# On the Input-Degradedness and Input-Equivalence Between Channels

## TL;DR

This paper characterizes input-degradedness and input-equivalence between channels, providing necessary and sufficient conditions, and explores the topological and continuity properties of these relationships in the space of channels.

## Contribution

It introduces new characterizations of input-degradedness, including a Blackwell-Sherman-Stein-like theorem, and analyzes the topologies and continuity of channel parameters under input-equivalence.

## Key findings

- A necessary and sufficient condition for input-degradedness.
- Any good decoder for one channel is also good for an input-degraded channel.
- Topological properties and continuity of channel parameters under input-equivalence.

## Abstract

A channel $W$ is said to be input-degraded from another channel $W'$ if $W$ can be simulated from $W'$ by randomization at the input. We provide a necessary and sufficient condition for a channel to be input-degraded from another one. We show that any decoder that is good for $W'$ is also good for $W$. We provide two characterizations for input-degradedness, one of which is similar to the Blackwell-Sherman-Stein theorem. We say that two channels are input-equivalent if they are input-degraded from each other. We study the topologies that can be constructed on the space of input-equivalent channels, and we investigate their properties. Moreover, we study the continuity of several channel parameters and operations under these topologies.