Tensor Product of Polymatroids and Common Information
Carles Padró
TL;DR
The paper investigates how common information extensions (CI extensions) and tensor products relate in polymatroids, focusing on tensoring with the uniform matroid $U_{2,3}$. It shows that the existence of a $U_{2,3}$ tensor product implies a CI extension for every pair of subsets, with rank-bounds governed by triple-intersection dimensions and a streamlined route to Ingleton's inequality. A direct vector-space construction provides a simple proof that Ingleton's inequality holds for all polymatroids admitting a $U_{2,3}$ tensor product, and a concrete method links CI extensions to tensor-product structure to yield a $1$-CI property. Together, these results unify two known necessary conditions for linear representability and offer a constructive framework for deriving CI extensions from tensor-product representations.
Abstract
A new connection between two different necessary conditions for a polymatroid to be linearly representable is presented. Specifically, we prove that the existence of a tensor product with the uniform matroid of rank two on three elements implies the existence of a common information extension for every pair of subsets of the ground set.