Vern Paulsen, University of Houston
A game where Alice and Bob are separated, forbidden
to comunicate, receive inputs from the same input set I, and produce
outputs from the same output set O is called synchronous provided that
any time Alice and Bob receive the same input, they are required to
produce the same output. Examples of synchronous games include the graph coloring game, the graph homomorphism game, and games related to fractional chromatic numbers. We prove that a quantum strategy for such a game is always a trace on the algebra generated by Alice's projective measurements. Conversely, every trace on an algebra of projections satisfying certain conditions, generates a quantum strategy. Moreover, for finite dimensional algebras, traces are just weighted sums of maximally entangled states, which simplifies computing the value of such games.