The stability of quantum Markov filters

TL;DR

This paper investigates conditions under which quantum filters become independent of initial states over time, establishing that observability ensures asymptotic stability in finite-dimensional quantum systems.

Contribution

It provides a rigorous criterion based on observability for the asymptotic independence of quantum filters from initial states, with explicit verification methods for finite-dimensional systems.

Findings

Quantum filters are asymptotically independent of initial states under observability.

The observability condition can be checked via a rank condition on the system's coefficients.

The results apply to absolutely continuous initial states in quantum stochastic models.

Abstract

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.