Courtade–Kumar Conjecture on most informative Boolean functions

Prove that for every Boolean function b:{0,1}^n→{0,1} and every binary symmetric channel with crossover probability α∈(0,1/2), the mutual information between b(X^n) and the channel output Y^n satisfies I(b(X^n);Y^n) ≤ 1−H(α), where H(α) is the binary entropy function.

Background

The conjecture asserts that dictatorship functions maximize mutual information through a binary symmetric channel. While special cases and high-noise regimes have been resolved, the general conjecture remains open.

The authors summarize prior progress and present partial advances (e.g., improved bounds in the high-noise regime) but explicitly note that the full conjecture is still unresolved.