text: We show that for any @math there is an XOR game @math with @math inputs for one player and @math inputs for the other player such that @math ebits are required for any strategy achieving bias that is at least a multiplicative factor @math from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent @math the scaling of our bound with @math is tight: for any XOR game there is an @math -optimal strategy using @math ebits, irrespective of the number of questions in the game.
  @cite_24: We give an explicit family of XOR games with O(n)-bit questions requiring 2n ebits to play near-optimally. More generally, we introduce a new technique for proving lower bounds on the amount of entanglement required by an XOR game: we show that near-optimal strategies for an XOR game G correspond to approximate representations of a certain C*-algebra associated to G. Our results extend an earlier theorem of Tsirelson characterising the set of quantum strategies which implement extremal quantum correlations.
  @cite_18: One Investigates inequalities for the probabilities and mathematical expectations which follow from the postulates of the local quantum theory. It turns out that the relation between the quantum and the classical correlation matrices is expressed] in terms of Grothendieck's known constant. It is also shown that the extremal quantum correlations characterize the Clifford algebra (i.e., canonical anticommutative relations).
summary: The general study of optimal strategies in XOR games was initiated by Tsirelson, who shows @cite_18 that for any XOR game with @math and @math inputs per party there is an optimal strategy that uses a maximally entangled state of dimension at most @math , where @math is the largest integer that satisfies @math and @math . To establish this Tsirelson first proves that to each player's input in the game can be associated a real @math -dimensional unit vector, @math for Alice and @math for Bob, such that the correlations @math achieve the optimal quantum bias in the game. Tsirelson then uses a clever construction, based on a representation of the Clifford algebra, to show that these vectors can be mapped to observables and a maximally entangled state in dimension @math that achieve precisely the same correlations. Slofstra @cite_24 shows that Tsirelson's bound is tight for a slight variant of the @math game.