强O值语境性：排除量子理论的离散非确定性替代方案
Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
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中文总结 AI 辅助
研究排除量子理论离散非确定性替代方案，通过构建有限测量配置，采用状态依赖哈迪型测试和广义KS定理两种方式，扩展层理论框架引入强\(O\)值语境性，探讨对多值逻辑及相关协议的影响。
中文摘要 AI 辅助
格里森定理通过对无限连续投影算子格的非语境性确定了玻恩规则，其推论科亨 - 斯佩克（KS）定理使用有限投影算子集排除了特定类别的确定性（{0,1}值）非语境模型。格里森定理还表明存在KS型有限向量构造以排除量子理论中除{0,1}情况之外的其他离散非确定性替代方案。在此，我们构建有限测量配置以排除结果概率取自任意有限子集\(O \subset [0,1]\)的非语境经验模型。我们通过两种方式做到这一点：（i）构建一族实验上可行的状态依赖哈迪型测试，以及（ii）为包括先前结果作为特殊情况的广泛类别的\(O\)证明一个广义KS定理。在阿布拉姆斯基和布兰登伯格的层理论框架中，已经建立了概率 - 可能性 - 强语境性的层次结构，将语境性量化为一种资源。我们通过引入强\(O\)值语境性扩展了这个框架，表明量子理论避开了所有有限值预层的全局截面。我们还讨论了该结果对有限多值逻辑作为量子理论的可行本体模型以及对基于语境性的（半）设备独立协议的影响。
英文摘要
Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ($\{0,1\}$-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the $\{0, 1\}$ case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset $O \subset [0,1]$. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of $O$ that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong $O$-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.
