Lyapunov-Based Stabilization and Control of the Stochastic Schrodinger Equation

TL;DR

This paper develops a Lyapunov-based stochastic stability theory for the control and stabilization of quantum systems described by the Stochastic Schrödinger Equation, facilitating quantum control under continuous measurement.

Contribution

It introduces a novel Lyapunov framework and extends Ito calculus to analyze and ensure stability of quantum systems driven by SSE.

Findings

Provides sufficient conditions for stochastic stability of SSE

Develops a mathematical framework for quantum control under continuous measurement

First to incorporate Lyapunov methods into quantum stochastic stability analysis

Abstract

This paper presents a detailed Lyapunov-based theory to control and stabilize continuously-measured quantum systems, which are driven by Stochastic Schrodinger Equation (SSE). Initially, equivalent classes of states of a quantum system are defined and their properties are presented. With the help of equivalence classes of states, we are able to consider global phase invariance of quantum states in our mathematical analysis. As the second mathematical modelling tool, the conventional Ito formula is further extended to non-differentiable complex functions. Based on this extended Ito formula, a detailed stochastic stability theory is developed to stabilize the SSE. Main results of this proposed theory are sufficient conditions for stochastic stability and asymptotic stochastic stability of the SSE. Based on the main results, a solid mathematical framework is provided for controlling and analyzing quantum system under continuous measurement, which is the first step towards implementing weak continuous feedback control for quantum computing purposes.