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 $ρ^W_p=p \proj{ψ^-}+(1-p){\one}/{4}$, we show that there is a local model for projective measurements if and only if $p\le 1/K_G(3)$, 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 $p\lesssim 0.66$, quite close to the current region of Bell violation, $p\sim 0.71$. We generalize this result to arbitrary quantum states.

6 pages, 1 figure

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