  Because of the fundamental importance of Bell's theorem, a loophole-free
demonstration of a violation of local realism (LR) is highly desirable. Here,
we study violations of LR involving photon pairs. We quantify the experimental
evidence against LR by using measures of statistical strength related to the
Kullback-Leibler (KL) divergence, as suggested by van Dam et al. [W. van Dam,
R. Gill and P. Grunwald, IEEE Trans. Inf. Theory. 51, 2812 (2005)].
Specifically, we analyze a test of LR with entangled states created from two
independent polarized photons passing through a polarizing beam splitter. We
numerically study the detection efficiency required to achieve a specified
statistical strength for the rejection of LR depending on whether photon
counters or detectors are used. Based on our results, we find that a test of LR
free of the detection loophole requires photon counters with efficiencies of at
least 89.71%, or photon detectors with efficiencies of at least 91.11%. For
comparison, we also perform this analysis with ideal unbalanced Bell states,
which are known to allow rejection of LR with detector efficiencies above 2/3.
