Optimal protocols for quantum metrology with noisy measurements

TL;DR

This paper investigates optimal quantum preprocessing protocols to enhance measurement precision in noisy quantum metrology, deriving analytical solutions and demonstrating near-noiseless limits in multi-probe systems.

Contribution

It introduces the quantum preprocessing-optimized Fisher information, formulates the optimization as a biconvex problem, and provides analytical solutions and practical control strategies.

Findings

Unitary controls are optimal for pure states.

Coarse-graining controls are optimal for certain mixed states.

Global controls can asymptotically recover noiseless precision in multi-probe systems.

Abstract

Measurement noise is a major source of noise in quantum metrology. Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision. We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols. First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases. Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases. Finally, we demonstrate that in multi-probe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements. Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls.