# A Characterization of the Shannon Ordering of Communication Channels

## TL;DR

This paper characterizes the Shannon ordering of communication channels, showing a specific structural condition involving convex-product channels, and explores topologies and continuity properties on the space of Shannon-equivalent channels.

## Contribution

It provides a new characterization of Shannon ordering using skew-composition and convex-product channels, extending the Blackwell-Sherman-Stein theorem.

## Key findings

- A channel contains another iff it is a skew-composition with a convex-product channel.
- Introduces the strong topology and BRM metric on Shannon-equivalent channels.
- Studies continuity of channel parameters under the strong topology.

## Abstract

The ordering of communication channels was first introduced by Shannon. In this paper, we aim to find a characterization of the Shannon ordering. We show that $W'$ contains $W$ if and only if $W$ is the skew-composition of $W'$ with a convex-product channel. This fact is used to derive a characterization of the Shannon ordering that is similar to the Blackwell-Sherman-Stein theorem. Two channels are said to be Shannon-equivalent if each one is contained in the other. We investigate the topologies that can be constructed on the space of Shannon-equivalent channels. We introduce the strong topology and the BRM metric on this space. Finally, we study the continuity of a few channel parameters and operations under the strong topology.