  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.
