On essentially conditional information inequalities

TL;DR

This paper explores the nature of conditional information inequalities, demonstrating that some are inherently conditional and cannot be derived from unconditional inequalities, with new results for Shannon entropy and Kolmogorov complexity.

Contribution

The paper proves a new essentially conditional information inequality for Shannon entropy and discusses the inherent conditionality of certain inequalities.

Findings

Some conditional inequalities are inherently non-extendable to unconditional form

A new essentially conditional inequality for Shannon entropy is established

Conditional inequalities for Kolmogorov complexity are discussed

Abstract

In 1997, Z.Zhang and R.W.Yeung found the first example of a conditional information inequality in four variables that is not "Shannon-type". This linear inequality for entropies is called conditional (or constraint) since it holds only under condition that some linear equations are satisfied for the involved entropies. Later, the same authors and other researchers discovered several unconditional information inequalities that do not follow from Shannon's inequalities for entropy. In this paper we show that some non Shannon-type conditional inequalities are "essentially" conditional, i.e., they cannot be extended to any unconditional inequality. We prove one new essentially conditional information inequality for Shannon's entropy and discuss conditional information inequalities for Kolmogorov complexity.