On quantumness in multi-parameter quantum estimation

TL;DR

This paper introduces a measure of quantumness in multi-parameter quantum estimation, linking the Uhlmann Curvature and Fisher Information to quantify quantum incompatibility affecting estimation precision.

Contribution

It proposes a novel ratio between Uhlmann Curvature and Fisher Information as a figure of merit for quantumness in multi-parameter estimation problems.

Findings

The ratio estimates quantum incompatibility affecting estimation precision.

Application to a quantum many-body system reveals phase diagram-dependent quantumness.

The measure quantifies the discrepancy between achievable and theoretical estimation precision.

Abstract

In this article we derive a measure of quantumness in quantum multi-parameter estimation problems. We can show that the ratio between the mean Uhlmann Curvature and the Fisher Information provides a figure of merit which estimates the amount of incompatibility arising from the quantum nature of the underlying physical system. This ratio accounts for the discrepancy between the attainable precision in the simultaneous estimation of multiple parameters and the precision predicted by the Cram\'er-Rao bound. As a testbed for this concept, we consider a quantum many-body system in thermal equilibrium, and explore the quantum compatibility of the model across its phase diagram.