  We study nonlocality tests in which each party performs photodetection and
homodyne measurements. The results of such measurements are dichotomized and a
Clauser-Horne-Shimony-Holt (CHSH) inequality is used. We prove that in this
scenario the maximal violation is attainable and fully characterize the set of
maximally violating states. If we restrict our search to states composed by at
most 2, 4, and 6 photons per mode, we find critical photodetection efficiencies
of 0.48, 0.36, and 0.29. We also found an entangled variation of the famous cat
states that has critical efficiency 0.32. These values are well within the
limit of current photodetector technology, which suggests the present approach
as a road for a loophole-free Bell experiment.
