The quantum multinomial distribution: a combinatorial formulation of multiphoton interference

TL;DR

This paper introduces a quantum generalization of the multinomial distribution for multiphoton interference in linear optical systems, providing a combinatorial framework that captures quantum coherence effects without Hilbert space formalism.

Contribution

It develops a novel combinatorial formulation of multiphoton interference that extends classical multinomial statistics to include quantum coherence effects, enabling new statistical witnesses for boson sampling verification.

Findings

Quantum multinomial distribution incorporates interference effects.

Departure from classical statistics appears in higher cumulants.

Phase-dependent covariances reveal bosonic interference signatures.

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.