Partial indistinguishability theory for multi-photon experiments in multiport devices

We generalize an approach for description of multi-photon experiments with multi-port unitary linear optical devices, initiated in \textit{Phys. Rev. A \textbf{89}, 022333 (2014)} for the case of single photons in mixed spectral states, to arbitrary (multi-photon) input and arbitrary photon detectors. We give a physical interpretation of a non-negative definite Hermitian matrix, the matrix of a quadratic form giving output probabilities, as the partial indistinguishability matrix. We show that output probabilities are \textit{always} given in terms of the matrix permanents of the Hadamard product of network matrix and matrices depending on spectral state of photons and spectral sensitivities of detectors. Moreover, in case of input with up to one photon per mode, the output probabilities are given by a sum (or integral) with each term being the absolute value squared of such a matrix permanent. We conjecture that, for an arbitrary multi-photon input, zero output probability of an output configuration is \textit{always} the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, \textit{does not depend} on coherence between only partially indistinguishable photons. The conjecture is supported by examples. Furthermore, we propose a measure of partial indistinguishability of photons which generalizes Mandel's observation, and find the law of degradation of quantum coherence in a realistic Boson-Sampling device with increase of the total number of photons and/or their "classicality parameter".