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 ${\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4$, we show that there is a local model for projective measurements if and only if $p\ensuremath{\leqslant}1∕{K}_{G}(3)$, where ${K}_{G}(3)$ is Grothendieck's constant of order 3. Known bounds on ${K}_{G}(3)$ prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary quantum states.