The quantum-to-classical graph homomorphism game

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].

v2: fixed errors in proofs of the old Theorem 4.7 and 5.6; removed Lemma 4.8

References in corpus (8)

• Connes' embedding problem and Tsirelson's problem
• Bigalois extensions and the graph isomorphism game
• Tsirelson's Problem
• Algebras, Synchronous Games and Chromatic Numbers of Graphs
• Quantum no-signalling correlations and non-local games
• Connes' Embedding Problem and Winning Strategies for Quantum XOR Games
• Chromatic numbers and a Lovász type inequality for non-commutative graphs
• Geometry of the set of synchronous quantum correlations

Cited by in corpus (9)

• Quantum no-signalling correlations and non-local games
• Quantum teleportation in the commuting operator framework
• Entanglement-symmetries of covariant channels
• A category of quantum posets
• Quantum graphs, subfactors and tensor categories I
• Repeated temperature measurements in quantum thermodynamics
• Algebraic connectedness and bipartiteness of quantum graphs
• Quantum Suplattices
• Weighted theta functions for non-commutative graphs