Conditional Information Inequalities for Entropic and Almost Entropic Points

TL;DR

This paper investigates the nature of conditional linear information inequalities, revealing their limitations, differences between entropic and almost entropic points, and exploring parallels in Kolmogorov complexity.

Contribution

It demonstrates that certain conditional inequalities cannot be extended unconditionally and compares their validity across entropic and almost entropic points.

Findings

Some conditional inequalities do not extend unconditionally.

Certain inequalities hold for almost entropic points but not for all.

Counterparts in Kolmogorov complexity are discussed.

Abstract

We study conditional linear information inequalities, i.e., linear inequalities for Shannon entropy that hold for distributions whose entropies meet some linear constraints. We prove that some conditional information inequalities cannot be extended to any unconditional linear inequalities. Some of these conditional inequalities hold for almost entropic points, while others do not. We also discuss some counterparts of conditional information inequalities for Kolmogorov complexity.