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

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.Comment: 34 pages, 3 figures, 2 table

A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems

This paper extends the Karhunen-Loeve representation from classical Gaussian random processes to quantum Wiener processes which model external bosonic fields for open quantum systems. The resulting expansion of the quantum Wiener process in the vacuum state is organised as a series of sinusoidal functions on a bounded time interval with statistically independent coefficients consisting of noncommuting position and momentum operators in a Gaussian quantum state. A similar representation is obtained for the solution of a linear quantum stochastic differential equation which governs the system variables of an open quantum harmonic oscillator. This expansion is applied to computing a quadratic-exponential functional arising as a performance criterion in the framework of risk-sensitive control for this class of open quantum systems.Comment: 8 pages, 1 figure, submitted to CDC 201

Risk-sensitive performance criteria and robustness of quantum systems with a relative entropy description of state uncertainty

This paper considers links between the original risk-sensitive performance criterion for quantum control systems and its recent quadratic-exponential counterpart. We discuss a connection between the minimization of these cost functionals and robustness with respect to uncertainty in system-environment quantum states whose deviation from a nominal state is described in terms of the quantum relative entropy. These relations are similar to those in minimax LQG control for classical systems. The results of the paper can be of use in providing a rational choice of the risk-sensitivity parameter in the context of robust quantum control with entropy theoretic quantification of statistical uncertainty in the system-field state.Comment: 8 pages, submitted to MTNS 201

Quadratic-exponential functionals of Gaussian quantum processes

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.Comment: 25 pages, submitted to Infinite Dimensional Analysis, Quantum Probability and Related Topic

Lie-algebraic connections between two classes of risk-sensitive performance criteria for linear quantum stochastic systems

This paper is concerned with the original risk-sensitive performance criterion for quantum stochastic systems and its recent quadratic-exponential counterpart. These functionals are of different structure because of the noncommutativity of quantum variables and have their own useful features such as tractability of evolution equations and robustness properties. We discuss a Lie algebraic connection between these two classes of cost functionals for open quantum harmonic oscillators using an apparatus of complex Hamiltonian kernels and symplectic factorizations. These results are aimed to extend useful properties from one of the classes of risk-sensitive costs to the other and develop state-space equations for computation and optimization of these criteria in quantum robust control and filtering problems.Comment: 8 pages, submitted to SIAM Conference on Control and Its Applications (CT19

A Girsanov type representation of quadratic-exponential cost functionals for linear quantum stochastic systems

This paper is concerned with multimode open quantum harmonic oscillators and quadratic-exponential functionals (QEFs) as quantum risk-sensitive performance criteria. Such systems are described by linear quantum stochastic differential equations driven by multichannel bosonic fields. We develop a finite-horizon expansion for the system variables using the eigenbasis of their two-point commutator kernel with noncommuting position-momentum pairs as coefficients. This quantum Karhunen-Loeve expansion is used in order to obtain a Girsanov type representation for the quadratic-exponential functions of the system variables. This representation is valid regardless of a particular system-field state and employs the averaging over an auxiliary classical Gaussian random process whose covariance operator is defined in terms of the quantum commutator kernel. We use this representation in order to relate the QEF to the moment-generating functional of the system variables. This result is also specified for the invariant multipoint Gaussian quantum state when the oscillator is driven by vacuum fields.Comment: 12 pages, submitted to the European Control Conference (ECC 2020), 12-15 May 2020, Saint Petersburg, Russi

A Karhunen-Loeve expansion for one-mode open quantum harmonic oscillators using the eigenbasis of the two-point commutator kernel

This paper considers one-mode open quantum harmonic oscillators with a pair of conjugate position and momentum variables driven by vacuum bosonic fields according to a linear quantum stochastic differential equation. Such systems model cavity resonators in quantum optical experiments. Assuming that the quadratic Hamiltonian of the oscillator is specified by a positive definite energy matrix, we consider a modified version of the quantum Karhunen-Loeve expansion of the system variables proposed recently. The expansion employs eigenvalues and eigenfunctions of the two-point commutator kernel for linearly transformed system variables. We take advantage of the specific structure of this eigenbasis in the one-mode case (including its connection with the classical Ornstein-Uhlenbeck process). These results are applied to computing quadratic-exponential cost functionals which provide robust performance criteria for risk-sensitive control of open quantum systems.Comment: 9 pages, accepted to the 2019 Australian and New Zealand Control Conference (ANZCC 2019), Auckland University of Technology, Auckland, New Zealand, 27-29 November 201

Frequency-domain computation of quadratic-exponential cost functionals for linear quantum stochastic systems

This paper is concerned with quadratic-exponential functionals (QEFs) as risk-sensitive performance criteria for linear quantum stochastic systems driven by multichannel bosonic fields. Such costs impose an exponential penalty on quadratic functions of the quantum system variables over a bounded time interval, and their minimization secures a number of robustness properties for the system. We use an integral operator representation of the QEF, obtained recently, in order to compute its asymptotic infinite-horizon growth rate in the invariant Gaussian state when the stable system is driven by vacuum input fields. The resulting frequency-domain formulas express the QEF growth rate in terms of two spectral functions associated with the real and imaginary parts of the quantum covariance kernel of the system variables. We also discuss the computation of the QEF growth rate using homotopy and contour integration techniques and provide two illustrations including a numerical example with a two-mode oscillator.Comment: 8 pages, 3 figures, submitted to the 21st IFAC World Congress, Berlin, Germany, July 12-17, 202

Invariant states of linear quantum stochastic systems under Weyl perturbations of the Hamiltonian and coupling operators

This paper is concerned with the sensitivity of invariant states in linear quantum stochastic systems with respect to nonlinear perturbations. The system variables are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation (QSDE) driven by quantum Wiener processes of external bosonic fields in the vacuum state. The quadratic system Hamiltonian and the linear system-field coupling operators, corresponding to a nominal open quantum harmonic oscillator, are subject to perturbations represented in a Weyl quantization form. Assuming that the nominal linear QSDE has a Hurwitz dynamics matrix and using the Wigner-Moyal phase-space framework, we carry out an infinitesimal perturbation analysis of the quasi-characteristic function for the invariant quantum state of the nonlinear perturbed system. The resulting correction of the invariant states in the spatial frequency domain may find applications to their approximate computation, analysis of relaxation dynamics and non-Gaussian state generation in nonlinear quantum stochastic systems.Comment: 30 pages, 6 figure

Measurement-based feedback control of linear quantum stochastic systems with quadratic-exponential criteria

This paper is concerned with a risk-sensitive optimal control problem for a feedback connection of a quantum plant with a measurement-based classical controller. The plant is a multimode open quantum harmonic oscillator driven by a multichannel quantum Wiener process, and the controller is a linear time invariant system governed by a stochastic differential equation. The control objective is to stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional (QEF) which penalizes the plant variables and the controller output. We combine a frequency-domain representation of the QEF growth rate, obtained recently, with variational techniques and establish first-order necessary conditions of optimality for the state-space matrices of the controller.Comment: 8 pages, 1 figure, submitted to the 21st IFAC World Congress, Berlin, Germany, July 12-17, 202