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Multi-photon split states, where each photon is in a different spatial mode,represent an essential resource for various quantum applications, yet theirefficient characterization remains an open problem. Here, we formulate thegeneral structure of their reduced spatial density matrices and identify thenumber of real and complex-valued independent coefficients, which in particularcompletely determine the distinguishability of all photons. Then, we show thatthis density matrix can be fully characterized by measuring correlations afterphoton interference in a static integrated circuit, where the required outputsscale sub-quadratically versus the number of photons. We present optimizedcircuit designs composed of segmented coupled waveguides, representing a linearoptical neural network, which minimize the reconstruction error and facilitaterobustness to fabrication deviations.

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