Practical Quantum Metrology in Noisy Environments

TL;DR

This paper investigates optimal phase estimation using two-level quantum probes in noisy environments, deriving bounds and analyzing the effects of noise, entanglement, and ancillas on measurement sensitivity.

Contribution

It introduces a simple noise model, derives a tight lower bound on phase sensitivity, and analyzes the role of entanglement and ancillas in noisy quantum metrology.

Findings

Optimal number of phase-imprinting applications $N_{opt}$ depends on noise properties.

Multi-probe entanglement offers limited advantage under local measurement restrictions.

Sensitivity initially grows quadratically with $N$, then decays after reaching $N_{opt}$.

Abstract

The problem of estimating an unknown phase $ \varphi $ using two-level probes in the presence of unital phase-covariant noise and using finite resources is investigated. We introduce a simple model in which the phase-imprinting operation on the probes is realized by a unitary transformation with a randomly sampled generator. We determine the optimal phase sensitivity in a sequential estimation protocol, and derive a general (tight-fitting) lower bound. The sensitivity grows quadratically with the number of applications $ N $ of the phase-imprinting operation, then attains a maximum at some $ N_\text{opt} $, and eventually decays to zero. We provide an estimate of $ N_\text{opt} $ in terms of accessible geometric properties of the noise and illustrate its usefulness as a guideline for optimizing the estimation protocol. The use of passive ancillas and of entangled probes in parallel to improve the phase sensitivity is also considered. We find that multi-probe entanglement may offer no practical advantage over single-probe coherence if the interrogation at the output is restricted to measuring local observables.