Revisiting Bell’s theorem for a class of down-conversion experiments

A certain class of parametric down-conversion Bell-type experiments has the following features. In the idealized perfect situation it is in only 50% of the cases that each observer receives a photon; in the other $50\%$ one observer receives both photons of a pair while the other observer receives none. The standard approach is to discard the events of the second type. Only the remaining ones are used as the data input to some Bell inequalities. This raises justified doubts about whether such experiments could ever be genuine tests of local realism. We propose to take into account these “unfavorable” cases and to analyze the entire pattern of polarization and localization correlations. This departure from the standard reasoning enables one to show that the experiments are indeed a true test of local realism.