A direct sum theorem in communication complexity via message compression

TL;DR

This paper establishes a new lower bound for the direct sum problem in classical communication complexity using message compression techniques, and discusses the limitations of such methods in quantum communication.

Contribution

It introduces a compression theorem linking information cost to communication cost in classical protocols, and explores the challenges in applying similar ideas to quantum protocols.

Findings

Classical message compression reduces communication proportional to information cost

The compression technique does not extend straightforwardly to quantum communication

Elementary information theoretic arguments derive the direct sum lower bounds

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, Shi, Wirth and Yao and refined further by Bar-Yossef, Jayram, Kumar and Sivakumar. Our main technical result is a `compression' theorem saying that, for any probability distribution $\mu$ 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. Hence, new techniques may be necessary to tackle the direct sum problem in quantum communication.