Complex structure and characterization of multi-photon split states in integrated circuits

Multi-photon split states, where each photon is in a different spatial mode, represent an essential resource for various quantum applications, yet their efficient characterization remains an open problem. Here, we formulate the general structure of their reduced spatial density matrices and identify the number of real and complex-valued independent coefficients, which in particular completely determine the distinguishability of all photons. Then, we show that this density matrix can be fully characterized by measuring correlations after photon interference in a static integrated circuit, where the required outputs scale sub-quadratically versus the number of photons. We present optimized circuit designs composed of segmented coupled waveguides, representing a linear optical neural network, which minimize the reconstruction error and facilitate robustness to fabrication deviations