A short proof of the G\'acs--K\"orner theorem

TL;DR

This paper provides a concise proof of the Gács–Körner theorem, establishing conditions under which the mutual information equals the extractable common information for two discrete variables.

Contribution

It introduces a new, shorter proof of a key information theory result, based on the properties of auxiliary random variables and their restrictions.

Findings

Characterizes when mutual information equals common information.

Shows existence of certain joint variables constrains all joint distributions.

Provides a more accessible proof of a fundamental theorem.

Abstract

We present a short proof of a celebrated result of G\'acs and K\"orner giving sufficient and necessary condition on the joint distribution of two discrete random variables $X$ and $Y$ for the case when their mutual information matches the extractable (in the limit) common information. Our proof is based on the observation that the mere existence of certain random variables jointly distributed with $X$ and $Y$ can impose restriction on all random variables jointly distributed with $X$ and $Y$.