A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

TL;DR

This paper introduces a quantum version of the pigeonhole principle and develops new semidefinite relaxation models for communication complexity, demonstrating their implications for solving certain computational games efficiently.

Contribution

It presents a novel quantum pigeonhole principle and constructs new semidefinite relaxation models for communication complexity, linking them to efficient solutions of Karchmer--Wigderson games.

Findings

Quantum pigeonhole principle is stronger than classical.

New models: γ₂ communication and quantum-lab protocols.

Protocols in these models can solve Karchmer--Wigderson games efficiently.

Abstract

We study semidefinite relaxations of $\Pi_1$ combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of $f$ is $> k$", we obtain new communication models, which we call "$\gamma_2$ communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models.