Probabilistic Monads, Domains and Classical Information

TL;DR

This paper extends classical information theory by applying probabilistic monads and domain theory to model and analyze classical channels, revealing new insights into channel capacity and structure.

Contribution

It introduces a novel domain-theoretic framework for classical channels, generalizing previous results and connecting to quantum channels and algebraic structures.

Findings

Classical channels with fixed input-output sizes form a domain via quotient compact ordered spaces.

The capacity map is Scott-continuous and factors through the quotient domain.

The framework relates to recent quantum channel and algebraic monoid discoveries.

Abstract

Shannon's classical information theory uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results of Martin, Allwein and Moskowitz for binary channels to show how some more modern tools - probabilistic monads and domain theory in particular - can be used to model classical channels. As initiated Martin, et al., the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the (n x n)-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.