The quantum multinomial distribution: a combinatorial formulation of multiphoton interference

Abstract

This paper presents a quantum generalization of the multinomial distribution for the transition probabilities of $m$ identical photons in a $k$-port linear optical interferometer: two multinomial coefficients (one for the input configuration, one for the output) times the squared modulus of a coherent sum over routing matrices, weighted by the multivariate hypergeometric distribution; no Hilbert space formalism is needed to state or evaluate it. The classical multinomial is recovered when all photons enter through a single port, the coherent sum degenerating to a single term with no interference; the quantum family is not a generalization in the Askey sense but a parallel family that departs from classical statistics through the coherence of the amplitude summation. The $r$-th factorial moment carries a squared multinomial coefficient in place of the classical single one, the extra factor arising from the two copies of the amplitude expansion whose indices the Fock state forces to agree; for the beam splitter, the third cumulant is invariant under bosonic interference and the quantum departure first appears in the fourth cumulant as negative excess kurtosis; for multiport interferometers, however, three-body interference breaks this invariance and the departure enters already at the third cumulant. Cross-mode covariances involve the phases of the scattering matrix through coherence terms that strengthen output anti-correlations beyond the classical value; together with the squared-coefficient signature in the single-mode moments, these provide low-order statistical witnesses for boson sampling verification without requiring the full permanent computation.