The Category-Theoretic Arithmetic of Information

TL;DR

This paper introduces a category-theoretic framework for understanding various measures of information flow, unifying discrete, continuous, and quantum information under a common axiomatic structure.

Contribution

It presents a novel axiomatic approach that models communication systems as morphisms, linking diverse information measures through category theory.

Findings

Framework includes measures from classical, continuous, and quantum information

Proves foundational results based on the axioms

Unifies mathematical constructs like vector space dimension with information measures

Abstract

We highlight the underlying category-theoretic structure of measures of information flow. We present an axiomatic framework in which communication systems are represented as morphisms, and information flow is characterized by its behavior when communication systems are combined. Our framework includes a variety of discrete, continuous, and, conjecturally, quantum information measures. It also includes some familiar mathematical constructs not normally associated with information, such as vector space dimension. We discuss these examples and prove basic results from the axioms.