Multi-point Gaussian states, quadratic-exponential cost functionals, and large deviations estimates for linear quantum stochastic systems

摘要

This paper is concerned with risk-sensitive performance analysis for linearquantum stochastic systems interacting with external bosonic fields. Weconsider a cost functional in the form of the exponential moment of theintegral of a quadratic polynomial of the system variables over a bounded timeinterval. An integro-differential equation is obtained for the time evolutionof this quadratic-exponential functional, which is compared with the originalquantum risk-sensitive performance criterion employed previously formeasurement-based quantum control and filtering problems. Using multi-pointGaussian quantum states for the past history of the system variables and theirfirst four moments, we discuss a quartic approximation of the cost functionaland its infinite-horizon asymptotic behaviour. The computation of theasymptotic growth rate of this approximation is reduced to solving twoalgebraic Lyapunov equations. We also outline further approximations of thecost functional, based on higher-order cumulants and their growth rates,together with large deviations estimates. For comparison, an auxiliaryclassical Gaussian Markov diffusion process is considered in a complexEuclidean space which reproduces the quantum system variables at the level ofcovariances but has different higher-order moments relevant to therisk-sensitive criteria. The results of the paper are also demonstrated by anumerical example and may find applications to coherent quantum risk-sensitivecontrol problems, where the plant and controller form a fully quantumclosed-loop system, and other settings with nonquadratic cost functionals.