Error Bounds on a Mixed Entropy Inequality

TL;DR

This paper derives sharp, quantitative bounds on the deficit in a mixed entropy inequality involving independent continuous and discrete variables, with a focus on Gaussian cases, revealing sub-Gaussian decay related to the Gaussian's standard deviation.

Contribution

It provides the first strong, sharp bounds on the entropy inequality deficit for Gaussian continuous variables, highlighting sub-Gaussian decay behavior.

Findings

Deficit decay is sub-Gaussian with respect to the reciprocal of the Gaussian's standard deviation.

Bounds are shown to be sharp up to rational factors.

Results are applicable to entropy computations in bit reset operations.

Abstract

Motivated by the entropy computations relevant to the evaluation of decrease in entropy in bit reset operations, the authors investigate the deficit in an entropic inequality involving two independent random variables, one continuous and the other discrete. In the case where the continuous random variable is Gaussian, we derive strong quantitative bounds on the deficit in the inequality. More explicitly it is shown that the decay of the deficit is sub-Gaussian with respect to the reciprocal of the standard deviation of the Gaussian variable. What is more, up to rational terms these results are shown to be sharp.