通过哈密顿扩展实现（逻辑）科亨 - 施佩克尔上下文相关性的出现与恢复
Emergence and Recovery of (logical) Kochen-Specker Contextuality via Hamilton Extension
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中文总结 AI 辅助
研究通过哈密顿扩展实现逻辑 KS 上下文相关性的出现与恢复，确定六向量是经此机制产生 KS 矛盾的最小父集，揭示新结构途径，为构建紧凑 KS 集提供框架，对相关量子信息协议和图论方法有影响。
中文摘要 AI 辅助
逻辑科亨 - 施佩克尔（KS）上下文相关性通常被视为特殊构造测量配置的固有属性。我们表明它可通过一种称为哈密顿扩展的构造过程从 KS 可着色向量集出现。哈密顿扩展为四维向量集定义，将每个实向量与一个测量上下文相关联，并在不同父向量的哈密顿扩展子向量之间诱导额外测量上下文。这些出现的上下文从根本上改变了兼容性结构，将 KS 可着色配置转变为 KS 不可着色配置，并恢复了在顶点增强下丢失的逻辑上下文相关性。我们确定了一个精确且最优的阈值——每个五向量父集的哈密顿扩展仍为 KS 可着色，而适当选择的六向量父集已产生逻辑 KS 矛盾。因此，六个向量构成了通过此机制能够产生 KS 矛盾的最小父集。我们的结果揭示了上下文相关性的新结构途径，为构建紧凑 KS 集提供了系统框架，并对基于上下文相关性的量子信息协议和非经典性的图论方法有影响。
英文摘要
Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.
