The quantum multinomial distribution: a combinatorial formulation of multiphoton interference

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.

💡 Research Summary

The paper introduces a purely combinatorial formulation for the transition probabilities of $m$ identical photons propagating through a loss‑less $k$‑port linear optical interferometer described by a unitary scattering matrix $U$. Traditionally, these probabilities are expressed as the squared permanent of an $m\times m$ sub‑matrix of $U$, a $#P$‑hard object that obscures the physical mechanism of bosonic interference. By reorganizing the permanent into a sum over “routing matrices’’ $\mathbf J$, the authors obtain a closed‑form expression that involves only multinomial coefficients, the unitary matrix elements, and a multivariate hypergeometric weight.

A routing matrix $\mathbf J$ is a non‑negative integer $k\times k$ matrix whose row sums equal the input occupation vector $\mathbf n=(n_1,\dots,n_k)$ and whose column sums equal the output occupation vector $\mathbf c=(c_1,\dots,c_k)$. The set of all such matrices, denoted $\mathcal T(\mathbf n,\mathbf c)$, is exactly the set of integer points in the transportation polytope. For each $\mathbf J$ the amplitude factor is
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