  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".
