We show an exponential gap between communication complexity and external information complexity, by analyzing a communication task suggested as a candidate by Braverman [Bra13]. Previously, only a separation of communication complexity and internal information complexity was known [GKR14,GKR15].

More precisely, we obtain an explicit example of a search problem with external information complexity $\leq O(k)$, with respect to any input distribution, and distributional communication complexity $\geq 2^k$, with respect to some input distribution. In particular, this shows that a communication protocol cannot always be compressed to its external information. By a result of Braverman [Bra12], our gap is the largest possible.

Moreover, since the upper bound of $O(k)$ on the external information complexity of the problem is obtained with respect to any input distribution, our result implies an exponential gap between communication complexity and information complexity (both internal and external) in the non-distributional setting of Braverman [Bra12]. In this setting, no gap was previously known, even for internal information complexity.