Thermodynamic uncertainty relations from the integral representation of quantum Fisher information

Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which mechanically conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for thermodynamically conjugate variables — specifically, a classical intensive parameter

and its corresponding extensive quantum operator

\hat{O} O ̂

— in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter

\theta θ

is quantified by its quantum Fisher information

\mathcal{F}_\theta ℱ θ

. The framework is based on an exact integral representation that relates

\mathcal{F}_{\theta} ℱ θ

to the power spectral density of

\hat{O} O ̂

. Using this representation, we demonstrate how fundamental information-theoretic limits naturally emerge as thermodynamic constraints. This yields a tight thermodynamic uncertainty relation:

\Delta\theta\,\overline{\Delta O} ≥ k_\text{B}T Δ θ Δ O ¯ ≥ k B T

, where

\overline{\Delta O}\equiv -\text{Tr}[\hat{O}\partial_\theta\hat{\rho}]\,\Delta\theta Δ O ¯ ≡ − Tr [ O ̂ ∂ θ ρ ̂ ] Δ θ

, and

T T

is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.