Tight bound on relative entropy by entropy difference

TL;DR

This paper establishes a tight lower bound on the relative entropy based on entropy difference and system dimension, with implications for thermodynamics and information theory.

Contribution

It introduces a new tight inequality relating relative entropy and entropy difference applicable to quantum and classical systems.

Findings

Derived a tight lower bound on relative entropy.

Provided a tight upper bound on the variance of surprisal.

Implications for thermodynamic reversibility and channel capacity.

Abstract

We prove a lower bound on the relative entropy between two finite-dimensional states in terms of their entropy difference and the dimension of the underlying space. The inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference, both for quantum and classical systems. We outline implications for information theory and thermodynamics, such as a necessary condition for a process to be close to thermodynamic reversibility, or an easily computable lower bound on the classical channel capacity. Furthermore, we derive a tight upper bound, uniform for all states of a given dimension, on the variance of the surprisal, whose thermodynamic meaning is that of heat capacity.