Maximal violations and efficiency requirements for Bell tests with photodetection and homodyne measurements

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 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 of 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.