Information Causality, Szemer\'{e}di-Trotter and Algebraic Variants of CHSH

TL;DR

This paper establishes new bounds on the classical and quantum success probabilities in the CHSH_q game, revealing a surprising link to geometric incidence theory and resolving a problem in Information Causality.

Contribution

It provides the first asymptotic and explicit bounds for the CHSH_q game values and connects these bounds to geometric incidence theory, also resolving a problem in Information Causality.

Findings

First asymptotic bounds on CHSH_q values

Explicit bounds on entangled and classical success probabilities

Connection between CHSH_q and geometric incidence theory

Abstract

In this work, we consider the following family of two prover one-round games. In the CHSH_q game, two parties are given x,y in F_q uniformly at random, and each must produce an output a,b in F_q without communicating with the other. The players' objective is to maximize the probability that their outputs satisfy a+b=xy in F_q. This game was introduced by Buhrman and Massar (PRA 2005) as a large alphabet generalization of the celebrated CHSH game---which is one of the most well-studied two-prover games in quantum information theory, and which has a large number of applications to quantum cryptography and quantum complexity. Our main contributions in this paper are the first asymptotic and explicit bounds on the entangled and classical values of CHSH_q, and the realization of a rather surprising connection between CHSH_q and geometric incidence theory. On the way to these results, we also resolve a problem of Pawlowski and Winter about pairwise independent Information Causality, which, beside being interesting on its own, gives as an application a short proof of our upper bound for the entangled value of CHSH_q.