State-space computation of quadratic-exponential functional rates for linear quantum stochastic systems

TL;DR

This paper develops a state-space method to compute the growth rates of quadratic-exponential functionals in linear quantum stochastic systems, enhancing robustness analysis through frequency and state-space techniques.

Contribution

It introduces a novel state-space approach for calculating QEF growth rates in quantum systems using algebraic equations, bridging frequency-domain and state-space methods.

Findings

The method accurately computes QEF growth rates for quantum systems.

Comparison with frequency-domain results validates the approach.

Numerical example demonstrates practical applicability.

Abstract

This paper is concerned with infinite-horizon growth rates of quadratic-exponential functionals (QEFs) for linear quantum stochastic systems driven by multichannel bosonic fields. Such risk-sensitive performance criteria impose an exponential penalty on the integral of a quadratic function of the system variables, and their minimization improves robustness properties of the system with respect to quantum statistical uncertainties and makes its behaviour more conservative in terms of tail distributions. We use a frequency-domain representation of the QEF growth rate for the invariant Gaussian quantum state of the system with vacuum input fields in order to compute it in state space. The QEF rate is related to a similar functional for a classical stationary Gaussian random process generated by an infinite cascade of linear systems. A truncation of this shaping filter allows the QEF rate to be computed with any accuracy by solving a recurrent sequence of algebraic Lyapunov equations together with an algebraic Riccati equation. The state-space computation of the QEF rate and its comparison with the frequency-domain results are demonstrated by a numerical example for an open quantum harmonic oscillator.