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arXiv:2103.09279 · Infinite Dimensional Analysis Quantum Probability and Related Topics

Quadratic-exponential functionals of Gaussian quantum processes

Igor G. Vladimirov, Ian R. Petersen, M. R. James

Published Dec 1, 2021• 7 citations
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Abstract

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.

DOI
10.1142/s0219025721500247

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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.
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.
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 infinite-horizon asymptotic growth rate in the invariant Gaussian state when the stable system is driven by vacuum input fields. The resulting frequency-domain formula expresses 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 an illustrative numerical example with a two-mode open quantum harmonic oscillator.
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.
This paper is concerned with quadratic-exponential moments (QEMs) for dynamic variables of quantum stochastic systems with position-momentum type canonical commutation relations. The QEMs play an important role for statistical “localisation” of the quantum dynamics in the form of upper bounds on the tail probability distribution for a positive definite quadratic function of the system variables. We employ a randomised representation of the QEMs in terms of the moment-generating function (MGF) of the system variables, which is averaged over its parameters using an auxiliary classical Gaussian random vector. This representation is combined with a family of weighted L 2 -norms of the MGF, leading to upper bounds for the QEMs of the system variables. These bounds are demonstrated for open quantum harmonic oscillators with vacuum input fields and non-Gaussian initial states.
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.
This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicable to recursive computation of such moments.