  Collective measurements on large quantum systems together with a majority
voting strategy can lead to a violation of the CHSH Bell inequality. In
presence of many entangled pairs, this violation decreases quickly with the
number of pairs, and vanishes for some critical pair number that is a function
of the noise present in the system. Here, we show that a different binning
strategy can lead to a more substantial Bell violation when the noise is
sufficiently small. Given the relation between the critical pair number and the
source noise, we then present an experiment where the critical pair number is
used to quantify the quality of a high visibility photon pair source. Our
results demonstrate nonlocal correlations using collective measurements
operating on clusters of more than 40 photon pairs.
