Lyapunov Stability Analysis for Invariant States of Quantum Systems

TL;DR

This paper introduces a Lyapunov stability method for analyzing the convergence of quantum system density operators by considering the set of invariant states within a Banach space framework.

Contribution

It presents a novel approach to quantum stability analysis by applying Lyapunov methods to the set of density operators in a Banach space, highlighting the convexity and closedness of invariant states.

Findings

Invariant density operators form a closed, convex set.

Multiple isolated invariant density operators cannot exist.

Lyapunov operators can be used to analyze stability of the set.

Abstract

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In contrast to many previously studied convergence analysis methods for invariant density operators which use weak convergence, in this article we analyze the convergence of density operators by considering the set of density operators as a subset of Banach space. We show that the set of invariant density operators is both closed and convex, which implies the impossibility of having multiple isolated invariant density operators. We then show how to analyze the stability of this set via a candidate Lyapunov operator.