Kumar–Courtade conjecture on Boolean functions and mutual information

Let $({\mathsf{x}},{\mathsf{y}})$ be dependent Rademacher random variables taking values in $\{-1,1\}$, and let

$$({\boldsymbol{{\mathsf{x}}}},{\boldsymbol{{\mathsf{y}}}})=({\mathsf{x}},{\mathsf{y}})^n$$

consist of $n$ independent, identically distributed copies. For a Boolean function $f\colon\{-1,1\}^n\to\{-1,1\}$, Kumar–Courtade conjecture.

$$I\bigl(f({\boldsymbol{{\mathsf{x}}}});{\boldsymbol{{\mathsf{y}}}}\bigr)\leq I({\mathsf{x}};{\mathsf{y}}).$$

Here $I(\cdot;\cdot)$ denotes mutual information. The conjecture asserts that no Boolean function of the first sequence can retain more mutual information with the second sequence than a single coordinate. The paper proves this inequality, resolving the conjecture.