Synchronicity for quantum non-local games

TL;DR

This paper introduces new classes of quantum non-local games with synchronous properties, providing algebraic characterizations of their perfect strategies and exploring their connections to quantum graph homomorphisms.

Contribution

It defines and analyzes concurrent quantum non-local games and their algebraic structures, extending the theory of quantum graph homomorphisms and strategies.

Findings

Characterization of perfect strategies in various correlation classes.

Algebraic structures for concurrent quantum non-local games.

Connection between quantum graph homomorphisms and entanglement-assisted classical homomorphisms.

Abstract

We introduce concurrent quantum non-local games, quantum output mirror games and concurrent classical-to-quantum non-local games, as quantum versions of synchronous non-local games, and provide tracial characterisations of their perfect strategies belonging to various correlation classes. We define *-algebras and C*-algebras of concurrent classical-to-quantum and concurrent quantum non-local games, and algebraic versions of the orthogonal rank of a graph. We show that quantum homomorphisms of quantum graphs can be viewed as entanglement assisted classical homomorphisms of the graphs, and give descriptions of the perfect quantum commuting and the perfect approximately quantum strategies for the quantum graph homomorphism game. We specialise the latter results to the case where the inputs of the game are based on a classical graph.