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

Abstract

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state ρ = v |ψ−ihψ−| + (1 − v)1/4 via a local hidden variable (LHV) model, where |ψ−i denotes the singlet state. We show analytically that these correlations are local for v = 999 × 689 × 10−6 cos4(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 constant KG(3) ≤ 1/v ' 1.4644. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v ' 0.4553. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.