Quantum Fisher Information for Entropy of Gibbs States

TL;DR

This paper derives the quantum Fisher information for entropy in Gibbs states, revealing a fundamental relation with heat capacity and thermodynamic coordinates, and identifies optimal measurement protocols.

Contribution

It establishes a duality-based uncertainty relation for entropy and temperature estimators and explores the geometric structure of thermodynamic state spaces.

Findings

Quantum Fisher information equals inverse heat capacity.

The product of entropy and temperature Fisher information is temperature-dependent only.

Energy measurement is optimal for entropy estimation.

Abstract

We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the R\'enyi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.