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Uncertainty Relation for Entropy and Temperature of Gibbs States

Francis J. Headley

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.

Uncertainty Relation for Entropy and Temperature of Gibbs States

Abstract

We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that , dual to the temperature Fisher information . Their product is independent of the Hamiltonian, yielding the universal uncertainty relation 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 and . We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where , and connect 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.
Paper Structure (18 sections, 38 equations, 1 figure, 1 table)

This paper contains 18 sections, 38 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Entropy Fisher information
  3. Universal uncertainty relation
  4. Two-level system
  5. Quantum harmonic oscillator
  6. Critical-point scaling
  7. Relation to thermodynamic geometry
  8. Other conjugate pairs
  9. Rényi entropy Fisher information
  10. Generalisations
  11. Discussion
  12. Reparametrisation validity
  13. Saturation by energy measurement
  14. Grand canonical ensemble
  15. Generalised Gibbs ensemble
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Fisher information for entropy and temperature estimation in (a) a two-level system with energy gap $\Delta$ and (b) a quantum harmonic oscillator with frequency $\omega$. Solid curves show $F_S = 1/C_{\!v}$ (blue) and $F_T = C_{\!v}/T^2$ (red); the dashed curve is their product $F_S \cdot F_T = 1/T^2$.