On the Power of Entangled Quantum Provers

We show that the value of a general two-prover quantum game cannot be computed by a semidefinite program of polynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NP-complete problem GAP-3D-MATCHING can be obtained by a 2-prover, 1-round quantum interactive proof system where the provers share entanglement, with perfect completeness and soundness s = 1 − 2, and such that the space of the verifier and the size of the messages are O(log n). This implies that QMIP∗log n,1,1−2−O(n) * P unless P = NP and provides the first non-trivial lower bound on the power of entangled quantum provers, albeit with an exponentially small gap. The gap achievable by our proof system might in fact be larger, provided a certain conjecture on almost commuting versus nearly commuting projector matrices is true.