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$D(f^{\oplus n}) \geq n\cdot(\frac{\Omega(D(f))}{\log \mathrm{r}\mathrm{k}(t)}-\log \text{rk}(f))$, where here$D(f), D(f^{\oplus n})$represent the deterministic communication complexity, and$\text{rk}(f)$is the rank of$f$. Our methods involve a new way to use information theory to reason about deterministic communication complexity.