Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$

Abstract

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state $\ddot{I}=v|\ddot{I}{ˆ}_{-}><\ddot{I}{ˆ}_{-}|+(1-v)\frac{1}{4}$ via a local hidden variable (LHV) model, where $|\ddot{I}{ˆ}_{-}>$ denotes the singlet state. We show analytically that these correlations are local for $v=999\times 689\times 10^{-6}$ ${\cos }^4(\ddot{I}€/50)\simeq 0.6829$. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant $K_G(3)\le 1/v\simeq 1.4644$. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for $v\simeq 0.4553$. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.

12 pages, typos corrected