On Characterization of Entropic Vectors at the Boundary of Almost Entropic Cones

TL;DR

This paper investigates the boundary of the entropy region for three variables, providing new outer bounds and demonstrating the non-tightness of existing inner bounds through explicit distribution constructions.

Contribution

It introduces new outer bounds for the entropy region's boundary and shows that some existing inner bounds are not tight for certain faces.

Findings

New outer bounds for the entropy region boundary.

Explicit distributions demonstrating non-tight inner bounds.

Insights into the structure of entropic vectors at the boundary.

Abstract

The entropy region is a fundamental object in information theory. An outer bound for the entropy region is defined by a minimal set of Shannon-type inequalities called elemental inequalities also referred to as the Shannon region. This paper focuses on characterization of the entropic points at the boundary of the Shannon region for three random variables. The proper faces of the Shannon region form its boundary. We give new outer bounds for the entropy region in certain faces and show by explicit construction of distributions that the existing inner bounds for the entropy region in certain faces are not tight.