On the Input-Degradedness and Input-Equivalence Between Channels

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.