Risk-sensitive Dissipativity of Linear Quantum Stochastic Systems under Lur'e Type Perturbations of Hamiltonians

TL;DR

This paper develops a risk-sensitive dissipativity framework for linear quantum stochastic systems with Hamiltonian perturbations, ensuring boundedness of system moments despite nonlinearities.

Contribution

It introduces a quantum risk-sensitive storage function approach for systems with Lur'e type Hamiltonian perturbations, extending classical dissipativity theory to quantum nonlinear dynamics.

Findings

Conditions for bounded quantum risk-sensitive storage functions.

Boundedness of moments of system variables of arbitrary order.

Application of noncommutative exponential and PDE techniques.

Abstract

This paper is concerned with a stochastic dissipativity theory using quadratic-exponential storage functions for open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations. The system is linearly coupled to external boson fields and has a quadratic Hamiltonian which is perturbed by nonquadratic functions of linear combinations of system variables. Such perturbations are similar to those in the classical Lur'e systems and make the quantum dynamics nonlinear. We study their effect on the quantum expectation of the exponential of a positive definite quadratic form of the system variables. This allows conditions to be established for the risk-sensitive stochastic storage function of the quantum system to remain bounded, thus securing boundedness for the moments of system variables of arbitrary order. These results employ a noncommutative analogue of the Doleans-Dade exponential and a multivariate partial differential version of the Gronwall-Bellman lemma.