A direct sum theorem in communication complexity via message compression

Abstract

We prove lower bounds for the direct sum problem for two-party bounded error randomised multiple-round communication protocols. Our proofs use the notion of information cost of a protocol, as defined by Chakrabarti et al. [CSWY01] and refined further by Bar-Yossef et al. [BJKS02]. Our main technical result is a 'compression' theorem saying that, for any probability distribution μ over the inputs, a k-round private coin bounded error protocol for a function f with information cost c can be converted into a k-round deterministic protocol for f with bounded distributional error and communication cost O(kc). We prove this result using a Substate Theorem about relative entropy and a rejection sampling argument. Our direct sum result follows from this 'compression' result via elementary information theoretic arguments. We also consider the direct sum problem in quantum communication. Using a probabilistic argument, we show that messages cannot be compressed in this manner even if they carry small information. © Springer-Verlag Berlin Heidelberg 2003.