A quantum protocol to win the graph colouring game on all Hadamard graphs

Abstract

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph $G_N$ with $N=c=16$. Their protocol applies only to Hadamard graphs where $N$ is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of $G_{12}$ having 1609 vertices, with $c=12$. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.

5page

References in corpus (1)

• The Impossibility of Pseudo-Telepathy Without Quantum Entanglement

Cited by in corpus (14)

• Estimating quantum chromatic numbers
• Graph Homomorphisms for Quantum Players
• A compositional approach to quantum functions
• Quantum source-channel coding and non-commutative graph theory
• A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
• Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
• Synchronous correlation matrices and Connes' embedding conjecture
• Violating the Shannon capacity of metric graphs with entanglement
• Graph-theoretic approach to Bell experiments with low detection efficiency
• Quantum chromatic numbers via operator systems
• Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number
• Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity
• Conic Formulations of Graph Homomorphisms
• Maximally entangled states in pseudo-telepathy games