A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

Pavel Dvořák; Bruno Loff; Suhail Sherif

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arXiv · 2409.04592

A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

Pavel Dvořák·Bruno Loff·Suhail Sherif

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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.

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Pavel Dvořák, Bruno Loff, Suhail Sherif. 2024-09-06. A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity. https://arxiv.org/abs/2409.04592

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