[论文解读] Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory

Carmen Constantin|arXiv (Cornell University)|Oct 9, 2015

Quantum Mechanics and Applications参考文献 109被引用 4

一句话总结

该论文运用层论与拓扑斯理论框架，分析量子基础现象，如互补性、非定域性和纠缠。论文提出了一套三层次的互补性分类体系，并证明除单量子比特态与贝尔态的张量积外，所有n量子比特纠缠态均为逻辑非定域性，且给出了识别见证可观测量的构造性算法。

ABSTRACT

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.

研究动机与目标

• 利用物理上有意义的上下文上的层论，理解量子与经典之间的基本差异——纠缠、互补性、非定域性。
• 通过层论互补性，对一类新型多体纠缠态（称为‘具有函数依赖关系的平衡态’）进行分类。
• 将层论洞察拓展至拓扑斯方法，为量子态、可观测量和命题提供统一的数学语言。
• 证明拓扑斯方法至少与传统量子逻辑一样具有表达力。
• 在拓扑斯框架内定义并发展上下文熵，统一经典与量子信息论中的熵概念。

提出的方法

• 使用偏序集上的层来建模量子系统，借助Abramsky-Brandenburger框架形式化互补性与非定域性。
• 通过层上同调中的全局截面障碍，定义三种互补性层次——弱互补性、逻辑非定域性与强互补性。
• 基于多项式构造对称性破缺态的分类，通过函数依赖度分析其互补性。
• 开发一种构造性算法，生成n+2个局部可观测量，以见证任意n量子比特纠缠态的逻辑非定域性。
• 应用拓扑斯方法，将量子态表示为谱预层上的测度，利用daseinisation与C*-代数的Gelfand对偶性。
• 将上下文熵定义为层论态上的测度，通过上下文上的部分迹与测度，推广香农熵与冯·诺依曼熵。

实验结果

研究问题

• RQ1 哪些多体纠缠态表现出逻辑非定域性？如何系统性地对其进行分类？
• RQ2 能否开发一种构造性算法，以识别任意n量子比特纠缠态中见证逻辑非定域性的局部可观测量？
• RQ3 拓扑斯理论语言在多大程度上至少与传统量子逻辑一样具有表达力？
• RQ4 如何定义上下文熵，并用于统一经典与量子信息论中的熵概念？
• RQ5 拓扑斯方法是否允许从层的全局截面重构纯态与任意量子态？

主要发现

• 除单量子比特态与两体最大纠缠态的张量积外，所有n量子比特纠缠态均为逻辑非定域性。
• 提供了构造性算法，可生成n+2个局部可观测量，以见证任意n量子比特纠缠态的逻辑非定域性。
• 具有函数依赖关系的平衡态类包含至少为逻辑非定域性的态，其中部分态表现出强互补性。
• 拓扑斯方法被证明至少与传统量子逻辑一样具有表达力，且从拓扑斯框架中形式化重构了命题逻辑。
• 上下文熵被定义为香农熵与冯·诺依曼熵的层论推广，其性质在部分迹下保持不变。
• 在纠缠上下文中构造了熵子可加性不成立的反例，表明在拓扑斯框架中，上下文熵的子可加性一般不成立。