The quantum multinomial distribution: a combinatorial formulation of multiphoton interference

TL;DR

This work introduces the quantum multinomial distribution to describe multiphoton interference in $k$-port interferometers by organizing transition amplitudes into routing matrices weighted by a multivariate hypergeometric distribution. The key idea is that $P(\mathbf{c}|\mathbf{n})$ equals a scaled squared coherent sum over routing configurations, $P(\mathbf{c}|\mathbf{n})=\binom{m}{\mathbf{n}}\binom{m}{\mathbf{c}}\left|\sum_{\mathbf{J}} w_{\mathbf{J}} a_{\mathbf{J}}\right|^2$, which makes explicit how quantum interference arises from the combinatorial structure of transport, while remaining a classical object in construction. In the two-port case this reduces to the quantum binomial distribution, the square of an orthonormal Krawtchouk function, with a nonfactorizable generating function and factorial moments featuring a $\binom{r}{j}^2$ pattern that signals coherent over incoherent averaging. For general $k$, routing matrices replace routing numbers and the multivariate hypergeometric replaces the univariate one, and the genuinely complex phases lead to rich phase structure and new cumulant signatures; the third cumulant becomes sensitive to interference for $k\ge 3$. The framework clarifies the connection to the permanent and reveals low-order statistical witnesses—such as cross-mode covariances—that can aid boson-sampling verification without computing the permanent. Overall, the quantum multinomial offers a clear, combinatorial lens on multiphoton statistics with practical implications for theory and experiments in quantum optics and boson sampling.

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.

Figures (1)

• Figure 1: The quantum binomial family $P_n(c)$ for $m = 6$ photons at a balanced beam splitter ($T = R = 1/2$), for input partitions $n = 0, 1, 2, 3$. The boundary case $n = 0$ is the classical binomial $\mathrm{Bin}(6, 1/2)$. As $n$ increases, interference produces oscillations and exact zeros; the balanced input $n = 3$ suppresses all odd output values.