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On the quantum chromatic number of a graph. (English) Zbl 1182.05054
Summary: We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colors necessary in a protocol on which two separated provers can convince a referee that they have a colouring of the graph.
After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 14.

MSC:
05C15 Coloring of graphs and hypergraphs
05C80 Random graphs (graph-theoretic aspects)
05C85 Graph algorithms (graph-theoretic aspects)
68M12 Network protocols
81P68 Quantum computation
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