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

Flavien Hirsch; Marco Túlio Quintino; Tamás Vértesi; Miguel Navascués; Nicolas Brunner

doi:10.22331/q-2017-04-25-3

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 Quintino^1,2, 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

Published:	2017-04-25, volume 1, page 3
Eprint:	arXiv:1609.06114v4
Doi:	https://doi.org/10.22331/q-2017-04-25-3
Citation:	Quantum 1, 3 (2017).

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Abstract

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.

► BibTeX data

@article{Hirsch2017betterlocalhidden,
  doi = {10.22331/q-2017-04-25-3},
  url = {https://doi.org/10.22331/q-2017-04-25-3},
  title = {Better local hidden variable models for two-qubit {W}erner states and an upper bound on the {G}rothendieck constant {$K_G(3)$}},
  author = {Hirsch, Flavien and Quintino, Marco T{\'{u}}lio and V{\'{e}}rtesi, Tam{\'{a}}s and Navascu{\'{e}}s, Miguel and Brunner, Nicolas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {3},
  month = apr,
  year = {2017}
}

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