The logic of contextuality
The logic of contextuality
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DOI：
10.4230/lipics.csl.2021.5
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发表时间：
2020-11
期刊：
影响因子：
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通讯作者：
S. Abramsky;Rui Soares Barbosa
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文献类型：
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作者：
S. Abramsky;Rui Soares Barbosa
背景性是量子非经典性的一个关键特征，它已被证明在使量子优势用于广泛的信息处理和计算任务方面发挥着核心作用。我们研究逻辑的上下文从结构的角度来看，在设置部分布尔代数介绍了Kochen和斯派克在他们的开创性工作。这些与传统的量子逻辑形成鲜明对比\`a la Birkhoff和冯诺依曼的操作，如合取和析取是部分的，只被定义在域中，它们是物理意义。我们研究这种设置如何与当前的工作上下文，如层理论和图论的方法。本文给出了一种推广部分布尔代数上可测性关系的一般自由构造，即二元逻辑运算的定义域。这种结构有着惊人的广泛用途。我们将其应用于一些问题的研究，包括：-建立上下文文献中研究的抽象测量场景与部分布尔代数的设置之间的联系;- 在这种情况下制定各种上下文属性，包括概率上下文以及由Kochen-Specker悖论给出的上下文的强的、状态独立的概念，- 研究逻辑排他性原则，以及它与概率排他性原则的关系，概率排他性原则在最近关于上下文的工作中被广泛研究，作为接近量子可实现相关性的一步;- 开展一些工作，对希尔伯特空间张量积的逻辑表示，使用逻辑排他性来捕捉它的一些显着的量子特征。
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic \`a la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.