Strong Parallel Repetition Theorem for Quantum XOR Proof Systems

TL;DR

This paper proves a perfect parallel repetition theorem for quantum XOR proof systems, showing that entangled provers' success probabilities multiply across multiple instances, unlike the classical case.

Contribution

It establishes a novel parallel repetition theorem for quantum XOR proof systems, utilizing semidefinite programming and Fourier analysis techniques.

Findings

Success probability equals product of individual probabilities in quantum case

Classical case does not exhibit this perfect parallel repetition property

Uses semidefinite programming and Fourier analysis for proof

Abstract

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.