Direct Sum Theorem for Bounded Round Quantum Communication Complexity

Abstract

We prove a direct sum theorem for bounded round entanglement-assisted quantum communication complexity. To do so, we use the fully quantum definition for information cost and complexity that we recently introduced, and use both the fact that information is a lower bound on the communication, and the fact that a direct sum property holds for quantum information complexity. We then give a protocol for compressing a single copy of a protocol down to its quantum information cost, up to terms depending on the number of rounds and the allowed increase in error. Two important tools to derive this protocol are a smooth conditional min-entropy bound for a one-shot quantum state redistribution protocol, and the quantum substate theorem of Jain, Radhakrishnan and Sen (FOCS'02) to transform this bound into a von Neumann conditional entropy bound. This result further establishes the newly introduced notions of quantum information cost and complexity as the correct quantum generalisations of the classical ones in the standard communication complexity setting. Finding such a quantum generalization of information complexity was one of the open problem recently raised by Braverman (STOC'12).

Incorrect statement of the substate theorem, affecting the main compression result. Thanks to Anurag Anshu and Rahul Jain for pointing this out

References in corpus (6)

• Quantum information can be negative
• The mother of all protocols: Restructuring quantum information's family tree
• The apex of the family tree of protocols: Optimal rates and resource inequalities
• Information Equals Amortized Communication
• Quantum Information Complexity and Amortized Communication
• A lower bound for bounded round quantum communication complexity of set disjointness

Cited by in corpus (1)

• Noisy quantum state redistribution with promise and the Alpha-bit