A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs

摘要

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players 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. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs G_N (N ≥ 12) and c = N yield pseudo-telepathy games.