Partial orders and contraction for BISO channels
Christoph Hirche, Oxana Shaya
TL;DR
The paper addresses quantifying information loss under noisy channels by leveraging contraction coefficients and channel partial orders, focusing on BISO channels. It derives a closed-form expression for the KL contraction coefficient $\\eta_{KL}$ of BISO channels and proves two extremality results: with equal $\\eta_{KL}$, BEC and BSC bound the less noisy order; with equal Doeblin coefficient (or max leakage), they bound the degradability order. It further shows that among BISO channels with output dimension up to three these orders induce comparability, while higher dimensions yield counterexamples, and extends the discussion to general binary channels. Applications include explicit secrecy-capacity formulas for wiretap scenarios, $f$-divergence bounds, and DPI under input constraints, illustrating practical implications of the extremal structure.
Abstract
A fundamental question in information theory is to quantify the loss of information under a noisy channel. Partial orders and contraction coefficients are typical tools to that end, however, they are often also challenging to evaluate. For the special class of binary input symmetric output (BISO) channels, Geng et al. showed that among channels with the same capacity, the binary symmetric channel (BSC) and binary erasure channel (BEC) are extremal with respect to the more capable order. Here, we show two main results. First, for channels with the same KL contraction coefficient, the same holds with respect to the less noisy order. Second, for channels with the same Dobrushin coefficient, or equiv. maximum leakage or Doeblin coefficient, the same holds with respect to the degradability order. In the process, we provide a closed-form expression for the contraction coefficients of BISO channels. We also discuss the comparability of BISO channels and extensions to binary channels in general.