Grothendieck’s constant and local models for noisy entangled quantum states

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ₏^W=p^-⟩⟨^-+ (1-p) 1∕4, we show that there is a local model for projective measurements if and only if p1∕K₆ (3), where K₆ (3) is Grothendieck's constant of order 3. Known bounds on K₆ (3) prove the existence of this model at least for p0. 66, quite close to the current region of Bell violation, p0. 71. We generalize this result to arbitrary quantum states.