Closing the detection loophole in the triangle network with high-dimensional photonic states

Bell nonlocality without input settings, e.g. in the triangle network, has been perceived to be particularly fragile, with low robustness to noise in physical implementations. Here we show to the contrary that nonlocality based on N00N states already for $N=2$ has an exceptionally high robustness to photon loss. For the dominant noise factor, single photon loss in the transmission channels, we can certify noise robustness up to $10\%$ loss, while for a realistic noise model we use neural network-based heuristics to observe $\sim 50\%$ robustness. Moreover we show that the robustness holds even for imperfect sources based on SPDC sources, where the heralding information of the sources can be used to avoid any global post-processing of the outcomes, such as discarding rounds when photons fail to arrive, and thus demonstrate how the detection loophole in the triangle network can be closed.