Uncertainty Relation for Entropy and Temperature of Gibbs States

• URL: http://arxiv.org/abs/2603.16456v2
• Date: Wed, 18 Mar 2026 14:26:00 GMT
• Title: Uncertainty Relation for Entropy and Temperature of Gibbs States
• Abstract summary: We derive the quantum Fisher information for entropy estimation in a Gibbs state.<n>Their product $F_Scdot F_T = 1/T2$ is independent of the Hamiltonian.<n>We identify energy measurement as the optimal protocol for entropy estimation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that $F_s = 1/C_v$, dual to the temperature Fisher information $F_S = C_v/T^2$. Their product $F_S\cdot F_T = 1/T^2$ is independent of the Hamiltonian, yielding the universal uncertainty relation $Δ^2 S\,Δ^2 T \geq T^2/n^2$ in which all system-specific quantities such as heat capacity, the Hamiltonian, and the number of degrees of freedom cancel identically. This is the metrological expression of the Legendre conjugacy between $S$ and $T$. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where $F_S \sim |t|^α\to 0$, and connect $F_S$ to the Ruppeiner metric in entropy coordinates. The uncertainty relation is shown to hold for all standard thermodynamic conjugate pairs, and we examine the distinguished role of the von~Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.

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