A lower bound on the value of entangled binary games

A two-player one-round binary game consists of two cooperative players who each one replies by one bit to a message that receives privately; they win the game if both questions and answers satisfy some predetermined property. A game is called entangled if the players are allowed to share a priori entanglement. It is well-known that the maximum winning probability (the value) of entangled XOR-games (binary games in which the predetermined property depends only on the XOR of the two output bits) can be computed by a semidefinite programming. In this paper we extend this result in the following sense; if a binary game is uniform, meaning that in the optimal strategy the marginal distribution of the output of each player is uniform, then its entangled value can be exactly computed by a semidefinite programming. We also introduce a lower bound on the entangled value of a general game; this bound depends on the size of the output set of each player and can be computed by a semidefinite programming. In particular, we show that if the game is binary, ωq is its entangled value, and ωsdp is the optimum value of the corresponding semidefinite program, then ωsdp ≥ ωq ≥ 0.68ωsdp.