An improved upper bound for the most informative boolean function conjecture

Suppose X is a uniformly distributed n-dimensional binary vector and Y is obtained by passing X through a binary symmetric channel with crossover probability α. A recent conjecture by Courtade and Kumar postulates that I(f(X); Y ) ≤ 1 - h(α) for any Boolean function f. So far, the best known upper bound was essentially I(f(X); Y ) ≤ (1 - 2α)
²
. In this paper, we derive a new upper bound that holds for all balanced functions, and improves upon the best known previous bound for α > 1 over 3.