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arXiv:2602.09894

The quantum multinomial distribution: a combinatorial formulation of multiphoton interference

Alfonso Martinez, Josep Font-Segura

quant-phmath.PRSubmitted Feb 10, 2026• Announced Feb 11, 2026• 0 citations
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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.

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This paper presents a quantum generalization of the multinomial distribution for the transition probabilities of mmm identical photons in a kkk-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 rrr-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.
We generalize an approach for description of multiphoton experiments with multiport unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral states, to arbitrary (multiphoton) input and arbitrary photon detectors. We show that output probabilities are always given in terms of the matrix permanents of the Hadamard product of a matrix built from the network matrix and matrices built from the spectral state of photons and spectral sensitivities of detectors. Moreover, in the 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 multiphoton input, zero output probability of an output configuration is always the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, is independent of 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.''
Multiphoton interference results in a wide variety of non-classical photon number statistics, including characteristic signatures of entanglement between two or more sets of optical modes. Here, we consider the photon number statistics observed after applying discrete Fourier transformations (DFTs) to bipartite entangled states generated using single photon sources and beam splitters. It is shown that the output photon number states of DFTs are eigenstates of a translational mode shifting operator in the input. The complex eigenvalues of the mode shift can be identified by a phase number K obtained from the output photon distribution. For each output distribution, the possible input states are limited to mode shift eigenstates with the same K-value. Using this mode shift rule, we can identify the quantum coherence between different photon number distributions in the input with experimentally observable K-values in the output of the DFT. In the case of multi-photon entanglement obtained by post-selection and beam splitting single photons, this coherence is non-local, resulting in correlated pairs of K-values that always sum up to zero. We can therefore observe both the correlations between the input photon number distributions and a complementary correlation between the output photon numbers of two DFTs to obtain a reliable characterization of the entanglement between the two multi-mode multi-photon systems. Importantly, the K-value allows a classification of large sets of possible photon number distributions, resulting in a significant simplification of the experimental evaluation of the multi-photon output statistics and opening up the road towards more efficient applications of non-classical multi-photon states
Multiphoton interference is at the very heart of quantum foundations and applications in quantum sensing and information processing. In particular, boson-sampling experiments have the potential to demonstrate quantum computational supremacy while relying on only multiphoton interference in linear optical interferometers. However, even when photonic losses are negligible, scalable experiments are challenged by the rapid decrease of the probability of the success of current schemes with probabilistic sources for a large number of single photons in each experimental sample. Remarkably, we show a boson sampling scheme in which the probability of success increases instead of decreasing with the number of input photons, eventually approaching a unit value even with nondeterministic sources. It is achieved by sampling measurements not only at the interferometer output but also over all the possible numbers of photons at each input port and, different from scattershot boson sampling, allowing a random number of occupied input ports from one experimental run to another. Therefore, these results provide an exciting route toward future demonstrations of quantum computational supremacy with scalable experimental resources.
We propose an algorithm to compute the full output distribution of photons passing through linear optical interferometers. More precisely, given n photons at the input of an m-mode interferometer, the proposed algorithm computes the probabilities of all the (n+m−1m−1)\binom{n + m - 1}{m - 1}(m−1n+m−1​) possible output states in a linear time complexity of O(n(n+m−1m−1))O\left( {n\binom{n + m - 1}{m - 1}} \right)O(n(m−1n+m−1​)), outperforming the naive method – computing one permanent for each state – by an exponential factor. The benchmarking has been carried out on a quantum machine learning use-case, to both illustrate the gain and give one possible application. Finally, as storing all the probabilities requires a memory space that is exponential in n and m, we discuss and give strong evidence that memory is far more limiting than the time for the computation of the full distribution.
Multiphoton interference is crucial to many photonic quantum technologies. In particular, interference forms the basis of optical quantum information processing platforms and can lead to significant computational advantages. It is therefore interesting to study the interference arising from various states of light in large interferometric networks. Here, we implement a quantum walk in a highly stable, low-loss, multiport interferometer with up to 24 ultrafast time bins. This time-bin interferometer comprises a sequence of birefringent crystals which produce pulses separated by 4.3\,ps, all along a single optical axis. Ultrafast Kerr gating in an optical fiber is employed to time-demultiplex the output from the quantum walk. We measure one-, two-, and three-photon interference arising from various input state combinations, including a heralded single-photon state, a thermal state, and an attenuated coherent state at one or more input ports. Our results demonstrate that ultrafast time bins are a promising platform to observe large-scale multiphoton interference.
Quantum interference of correlated particles is a fundamental quantum phenomenon which carries signatures of the statistics properties of the particles, such as bunching or antibunching. In the presence of particular symmetries, interference effects take place with high visibility, one of the simplest cases being the suppression of coincident detection in the Hong-Ou-Mandel effect. Tichy et al., [Phys. Rev. Lett. 104, 220405 (2010)] recently demonstrated a simple sufficient criterion for the suppression of output events in the more general case of Fourier multiport beam splitters. Here we study the case in which 2q{2}^{q}2q particles (either bosonic or fermionic) are injected simultaneously in different ports of a Sylvester interferometer with ${2}^{p}\ensuremath{\ge}{2}^{q}$ modes. In particular, we prove a necessary and sufficient criterion for a significant fraction of output states to be suppressed, for specific input configurations. This may find application in assessing the indistinguishability of multiple single-photon sources and in the validation of boson sampling machines.