Semi-Global Approximate stabilization of an infinite dimensional quantum stochastic system

TL;DR

This paper develops a semi-global approximate feedback control method for stabilizing an infinite-dimensional quantum stochastic system, ensuring convergence to a target state with high probability using Lyapunov functions and probabilistic techniques.

Contribution

It introduces a novel stabilization approach for quantum systems modeled by infinite-dimensional Markov chains, utilizing Lyapunov functions and probabilistic convergence methods.

Findings

Proves convergence of the quantum system to the target state with high probability.

Develops a stabilization method applicable to infinite-dimensional quantum systems.

Uses Lyapunov functions and Prohorov's theorem for convergence analysis.

Abstract

In this paper we study the semi-global (approximate) state feedback stabilization of an infinite dimensional quantum stochastic system towards a target state. A discrete-time Markov chain on an infinite-dimensional Hilbert space is used to model the dynamics of a quantum optical cavity. We can choose an (unbounded) strict Lyapunov function that is minimized at each time-step in order to prove (weak-$\ast$) convergence of probability measures to a final state that is concentrated on the target state with (a pre-specified) probability that may be made arbitrarily close to 1. The feedback parameters and the Lyapunov function are chosen so that the stochastic flow that describes the Markov process may be shown to be tight (concentrated on a compact set with probability arbitrarily close to 1). We then use Prohorov's theorem and properties of the Lyapunov function to prove the desired convergence result.