The maximum mutual information between the output of a binary symmetric channel and a Boolean function of its input

We prove the Courtade-Kumar conjecture, which states that the mutual information between any Boolean function of an n-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the corresponding vector of outputs is upper-bounded by 1 − H(p), where H(p) represents the binary entropy function. That is, let X = [X1 . . . Xn] be a vector of independent and identically distributed Bernoulli( 1 2 ) random variables, which are the input to a memoryless binary symmetric channel, with the error probability equal to 0 ≤ p ≤ 1 2 , and Y = [Y1 . . . Yn] the corresponding output. Let f : {0, 1} → {0, 1} be an n-dimensional Boolean function. Then, MI(f(X),Y) ≤ 1 − H(p). We provide the proof for the most general case of the conjecture, that is for any n-dimensional Boolean function f and for any value of the error probability of the binary symmetric channel, 0 ≤ p ≤ 1 2 . Our proof employs only basic concepts from information theory, probability theory and transformations of random variables and vectors.