Finite-Temperature de Bruijn Identities: Fisher Information as the Spectral Gap of Blahut--Arimoto Dynamics

2026-06-02

AI Summary

This work addresses the limitation of the classical de Bruijn identity, which applies only to Gaussian convolutions and lacks a finite-temperature generalization, thereby obscuring deeper structural insights in information-theoretic optimization. By leveraging the spectral theory of Blahut–Arimoto (BA) dynamics within a finite-temperature statistical mechanics framework, the paper establishes an exact identity between differential entropy and Fisher information. Its key contribution lies in interpreting the spectral gap of the BA relaxation kernel as a finite-temperature regularization of Fisher information and proving that it equals the partial derivative of the BA free energy with respect to noise variance. This result not only recovers the classical de Bruijn identity in the zero-temperature limit but also unifies geometric perspectives on information inequalities and yields new insights into entropy power inequalities and χ² dissipation structures.

Abstract

We uncover a finite-temperature extension of de Bruijn's identity -- the classical relation $\frac{d}{dt}h(X+\sqrt{t}Z)=\frac{1}{2}J(X)$ connecting differential entropy and Fisher information. Our framework is the spectral theory of Blahut--Arimoto (BA) dynamics, recently developed by Wang~\cite{Wang2026} for the analysis of rate-distortion optimization. The central observation is elementary yet profound: for Gaussian sources, the spectral gap $\lam$ of the BA relaxation kernel $\G$ satisfies $\lam = 1/(2βσ^2)$~\cite{Wang2026}, while the Fisher information of the source is $J = 1/σ^2$. Hence \[ {\lam = \frac{J}{2β}} \] for all inverse temperatures $β> 1/(2σ^2)$. This identifies the BA spectral gap as a \emph{finite-temperature regularization of Fisher information}. From this observation we derive an exact finite-temperature de Bruijn identity: \[ \frac{\partial F_β}{\partial σ^2} = \frac{1}{2βσ^2} = \lam, \] where $F_β$ is the BA free energy. This identity holds for all finite $β$ without any limit procedure. The classical de Bruijn identity follows as the exact consequence $β\,\partial F_β/\partialσ^2 = J/2$. The significance is structural: classical de Bruijn is not an isolated fact about Gaussian convolutions, but the $β\to\infty$ shadow of a one-parameter family of exact identities living in the spectral geometry of rate-distortion optimization. We discuss implications for the entropy power inequality, the $χ^2$-dissipation structure of BA dynamics, and the geometric unification of information inequalities.

Problem

• de Bruijn identity
• Fisher information
• Blahut–Arimoto dynamics
• spectral gap
• finite-temperature

Innovation

• de Bruijn identity
• Blahut–Arimoto dynamics
• spectral gap
• Fisher information
• finite-temperature regularization