A Sharp Mutual Information Bound for Binary Pairs

We prove a sharp upper bound on the mutual information between two binary random variables in terms of their distribution-dependent KL contraction coefficient. This resolves the binary-pair conjecture of Anantharam, Gohari, Kamath, and Nair. Equality holds if and only if the variables are independent or form a doubly symmetric binary source. The proof uses supporting tangents of an entropy curve to vary the input distribution and the channel while preserving the contraction coefficient. Together with tensorization and data processing, the result also bounds the mutual information between binary outputs obtained separately from two correlated observation sequences. When the observation pairs are independent copies of a finite-alphabet pair, this bound depends only on the law of one pair and not on the number of observations.