Separating Abelian and Homomorphic Entropy Cones

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 $\widetilde\Gamma^{\mathrm{Abl}}_n$ and the homomorphic cone $\widetilde\Gamma^{\mathrm{Hom}}_n$, the latter generated by coset systems of normal subgroups. We prove \[ \widetilde\Gamma^{\mathrm{Abl}}_{16}\subsetneq\widetilde\Gamma^{\mathrm{Hom}}_{16}, \] 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 P\'alfy--Szab\'o 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.