# Comment on the Equality Condition for the I-MMSE Proof of Entropy Power Inequality

## TL;DR

This paper clarifies the conditions under which the I-MMSE proof of the entropy power inequality achieves equality, by deriving an exact expression for the deficit and linking it to the Cauchy functional equation.

## Contribution

It provides the first precise characterization of the equality condition in the I-MMSE proof of the EPI, connecting it to the Cauchy functional equation.

## Key findings

- Derived an exact expression for the EPI deficit.
- Identified a necessary condition for equality involving the Cauchy functional equation.
- Enhanced understanding of the equality case in the I-MMSE proof.

## Abstract

The paper establishes the equality condition in the I-MMSE proof of the entropy power inequality (EPI). This is done by establishing an exact expression for the deficit between the two sides of the EPI. Interestingly, a necessary condition for the equality is established by making a connection to the famous Cauchy functional equation.