Quantum games and synchronicity

TL;DR

This paper extends nonlocal quantum games to include quantum questions and answers using categorical quantum mechanics, introducing new definitions and results related to synchronicity and quantum graph homomorphisms.

Contribution

It develops a diagrammatic calculus for quantum questions and answers, extending classical game concepts to a quantum setting with new synchronicity results.

Findings

Quantum graph homomorphism game is shown to be synchronous.

Extended definitions of strategies and correlations align with existing literature.

Connected perfect strategies to quantum graph isomorphisms.

Abstract

In the flavour of categorical quantum mechanics, we extend nonlocal games to allow quantum questions and answers, using quantum sets (special symmetric dagger Frobenius algebras) and the quantum functions of Musto, Reutter, and Verdon [arXiv:1711.07945]. Equations are presented using a diagrammatic calculus for tensor categories. To this quantum question and answer setting, we extend the standard definitions, including strategies, correlations, and synchronicity, and we use these definitions to extend results about synchronicity. We extend the graph homomorphism (isomorphism) game to quantum graphs, and show it is synchronous (bisynchronous) and connect its perfect (bi)strategies to quantum graph homomorphisms (isomorphisms). Our extended definitions agree with the existing quantum games literature, except in the case of synchronicity.