Capacity of Noisy Permutation Channels

We establish the capacity of a class of communication channels introduced by Makur. The n-letter input from a finite alphabet is passed through a discrete memoryless channel P-Z|X and then the output n-letter sequence is uniformly permuted. We show that the maximal communication rate (normalized by log n) equals 1/2 (rank(P-Z|X) - 1) whenever P-Z|X is strictly positive. This is done by establishing a converse bound matching the achievability of Makur. The two main ingredients of our proof are: 1) a sharp bound on the Kullback-Leibler divergence of a uniformly sampled vector from a type class and observed through a DMC to an iid vector; and 2) the covering e-net of a probability simplex with Kullback-Leibler divergence as a metric. In addition to strictly positive DMC we also find the noisy permutation capacity for q-ary erasure channels, the Z-channel and others.