Algebraic Representations of Entropy and Fixed-Parity Information Quantities

TL;DR

This paper introduces an algebraic framework for representing and analyzing fixed-parity information quantities, refining previous measure spaces to better distinguish between different systems and proving a key property of XOR systems.

Contribution

It presents a natural algebraic structure for fixed-parity information quantities using ideals, enabling new counting arguments and characterizations, including a proof about XOR systems.

Findings

Refined measure space can distinguish more systems than previous models.

Algebraic behaviour of these quantities is characterized using ideals as upper-sets.

Proved that the XOR gate is the only completely synergistic three-variable system.

Abstract

Many information-theoretic quantities have corresponding representations in terms of sets. The prevailing signed measure space for characterising entropy, the $I$-measure of Yeung, is occasionally unable to discern between qualitatively distinct systems. In previous work, we presented a refinement of this signed measure space and demonstrated its capability to represent many quantities, which we called logarithmically decomposable quantities. In the present work we demonstrate that this framework has natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper-sets), and we show that this behaviour allows us to make various counting arguments and characterise many fixed-parity information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables $X$, $Y$ and $Z = f(X,Y)$ is the XOR gate.