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

The nonclassical properties of the noisy three-qubit Greenberger-Horne-Zeilinger (GHZ) states, ρv=v|GHZ><GHZ|+(1-v)1/8 parameterized by the visibility 0≤v≤1 are investigated. Based on the violation of the 2x2x2-setting Mermin inequality, ρv is nonclassical for the parameter range 1/2<v≤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=0.496 057, attainable with 5x5x5 settings, whereas the most economical one in number of settings corresponds to 3x3x4 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.