On the role of symmetry and geometry in global quantum sensing

TL;DR

This paper compares invariance-based and geometry-based Bayesian approaches in global quantum sensing, showing how to unify them into a versatile framework for designing high-precision quantum sensors.

Contribution

It introduces a unified framework combining invariance and geometry principles for optimal quantum sensing, guiding the choice of assumptions for different estimation tasks.

Findings

Invariance-based methods often yield simpler priors and estimators.

Geometry-based approaches can achieve faster posterior convergence.

Unified framework guides optimal sensor design in noisy quantum environments.

Abstract

Global quantum sensing enables parameter estimation across arbitrary ranges with a finite number of measurements. Among the various existing formulations, the Bayesian paradigm stands as a flexible approach for optimal protocol design under minimal assumptions. Within this paradigm, however, there are two fundamentally different ways to capture prior ignorance and uninformed estimation; namely, requiring invariance of the prior distribution under specific parameter transformations, or adhering to the geometry of a state space. In this paper we carefully examine the practical consequences of both the invariance-based and the geometry-based approaches, and show how to apply them in relevant examples of rate and coherence estimation in noisy settings. We find that, while the invariance-based approach often leads to simpler priors and estimators and is more broadly applicable in adaptive scenarios, the geometry-based one can lead to faster posterior convergence in a well-defined measurement setting. Crucially, by employing the notion of location-isomorphic parameters, we are able to unify the two formulations into a single practical and versatile framework for optimal global quantum sensing, detailing when and how each set of assumptions should be employed to tackle any given estimation task. We thus provide a blueprint for the design of novel high-precision quantum sensors.