Separating Abelian and Homomorphic Entropy Cones
2026-08-10 • Information Theory
Information Theory
AI summaryⓘ
The authors compare two special sets of entropy inequalities related to groups: one based on Abelian groups and one based on bigger structures called homomorphic groups. They prove that for 16 variables, the homomorphic group-based set is strictly larger than the Abelian one, and they narrow down the smallest number of variables where this difference appears to between 6 and 16. They find a specific inequality that holds for Abelian groups but not universally, showing a clear distinction between these sets. Their example uses complex group constructions to precisely measure differences in entropy behaviors, highlighting how these different group-based models capture different information patterns.
entropy coneAbelian groupshomomorphic groupsinformation inequalityfinite groupsnormal subgroupscoset systemsPálfy–Szabó identityclass-two 2-groupentropy functions
Authors
Shahram Khazaei
Abstract
Chan and Yeung showed that finite groups suffice to determine which homogeneous linear information inequalities are universally valid. We compare two restricted group-characterizable entropy cones: the Abelian cone $\widetildeΓ^{\mathrm{Abl}}_n$ and the homomorphic cone $\widetildeΓ^{\mathrm{Hom}}_n$, the latter generated by coset systems of normal subgroups. We prove \[ \widetildeΓ^{\mathrm{Abl}}_{16}\subsetneq\widetildeΓ^{\mathrm{Hom}}_{16}, \] and, if $n_{\rm AH}$ is the least number of variables for which these cones differ, we show $6\le n_{\rm AH}\le16$. The separating functional is a class-restricted entropy inequality: it is valid on the Abelian cone but is not a universal information inequality. It is obtained by lifting the order dual of the Pálfy--Szabó six-cross identity while quantifying errors at inexact subgroup joins. We then construct sixteen normal subgroups of a class-two $2$-group of order $2^{43}$ for which every join error vanishes while the endpoint containment fails by one bit. Since mixed-linear random variables are Abelian, the same example also separates the mixed-linear and homomorphic entropy cones.