Probabilistic bounds with quadratic-exponential moments for quantum stochastic systems

TL;DR

This paper develops probabilistic bounds for quantum stochastic systems using quadratic-exponential moments, employing a novel randomized representation involving Gaussian averaging, and demonstrates these bounds on quantum harmonic oscillators.

Contribution

It introduces a new randomized approach to bound quadratic-exponential moments in quantum systems, enhancing understanding of their statistical localization.

Findings

Derived upper bounds for QEMs of quantum variables.

Applied bounds to open quantum harmonic oscillators with non-Gaussian states.

Demonstrated effectiveness of the method through specific quantum system examples.

Abstract

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.