Maximal violation of the Collins-Gisin-Linden-Massar-Popescu inequality for infinite dimensional states

We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the 2×2×d Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2×2 8 scenario.