Nonclassicality threshold for the three-qubit Greenberger-Horne-Zeilinger state

Abstract

The nonclassical properties of the noisy three-qubit Greenberger-Horne-Zeilinger (GHZ) states, ${\ensuremath{\rho}}_{v}=v|\text{GHZ}\ensuremath{\rangle}\ensuremath{\langle}\text{GHZ}|+(1\ensuremath{-}v)\mathbb{1}/8$ parameterized by the visibility $0\ensuremath{\le}v\ensuremath{\le}1$ are investigated. Based on the violation of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$-setting Mermin inequality, ${\ensuremath{\rho}}_{v}$ is nonclassical for the parameter range $1/2<v\ensuremath{\le}1$. It has been posed whether additional settings would allow to lower the threshold visibility. Here we report on Bell inequalities giving a threshold value smaller than $v=1/2$. This rules out the possibility of a local hidden variable model in the limit of $v=1/2$. In particular, the lowest threshold visibility we found is $v=0.496\phantom{\rule{0.16em}{0ex}}057$, attainable with $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5$ settings, whereas the most economical one in number of settings corresponds to $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ settings. The method which enabled us to obtain these results, and in particular the about 10 000 tight Bell inequalities giving $v<1/2$, are also discussed in detail.