  We consider a class of two-prover interactive proof systems where each prover
returns a single bit to the verifier and the verifier's verdict is a function
of the XOR of the two bits received. We show that, when the provers are allowed
to coordinate their behavior using a shared entangled quantum state, a perfect
parallel repetition theorem holds in the following sense. The prover's optimal
success probability for simultaneously playing a collection of XOR proof
systems is exactly the product of the individual optimal success probabilities.
This property is remarkable in view of the fact that, in the classical case
(where the provers can only utilize classical information), it does not hold.
The theorem is proved by analyzing parities of XOR proof systems using
semidefinite programming techniques, which we then relate to parallel
repetitions of XOR games via Fourier analysis.
