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

Figure 1

Nonlocal properties of two-qubit Werner states ${\rho }_p^W$. Werner’s local model works up to $p=1∕2$, while the CHSH inequality is violated when $p>2^{-1∕2}\sim 0.71$. Here, we prove the existence of a local model for projective measurements when $p\lesssim 0.66$.