带有复态射权重的范畴上玻恩规则的局部唯一性
Local Uniqueness of the Born Rule on Categories with Complex-Weighted Morphisms
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中文总结 AI 辅助
本文在带复态射权重的范畴框架下，基于五个条件证明玻恩规则P(z)=|z|²的局部唯一性，将其与量子概率重构工作关联，指出整体相干性扩展是开放问题。
中文摘要 AI 辅助
本文在配备复态射权重与路径振幅概率泛函的小范畴框架下，证明了玻恩规则的局部唯一性定理。给定五个条件：（i）非负性、（ii）有界总次数的多项式性、（iii）整体U(1)不变性、（iv）互斥路径的经典极限可加性、（v）归一化，本文证明概率赋值P: C→R≥0被唯一确定为P(z)=|z|²。互斥路径的概念被精确表述为不存在通过任意公共态射的公共因子分解。本文将该结果与Gleason、Hardy及Chiribella-D'Ariano-Perinotti的量子概率重构工作相联系，并指出将其扩展至态射复合下的整体相干性是与马尔可夫范畴中合成概率理论相关的开放问题。玻恩规则作为复振幅上唯一局部一致的概率定律出现，仅由相位不变性与经典极限行为确定，独立于任何希尔伯特空间框架。
英文摘要
I prove a local uniqueness theorem for the Born rule in the setting of quiver-generated categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) invariance of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P: C -> R>=0 is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, Chiribella-D'Ariano-Perinotti, and several further, more recent reconstructions, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed by phase invariance and classical-limit behavior together with polynomiality and normalization, independent of any Hilbert-space framework.
