Proof of the Most Informative Boolean Function Conjecture

Suppose $\XX{N}$ is a uniformly distributed $N$-dimensional binary vector and $\YY{N}$ is obtained by passing $\XX{N}$ through a binary symmetric channel with crossover probability $\alpha$. Recently, Courtade and Kumar postulates that $I(f(\XX{N});\YY{N})\leq 1-\Be(\alpha)$ for any Boolean function $f$ \cite{courtade}. In this paper, we provide a proof of the correctness of this conjecture.