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Mathematics  2005 

Colouring an Orthogonality Graph

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We deal with a graph colouring problem that arises in quantum information theory. Alice and Bob are each given a $\pm1$-vector of length $k$, and are to respond with $k$ bits. Their responses must be equal if they are given equal inputs, and distinct if they are given orthogonal inputs; however, they are not allowed to communicate any information about their inputs. They can always succeed using quantum entanglement, but their ability to succeed using only classical physics is equivalent to a graph colouring problem. We resolve the graph colouring problem, thus determining that they can succeed without entanglement exactly when $k\leq3$.


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