效应值测量模型与情境性

中文摘要 AI 辅助

我们通过用凸效应代数\(A\)中的值分布取代概率分布，推广了阿布拉姆斯基 - 勃兰登堡对情境性的层理论处理方法。这产生了一个\(A\)值测量模型的概念，将概率、确定性和量子测量结构包含在一个框架内。态\(\sigma:A\to[0,1]\)诱导普通经验模型，使可观测行为可视为源于效应值测量数据。这导致了\(A\)模型的内部情境性与态评估后的可观测情境性之间的区别。我们分析了这些概念之间的关系，并确定了可观测经典解释组合成内部解释的相干条件。利用效应模的有序向量空间表示，我们表明非情境性由相关锥规划的可行性来表征，推广了概率情形下情境性的线性规划表述。我们还引入了效应值情境分数并研究其与可观测情境见证的关系。我们展示了锥对偶性如何导致情境性的贝尔见证概念，作为贝尔不等式的直接推广。最后，我们分析了测量模型的尖锐可实现性和均匀扩张，阐明了一般正算子值测量与投影测量之间的关系，并展示了资源索引效应结构如何诱导推广量子单子的分级单子。

英文摘要

We generalize the Abramsky--Brandenburger sheaf-theoretic treatment of contextuality by replacing probability distributions with distributions valued in a convex effect algebra \(A\). This yields a notion of \(A\)-valued measurement model encompassing probabilistic, deterministic, and quantum measurement structures within a single framework. States \(σ:A\to[0,1]\) induce ordinary empirical models, allowing observable behaviour to be viewed as arising from effect-valued measurement data. This leads to a distinction between internal contextuality of \(A\)-models and observable contextuality after state evaluation. We analyze the relationship between these notions and identify coherence conditions under which observable classical explanations assemble into internal ones. Using the ordered-vector-space representation of effect modules, we show that non-contextuality is characterized by feasibility of an associated cone program, generalizing the linear-programming formulation of contextuality in the probabilistic case. We also introduce an effect-valued contextual fraction and study its relation to observable contextuality witnesses. We show how cone duality leads to a notion of Bell witnesses for contextuality as a direct generalization of Bell inequalities. Finally, we analyze sharp realizability and uniform dilation for measurement models, clarifying the relationship between general POVMs and projective measurements, and show how resource-indexed effect structures induce graded monads generalizing the quantum monad.

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37 pages