量子多项分布：多光子干涉的组合公式
The quantum multinomial distribution: a combinatorial formulation of multiphoton interference
AI总结：
本文提出了一种量子多项分布的泛化，用于描述多光子干涉的统计特性，通过相干求和和多变量超几何分布的结合，提供了验证玻色采样的一种低阶统计见证。
摘要：
本文提出了一种量子多项分布的泛化，用于描述 $ m $ 个相同光子在 $ k $ 端口线性光学干涉仪中的传输概率：两个多项系数（一个用于输入配置，一个用于输出）乘以相干求和过路由矩阵的平方模，权重由多变量超几何分布给出；无需希尔伯特空间形式化来陈述或评估该结果。经典多项分布在所有光子通过单个端口时恢复，相干求和退化为单个项无干涉；量子家族并非在Askey意义上的一般化，而是一个并行家族，通过幅度求和的相干性偏离经典统计。第 $ r $-阶阶乘矩携带一个平方多项系数代替经典单一系数，额外因子来自两个幅度展开的副本，其索引由福克态强制一致；对于光子束 splitter，第三累积量在玻色干涉下不变，而量子偏离首次出现在第四累积量中作为负超额峰度；然而对于多端口干涉仪，三体干涉破坏了这种不变性，偏离在第三累积量中就已出现。跨模式协方差涉及散射矩阵的相位通过相干项，这些项强化了输出抗相关性，超越经典值；结合单模矩中的平方系数特征，这些提供了无需完整永久计算即可验证玻色采样的低阶统计见证。
英文摘要：
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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