states and how graph structure influences quantum coherence over time.

Contribution

It introduces CTOQW as a framework for quantum dynamical semigroups on graphs and analyzes how graph properties affect steady states and quantum coherence.

Findings

CTOQW always converges to a steady state on connected graphs.

Steady state is maximally mixed for connected regular graphs.

Graph topology influences persistence of quantum coherence.

Abstract

Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. As shown by the examples in this article, the steady states of CTOQW can be very unusual and complicated even though the underlying graphs are simple. The examples demonstrate that the structure of a graph can affect quantum coherence in CTOQW through a long time run. Precisely, the quantum coherence persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, the quantum coherence will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).