Quantum One-Way Communication is Exponentially Stronger Than Classical Communication

In STOC 1999, Raz presented a (partial) function for which there is a quantum protocol communicating only $O(\log n)$ qubits, but for which any classical (randomized, bounded-error) protocol requires $\text{poly}(n)$ bits of communication. That quantum protocol requires two rounds of communication. Ever since Raz’s paper it was open whether the same exponential separation can be achieved with a quantum protocol that uses only one round of communication. Here we settle this question in the affirmative.