Title: Contextuality beyond the Kochen-Specker theorem Show Chinese title

Title: 超越Kochen-Specker定理的上下文性

Abstract: When it isn't possible to tell two distinct experimental procedures apart purely from their input/output statistics, then it seems a plausible hypothesis that the two procedures must be physically identical. We call such a hypothesis "noncontextuality", an instance of Leibniz's principle of the identity of indiscernibles. Read in the contrapositive, this hypothesis entails that any physical distinctions between two experimental procedures must necessarily lead to a difference in their operational statistics. The results I present in this thesis concern the failure of this hypothesis -- a failure dubbed "contextuality" -- when one tries to embed an operational theory (such as quantum theory) in the ontological models framework. The Kochen-Specker theorem demonstrates the failure of noncontextuality for deterministic ontological models of quantum theory, i.e., those ontological models where the ontic/physical state of the system fixes the outcome of any projective measurement on the system in a deterministic manner. This thesis goes beyond the Kochen-Specker (KS) theorem by asking what operational facts must be verified in experiments to conclude that Nature does not admit noncontextual ontological models, not even indeterministic ones. This leads to noncontextuality inequalities that are robust to noise in the preparations and measurements. In the particular case of quantum theory, these inequalities are meaningful even when unsharp measurements (or POVMs) are allowed, a feature not shared by the traditional approach to KS-noncontextuality where unsharp measurements are excluded by fiat: allowing them renders even trivial POVMs (proportional to identity) maximally KS-contextual. The sense in which trivial POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they are simply too noisy to lead to violation of our noncontextuality inequalities. Show Chinese abstract

Abstract: 当仅从输入/输出统计无法区分两个不同的实验程序时，似乎合理的假设是这两个程序必须在物理上是相同的。 我们将这种假设称为“非上下文性”，这是莱布尼茨不可分辨物同一性原则的一个实例。 反向解读这个假设，则意味着两个实验程序之间的任何物理差异必然会导致其操作统计上的差异。 本论文中的结果涉及这一假设的失败——即所谓的“上下文性”——当我们尝试将操作理论（如量子理论）嵌入本体论模型框架时。 Kochen-Specker 定理展示了量子理论确定性本体论模型中非上下文性的失败，即那些本体论模型中系统的本征/物理状态以确定的方式固定了系统上任何投影测量的结果。 本文超出了 Kochen-Specker (KS) 定理，探讨了为了得出自然界不承认非上下文本体论模型（甚至非确定性模型）需要在实验中验证的操作事实。 这导致了对制备和测量中的噪声具有鲁棒性的非上下文性不等式。 在量子理论的特定情况下，即使允许使用模糊测量（或 POVM），这些不等式仍然有意义，而传统处理 KS 非上下文性方法则排除了模糊测量：允许它们甚至会使平凡 POVM（与单位矩阵成比例）变得完全 KS 上下文。 在我们的方法中，平凡 POVM 确实是“平凡的”（或“非上下文的”）这一点很明确：它们只是太嘈杂，不会违反我们的非上下文性不等式。