The Hilbertian tensor norm and entangled two-prover games

Abstract

We study tensor norms over Banach spaces and their relation to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm γ2 and its dual γ2* that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.