Extending and Characterizing Quantum Magic Games

TL;DR

This paper generalizes quantum magic games to arbitrary arrangements, characterizes when they exhibit quantum pseudo-telepathy, and shows that sharing three Bell pairs and using Pauli measurements suffices for winning strategies.

Contribution

It introduces a framework for generalized magic games, characterizes pseudo-telepathy conditions, and provides minimal entanglement and measurement requirements for winning strategies.

Findings

Characterization of when generalized magic games exhibit quantum pseudo-telepathy

Winning strategies require only three Bell pairs of entanglement

Measurements from the three-qubit Pauli group are sufficient

Abstract

The Mermin-Peres magic square game is a cooperative two-player nonlocal game in which shared quantum entanglement allows the players to win with certainty, while players limited to classical operations cannot do so, a phenomenon dubbed "quantum pseudo-telepathy". The game has a referee separately ask each player to color a subset of a 3x3 grid. The referee checks that their colorings satisfy certain parity constraints that can't all be simultaneously realized. We define a generalization of these games to be played on an arbitrary arrangement of intersecting sets of elements. We characterize exactly which of these games exhibit quantum pseudo-telepathy, and give quantum winning strategies for those that do. In doing so, we show that it suffices for the players to share three Bell pairs of entanglement even for games on arbitrarily large arrangements. Moreover, it suffices for Alice and Bob to use measurements from the three-qubit Pauli group. The proof technique uses a novel connection of Mermin-style games to graph planarity.