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

Flavien Hirsch1, Marco Túlio Quintino1,2, Tamás Vértesi3, Miguel Navascués4, and Nicolas Brunner1

1Groupe de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland
2Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
3Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary
4Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state $\rho = v |\psi_- \rangle \langle\psi_- | + (1- v ) \frac{1}{4}$ via a local hidden variable (LHV) model, where $|\psi_- \rangle$ denotes the singlet state. We show analytically that these correlations are local for $ v = 999\times689\times{10^{-6}}$ $\cos^2(\pi/50) \simeq 0.6829$. 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 $K_G(3) \leq 1/v \simeq 1.4644$. 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.

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