Strong Parallel Repetition Theorem for Quantum XOR Proof Systems

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. Such proof systems, called XOR proof systems, have previously been shown to characterize MIP (= NEXP) in the case of classical provers but to reside in EXP in the case of quantum provers (who are allowed to share a priori entanglement). We show that, in the quantum case, a perfect parallel repetition theorem holds for such proof systems 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, it does not hold. The theorem is proved by analyzing an XOR operation on XOR proof systems. Using semidefinite programming techniques, we show that this operation satisfies a certain additivity property, which we then relate to parallel repetitions of XOR games.