Convexity and uncertainty in operational quantum foundations
Convexity and uncertainty in operational quantum foundations
复制标题
操作量子基础中的凸性和不确定性
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Ryo Takakura
中科院分区：
文献类型：
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作者：
Ryo Takakura
发现量子理论的本质，不仅是理论研究的重要问题，也是量子技术应用的重要问题。在这些关于量子基础的研究中，不确定性的概念在量子理论的几个令人惊叹的特征中扮演着主要角色。这篇论文的目的是研究不确定性的基本方面。具体地说，我们着重于凸性来解决这个问题，凸性是有操作根源的。我们首先试图揭示为什么在量子理论中，两种不确定关系，即准备不确定关系和测量不确定关系，往往得到类似的界限。为了做到这一点，我们在被称为广义概率理论(GPTS)的最一般的物理学框架中考虑不确定性关系。证明了在GPTS中，某些态的几何结构将这两种不确定关系联系在一起，它们之间的关系可以用几个表达式来表示，如熵1。我们的结果暗示了这些不确定关系之间的密切关系所必需的。然后我们考虑量子理论中不确定性的一种更广泛的表达方式，称为量子不相容。受操作直觉的启发，我们提出并研究了与态的凸性直接相关的不相容的新的量化。结果还表明，即使在最简单的互不相容的情况下，也可以观察到两个相互无偏的量子比特可观测量的显著现象。最后，我们在量子理论中研究了混合的热力学熵，它也可以被看作是不确定性的量化。我们考虑它在操作上对GPTS的自然扩展，然后试图刻画量子理论中的熵的具体程度。结果表明，在一类被称为正多边形理论的GPT中，只有经典的和类量子的理论中才允许操作上的自然熵存在。
To find the essential nature of quantum theory has been an important problem for not only theoretical interest but also applications to quantum technologies. In those studies on quantum foundations, the notion of uncertainty plays a primary role among several stunning features of quantum theory. The purpose of this thesis is to investigate fundamental aspects of uncertainty. In particular, we address this problem focusing on convexity, which has an operational origin. We first try to reveal why in quantum theory similar bounds are often obtained for two types of uncertainty relations, namely, preparation and measurement uncertainty relations. To do this, we consider uncertainty relations in the most general framework of physics called generalized probabilistic theories (GPTs). It is proven that some geometric structures of states connect those two types of uncertainty relations in GPTs in terms of several expressions such as entropic one. Our result implies what is essential for the close relation between those uncertainty relations. Then we consider a broader expression of uncertainty in quantum theory called quantum incompatibility. Motivated by an operational intuition, we propose and investigate new quantifications of incompatibility which are related directly to the convexity of states. It is also shown that there can be observed a notable phenomenon for those quantities even in the simplest incompatibility for a pair of mutually unbiased qubit observables. Finally, we study thermodynamical entropy of mixing in quantum theory, which also can be seen as a quantification of uncertainty. We consider its operationally natural extension to GPTs, and then try to characterize how specific the entropy in quantum theory is. It is shown that the operationally natural entropy is allowed to exist only in classical and quantum-like theories among a class of GPTs called regular polygon theories.