标题： 层论方法在量子力学和量子信息论中的应用 显示英文标题

标题： Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory

摘要： 本论文中，我们利用层论的语言，以期深入理解量子物理和经典物理之间的一些根本差异——例如纠缠、上下文性和非定域性。 首先，基于 Abramsky 和 Brandenburger 的工作，我们展示如何通过定义在某些具有物理意义的上下文偏序集上的层来自然地刻画和分析重要的量子力学现象，例如量子非定域性和上下文性。 我们也描述了这一框架如何自然地导出量子上下文性的三个层级：弱上下文性、逻辑非定域性和强上下文性。 我们利用这些洞见来分类某些多部分纠缠态。几乎所有的这些态都至少是逻辑非定域的，其中一些甚至表现出强上下文性。 我们进一步扩展这一结果，证明除了单量子比特和双部分最大纠缠态的张量积外的所有 n 量子比特纠缠态都是逻辑非定域的。我们的证明是构造性的：对于任意给定的 n 量子比特态，我们提出一个算法生成 n+2 个局部可观测量来见证其逻辑非定域性。 在论文的后半部分，我们使用同样基于物理意义上下文的层的基本原理，提出一种优雅的数学语言，即所谓的拓扑方法，在这种语言中可以表达许多量子力学概念，例如态、可观测量以及关于这些的命题。 然后我们展示拓扑方法的语言在逻辑上至少与传统的量子逻辑一样富有表现力。 最后，从拓扑学的角度出发，我们发展了上下文熵的构造，以统一经典和量子信息论熵的概念。 显示英文摘要

摘要： In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.