Quantum Advantages in Hypercube Game

TL;DR

This paper introduces a multiplayer hypercube game generalizing CHSH, analyzing classical, quantum, and no-signalling strategies, revealing that quantum advantage diminishes exponentially with more players, unlike no-signalling strategies.

Contribution

It generalizes the CHSH game to multiple players, providing a complete characterization of classical, quantum, and no-signalling winning probabilities.

Findings

Quantum advantage decreases exponentially with more players.

No-signalling strategies always achieve perfect success.

Quantum value approaches classical value as number of players increases.

Abstract

We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all $m$ players are required to assign values to vertices on corresponding facets of an $m$-dimensional hypercube. The players win if and only if their answers satisfy both parity and consistency conditions. We completely characterize the maximum winning probabilities (game value) under classical, quantum and no-signalling strategies, respectively. In contrast to the original CHSH game designed to demonstrate the superiority of quantumness, we find that the quantum advantages in the Hypercube game significantly decrease as the number of players increase. Notably, the quantum value decays exponentially fast to the classical value as $m$ increases, while the no-signalling value always remains to be one.