Entanglement of approximate quantum strategies in XOR games

We characterize the amount of entanglement that is sufficient to play any XOR game near-optimally. We show that for any XOR game G and epsilon > 0 there is an epsilon-optimal strategy for G using inverted right perpendicular epsilon(-1)inverted right perpendicular ebits of entanglement, irrespective of the number of questions in the game. By considering the family of XOR games CHSH(n) introduced by Slofstra (Jour. Math. Phys. 2011), we show that this bound is nearly tight: for any epsilon > 0 there is an n = Theta(epsilon(-1/5)) such that Omega(epsilon(-1/5)) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1 - epsilon) from optimal in CHSH(n).