Exponential Stability of Subspaces for Quantum Stochastic Master Equations

TL;DR

This paper investigates the stability of quantum states and subspaces under stochastic dynamics, establishing conditions for invariance and asymptotic stability, and analyzing Lyapunov functions and stability rates in monitored quantum systems.

Contribution

It provides a comprehensive analysis of stability conditions for quantum subspaces, linking almost sure and average invariance, and introduces bounds on Lyapunov exponents considering measurement effects.

Findings

Almost sure invariance iff average invariance.

Existence of strict linear Lyapunov functions for average evolution.

Measurement can improve stability rate bounds.

Abstract

We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously--monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution, and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant.