Directly Coupled Observers for Quantum Harmonic Oscillators with Discounted Mean Square Cost Functionals and Penalized Back-action

TL;DR

This paper develops a quantum filtering approach for harmonic oscillators, optimizing a discounted mean square error while penalizing back-action, using algebraic matrix equations and Lie-algebraic techniques.

Contribution

It introduces a novel optimal control framework for quantum observers that accounts for back-action effects with discounted performance criteria.

Findings

Derived first-order optimality conditions for the quantum filtering problem.

Represented the optimality equations explicitly for equal-dimension plant and observer.

Utilized Lie-algebraic methods to analyze the Hamiltonian structure of the system.

Abstract

This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-order moments of the system variables. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action of the observer on the covariance dynamics of the plant. For the discounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and related Lie-algebraic techniques, we represent this set of equations in a more explicit form in the case of equally dimensioned plant and observer.