Quantum games and synchronicity
Adina Goldberg
TL;DR
This work extends nonlocal games to quantum questions and answers within a diagrammatic, categorical framework, defining quantum games via quantum sets and quantum functions in the setting of finite-dimensional Hilbert spaces. It introduces a novel synchronicity notion based on sharing in the $X\\otimes X^{\\mathrm{op}}$ picture and proves that perfect synchronous correlations and strategies are themselves synchronous, also giving a diagrammatic Cauchy–Schwarz inequality and a notion of classical dimension. The paper then develops bisynchronicity, analyzes its consequences for dimensions and quantum isomorphisms, and applies the theory to games on quantum graphs, including quantum graph homomorphism and isomorphism games with links to quantum graph bijections, situating these results alongside classical graph theory. It positions these quantum diagrammatic constructions relative to existing approaches (LTW, MRV, BKS21, BHTT21/23, TT24) and demonstrates how quantum graphs can be manipulated and compared in this unified framework. Overall, the results provide a cohesive, graphical formalism for synchronous properties in quantum question/answer games and establish connections to quantum graph theory with potential implications for quantum networks and operator-algebraic perspectives.
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.