Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

Abstract

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.Keywords Linear quantum stochastic system · Gaussian quantum state · Risk-sensitive quantum control Mathematics Subject Classification (2010) 81S25 · 81S05 · 81S22 · 81P16 · 81P40 · 81Q93 · 81Q10 · 60G15

A Quantum Karhunen-Loeve Expansion and Quadratic-Exponential Functionals for Linear Quantum Stochastic Systems

“…In combination with the commutativity [dW (v), X(s) T ] = 0 between the forward increments of the quantum Wiener process and the past system variables for all v s 0, and the CCRs (4), the relation (10) yields the two-point CCRs [32]:…”

“…A more general form T 0 X(t) T ΠX(t)dt of such a function, specified by a real positive definite symmetric matrix Π of order n, is reduced to (58) by transforming the OQHO as in (8), (9) with S := √ Π. The performance criteria in linear quadratic Gaussian control and filtering problems [17], [18], [39] are concerned with the minimization of quadratic costs EQ := Tr(ρQ), where E(·) is the quantum expectation over an underlying density operator ρ on the system-field space H. However, imposition of an exponential penalty on Q in (58) through the QEF [32] Ξ := Ee…”

“…as a risk-sensitive cost to be minimised (with θ > 0 quantifying the risk sensitivity), leads to additional properties of the quantum system (the 1 2 factor in (59) is introduced for convenience). These properties pertain to the large deviations of quantum trajectories [32] and robustness to quantum statistical uncertainties with a relative entropy description [20], [38] with respect to the nominal system-field state [33].…”

“…The QEF is the exponential of an integral-of-quadratic function of system variables, which is an alternative version of the original quantum risk-sensitive cost [13], [14] based on time-ordered exponentials. In addition to the robust performance bounds, the QEF is also related to the large deviations of quantum trajectories [32], so that its minimization (by varying the admissible controller parameters for a given quantum plant) can be used in order to secure more conservative closed-loop system dynamics. The QEF criteria and the original quantum risk-sensitive costs can share useful properties due to the Lie-algebraic links [35] between these two classes of cost functionals.…”

“…The Girsanov type representation, which involves only the commutation structure of the quantum dynamics regardless of the system-field state, allows the QEF to be related to the moment-generating functional (MGF) of the system variables. This result is then specified for the invariant multipoint Gaussian quantum state [32] when the oscillator is driven by vacuum fields. In this case, the QEF is reduced to a quadratic-exponential moment for the auxiliary Gaussian random process.…”