All tight correlation Bell inequalities have quantum violations
It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimension of these faces, bounding it far away from the maximum. In the special case of non-local computation games, it had been shown before that they are not facet-defining; our result generalises and improves this. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets.