2021/03/16 by Igor G. Vladimirov, Vladimirov, Igor G., Ian R. Petersen +3
Computer Science · Physics and Astronomy · #37L40 #47B35 #60G15 #81P16 #81Q10 #81Q93 #81R15 #81S05 #81S22 #81S25 #93B35 #94A17 #Advanced Thermodynamics and Statistical Mechanics #FOS: Electrical engineering #FOS: Mathematics #FOS: Physical sciences #Optimization and Control (math.OC) #Probability (math.PR) #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2103.09279
openalex publication_date 2021/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.