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

Abstract

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner stateρ=v|ψ−⟩⟨ψ−|+(1−v)14via a local hidden variable (LHV) model, where|ψ−⟩denotes the singlet state. We show analytically that these correlations are local forv=999×689×10−6cos2⁡(π/50)≃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 constantKG(3)≤1/v≃1.4644. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state forv≃0.4553. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.