Characterizations of Two Channel Orderings: Input-Degradedness and the Shannon Ordering

The ordering of communication channels was first introduced by Shannon. In this paper, we aim to find characterizations of two orderings: input-degradedness and the Shannon ordering. 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 show that W' contains W (in the Shannon ordering sense) if and only if W is the skew-composition of W' with a convex-product n channel. This fact is used to derive a characterization of the Shannon ordering that is similar to the Blackwell-Sherman-Stein theorem.