Moment dynamics and observer design for a class of quasilinear quantum stochastic systems

TL;DR

This paper analyzes a class of open quantum systems with algebraic structure, deriving moment dynamics, invariant state characteristics, and designing a Kalman-like quantum filter for state estimation.

Contribution

It introduces a tractable approach to moment dynamics and observer design for quasilinear quantum stochastic systems with algebraic structure.

Findings

Closed-form quasi-characteristic function of invariant state

Explicit infinite-horizon growth rates for cost functionals

Kalman-like quantum filter for state estimation

Abstract

This paper is concerned with a class of open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices pertaining to finite-level systems. The system interacts with external bosonic fields, and its Hamiltonian and coupling operators depend linearly on the system variables. This results in a Hudson-Parthasarathy quantum stochastic differential equation (QSDE) whose drift and dispersion terms are affine and linear functions of the system variables. The quasilinearity of the QSDE leads to tractable dynamics of mean values and higher-order multi-point moments of the system variables driven by vacuum input fields. This allows for the closed-form computation of the quasi-characteristic function of the invariant quantum state of the system and infinite-horizon asymptotic growth rates for a class of cost functionals. The tractability of the moment dynamics is also used for mean square optimal Luenberger observer design in a measurement-based filtering problem for a quasilinear quantum plant, which leads to a Kalman-like quantum filter.