This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. An integro-differential equation is obtained for the time evolution of this quadratic-exponential functional, which is compared with the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. We also outline further approximations of the cost functional, based on higher-order cumulants and their growth rates, together with large deviations estimates. For comparison, an auxiliary classical Gaussian Markov diffusion process is considered in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different higher-order moments relevant to the risk-sensitive criteria. The results of the paper are also demonstrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals.
A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems
Research School of Electrical, Energy and Materials Engineering, College of Engineering and Computer Science, Australian National University（澳大利亚国立大学工程与计算机科学学院电气、能源与材料工程研究学院）
Quadratic Mean-Field BSDEs and Exponential Utility Maximization
School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）；School of Mathematics, Monash University（蒙纳士大学数学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
