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

Abstract

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 =p ψ- ψ- + (1-p) 1, we show that there is a local model for projective measurements if and only if p≤1 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. © 2006 The American Physical Society.