Stability Analysis of Quantum Systems: a Lyapunov Criterion and an Invariance Principle

TL;DR

This paper introduces a Lyapunov stability method for quantum systems, analyzing the convergence of density operators and characterizing invariant sets using quantum analogs of classical stability theorems.

Contribution

It develops a Lyapunov-based framework and invariance principles for stability analysis of quantum systems, extending classical methods to the quantum domain.

Findings

Invariant density operators form a closed convex set.

A Lyapunov operator can analyze stability of quantum states.

Quantum Barbashin-Krasovskii-La Salle theorem is established.

Abstract

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In analog to the classical probability measure for Markovian processes, we show that the set of invariant density operators is both closed and convex. We then show how to analyze the stability of this set via a candidate Lyapunov operator. We complete our analysis of the set of invariant density operators by introducing an analog of the Barbashin-Krasovskii-La Salle theorem on the dynamics of quantum systems.