An XOR Lemma for Deterministic Communication Complexity

We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot \Big(\frac{\Omega(D(f))}{\log \mathsf{rk}(f)} -\log \mathsf{rk}(f)\Big )$, where here $D(f), D(f^{\oplus n})$ represent the deterministic communication complexity, and $\mathsf{rk}(f)$ is the rank of $f$. Our methods involve a new way to use information theory to reason about deterministic communication complexity.