Perfect 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. 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.

Authors and Affiliations

1. School of Computer Science and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

   Richard Cleve

2. Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, N2L 2Y5, Canada

   Richard Cleve

3. Department of Mathematics, University of California at Berkeley, Berkeley, CA, 94720-3840, USA

   William Slofstra

4. CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

   Falk Unger

5. School of Computer Science and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

   Sarvagya Upadhyay

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