The quantum-to-classical graph homomorphism game

TL;DR

This paper introduces a quantum-to-classical graph homomorphism game that generalizes classical graph homomorphisms to quantum settings, connecting non-local games, quantum colorings, and algebraic structures.

Contribution

It defines a new quantum-classical game framework, relates winning strategies to non-commutative graph homomorphisms, and extends algebraic results to quantum graphs.

Findings

Quantum colorings of all quantum complete graphs are explicitly constructed.

The algebra of the 4-coloring game for quantum graphs is always non-trivial.

The framework generalizes classical graph homomorphism concepts to quantum settings.

Abstract

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analogue of the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the $4$-coloring game for a quantum graph is always non-trivial, extending a result of [22].