Tensor Product of Polymatroids and Common Information

TL;DR

This paper establishes a novel link between linear representability conditions of polymatroids and the existence of common information extensions, using tensor products with uniform matroids.

Contribution

It introduces a new connection between necessary conditions for polymatroid representability and common information, expanding understanding of polymatroid structure.

Findings

Tensor product with uniform matroid implies common information extension.

New necessary condition for polymatroid linear representability.

Bridges between matroid theory and information theory.

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.