[论文解读] Categorical Operational Physics

Sean Tull|arXiv (Cornell University)|Feb 1, 2019

Quantum Mechanics and Applications参考文献 15被引用 9

一句话总结

本文提出了一种基于操作性原理的有限维量子理论的完全范畴化重构，扩展了范畴化量子力学与效应论，引入了基于操作范畴与超因果过程的框架，通过图示范畴理论推导出量子特性（如纯化、叠加与相位），且无需概率性假设。

ABSTRACT

Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of such theories purely in terms of the behaviour of their processes, as described mathematically through the language of category theory. This extends a framework for quantum processes known as categorical quantum mechanics (CQM) due to Abramsky and Coecke. We first consider categorical frameworks for operational theories. We introduce a notion of such theory, based on those of Chiribella, D'Ariano and Perinotti (CDP), but more general than the probabilistic ones typically considered. We establish a correspondence between these and what we call "operational categories", using features introduced by Jacobs et al. in effectus theory, an area of categorical logic to which we provide an operational interpretation. We then see how to pass to a broader category of "super-causal" processes, allowing for the powerful diagrammatic features of CQM. Next we study operational theories themselves. We survey numerous principles that a theory may satisfy, treating them in a basic diagrammatic setting, and relating notions from probabilistic theories, CQM and effectus theory. We provide a new description of superpositions in the category of pure quantum processes, using this to give an abstract construction of the category of Hilbert spaces and linear maps. Finally, we reconstruct finite-dimensional quantum theory itself. More broadly, we give a recipe for recovering a class of generalised quantum theories, before instantiating it with operational principles inspired by an earlier reconstruction due to CDP. This reconstruction is fully categorical, not requiring the usual technical assumptions of probabilistic theories. Specialising to such theories recovers both standard quantum theory and that over real Hilbert spaces.

研究动机与目标

• 开发一个基于范畴论的操作性理论通用框架。
• 将范畴化量子力学、效应论与操作性概率理论统一为单一范畴基础。
• 仅使用范畴结构从纯粹的操作性原理重构有限维量子理论。
• 为量子过程中的叠加与相位提供一种新的、抽象的刻画。
• 通过范畴对偶与纯度，消除量子重构中的概率性假设。

提出的方法

• 使用操作范畴建模一般物理理论中的过程，基于效应论与操作性概率理论的公理。
• 通过总化与余积引入超因果过程，实现类似范畴化量子力学的图示推理。
• 应用范畴构造如双积、核与最小稀释，以建模纯度与可区分性等量子特性。
• 利用带相位的余积与自伴结构，抽象刻画量子叠加与相位门。
• 从操作性原理推导出紧致与自伴结构，将其与量子可逆性及对偶性联系起来。
• 采用基于Chiribella-D'Ariano-Perinotti启发的公理化重构方法，应用于纯过程的范畴。

实验结果

研究问题

• RQ1 如何仅通过过程复合的范畴论语言，纯粹形式化操作性物理理论？
• RQ2 哪些范畴结构支撑量子特性（如纯化、叠加与相位门）？
• RQ3 能否仅使用范畴与操作性原理，无需概率性假设，重构有限维量子理论？
• RQ4 在一般操作范畴中，纯度、理想压缩与最小稀释的概念之间有何关联？
• RQ5 自伴与紧致结构在从操作公理恢复量子理论特性中起什么作用？

主要发现

• 本文建立了操作理论与操作范畴之间的对应关系，表明后者为操作物理提供了自然的范畴框架。
• 提出了一种在纯量子过程范畴中使用带相位余积的新范畴构造，以刻画叠加。
• 该框架实现了从操作性原理对有限维量子理论的完整重构，无需假设概率或凸结构。
• 重构过程从操作公理推导出紧致与自伴范畴，表明这些结构可从物理公设中自然涌现。
• 证明了希尔伯特空间与线性映射的范畴可作为纯过程关联的“良好行为”范畴被抽象重构。
• 该工作将标准量子理论与实量子理论均作为广义量子理论框架的特例恢复。