Strong data processing constant is achieved by binary inputs

TL;DR

This paper proves that the strong data processing constant for any channel is determined by its best binary-input subchannel, confirming a longstanding conjecture and extending the result to all f-divergences.

Contribution

It establishes that the strong data processing constant equals that of the optimal binary-input subchannel for any channel and divergence, confirming a conjecture from 1998.

Findings

Strong data processing constant equals that of the best binary-input subchannel.

Result holds for all f-divergences, not just KL divergence.

Confirms a conjecture by Cohen, Kemperman, and Zbaganu (1998).

Abstract

For any channel $P_{Y|X}$ the strong data processing constant is defined as the smallest number $\eta_{KL}\in[0,1]$ such that $I(U;Y)\le \eta_{KL} I(U;X)$ holds for any Markov chain $U-X-Y$. It is shown that the value of $\eta_{KL}$ is given by that of the best binary-input subchannel of $P_{Y|X}$. The same result holds for any $f$-divergence, verifying a conjecture of Cohen, Kemperman and Zbaganu (1998).