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Simple proof links group theory to advanced entropy inequalities

A simple proof of the group-theoretic Zhang-Yeung inequality

Abstract: Zhang and Yeung (IEEE Trans. Inf. Theory, 1998) established the first non-Shannon-type inequality that holds for all entropic vectors. Chan and Yeung (IEEE Trans. Inf. Theory, 2002) showed that there is a one-to-one relation between linear entropy inequalities and multiplicative inequalities involving cardinalities of subgroups of a finite group. The Shannon inequality admits a simple proof in the group theoretic setting, but a direct proof of the translation of Zhang--Yeung's inequality to the group-theoretic setting remained elusive. We resolve this open problem, giving a direct proof of the group-theoretic Zhang--Yeung inequality for cardinalities of subgroups of finite groups, using elementary group-theoretic and counting arguments.

Mon 28 SeptInformation Theory

The gist

Some mathematical rules about measuring information, called entropy inequalities, are tricky to prove. Zhang and Yeung found a special inequality that wasn't explained by older rules. The authors solved a puzzle that translates this special rule into a group theory context, using basic group math and counting. This direct proof was missing for over twenty years and strengthens the connection between abstract algebra and information theory.

Open → 2609.35451v1