AI 中文总结
本研究对比阿贝尔熵锥与同态熵锥，证明16变量下前者真包含于后者，确定两锥首次不同的最少变量数，构造出类2p-群的分离实例，同时实现混合线性与同态熵锥的分离。
AI 中文摘要
Chan与Yeung证明，有限群足以判定哪些齐次线性信息不等式具有普适有效性。本文对比两类受限的群可表征熵锥：阿贝尔锥$\widetildeΓ^{\mathrm{Abl}}_n$与同态锥$\widetildeΓ^{\mathrm{Hom}}_n$，后者由正规子群的陪集系统生成。我们证明\\[\n \widetildeΓ^{\mathrm{Abl}}_{16}\subsetneq\widetildeΓ^{\mathrm{Hom}}_{16}, \\]且若$n_{\rm AH}$为两锥首次出现差异的最少变量数，我们给出$6\le n_{\rm AH}\le16$。该分离泛函是一类受限熵不等式：它在阿贝尔锥上成立，但并非普适信息不等式。它通过提升Pálfy–Szabó六交叉恒等式的序对偶，并量化非精确子群并处的误差得到。随后我们构造了一个阶为$2^{43}$的2类$2$-群的16个正规子群，其所有并误差均为零，而端点包含关系相差1比特不成立。由于混合线性随机变量是阿贝尔的，该实例同样实现了混合线性熵锥与同态熵锥的分离。
英文摘要
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.
