Loss-tolerant hybrid measurement test of CHSH inequality with weakly amplified N00N states

Although our understanding of Bell's theorem and experimental techniques to test it have improved over the last 40 years, thus far all Bell tests have suffered at least from the detection or the locality loophole. Most photonic Bell tests rely on inefficient discrete-outcome measurements, often provided by photon counting detection. One possible way to close the detection loophole in photonic Bell tests is to involve efficient continuous-variable measurements instead, such as homodyne detection. Here, we propose a test of the Clauser-Horne-Shimony-Holt inequality that applies photon counting and homodyne detection on weakly amplified two-photon N00N states. The scheme suggested is remarkably robust against experimental imperfections and suits the limits of current technology. As amplified quantum states are considered, our work also contributes to the exploration of entangled macroscopic quantum systems. Further, it may constitute an alternative platform for a loophole-free Bell test, which is also important for quantum-technological applications.