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 ρₚW=pψ^-⟩⟨ψ^-+(1-p)1∕4, we show that there is a local model for projective measurements if and only if p1∕KG(3), where KG(3) is Grothendieck's constant of order 3. Known bounds on KG(3) prove the existence of this model at least for p0.66, quite close to the current region of Bell violation, p~0.71. We generalize this result to arbitrary quantum states.