Application of Pontryagin's Maximum Principle to Quantum Metrology in Dissipative Systems

TL;DR

This paper applies Pontryagin's Maximum Principle to optimize quantum parameter estimation in dissipative systems, revealing how decoherence influences optimal control strategies and Fisher information.

Contribution

It introduces an efficient gradient computation method for quantum Fisher information and analyzes the impact of decoherence on optimal control protocols.

Findings

Optimal control maximizes quantum Fisher information in dissipative systems.

Decoherence causes the optimal control to become unbounded without constraints.

Maximum quantum Fisher information occurs at finite time due to decoherence.

Abstract

Optimal control theory, also known as Pontryagin's Maximum Principle, is applied to the quantum parameter estimation in the presence of decoherence. An efficient procedure is devised to compute the gradient of quantum Fisher information with respect to the control parameters and is used to construct the optimal control protocol. The proposed procedure keeps the control problem in the time-invariant form so that both first-order and second-order optimality conditions derived from Pontryagin's Maximum Principle apply; the second-order condition turns out to be crucial when the optimal control contains singular arcs. Concretely we look for the optimal control that maximizes quantum Fisher information for "twist and turn" problem. We find that the optimal control is singular without dissipation but can become unbounded once the quantum decoherence is introduced. An amplitude constraint is needed to guarantee a bounded solution. With quantum decoherence, the maximum quantum Fisher information happens at a finite time due to the decoherence, and the asymptotic value depends on the specific decoherence channel and the control of consideration.