Exponential Separation of Information and Communication

We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity <= O(k), and distributional communication complexity >= 2(k). This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [1], our gap is the largest possible. By a result of Braverman and Rao [2], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold.