关于狄拉克型相关性
On Dirac-type correlations
AI总结：
本文提出局部密度算符理论，扩展了格莱森定理至非空间状分离系统，建立了狄拉克测度与局部密度算符的对应关系。
摘要：
量子相关性往往无法用经典物理的基本概念如因果性、局域性和现实性来解释。虽然自1930年代以来，关于空间状分离系统之间量子相关性的数学理论已经建立得很完善，但关于非空间状分离系统之间相关性的数学理论却发展得较少。在本工作中，我们发展了所谓的“局部密度算符”理论，我们将其视为可能非空间状分离量子系统的联合状态。局部密度算符是单位迹算符，其边缘是真实的密度算符，我们证明不仅包含了密度算符的概念，还包含了将密度算符扩展到时空域的几种扩展，如伪密度算符和时间上的量子状态。更重要的是，我们证明了一个结果，建立了局部密度算符与我们称为“狄拉克测度”的一一对应关系，后者是与两个量子系统相关的可分离投影子空间上的复值测度。当其中一个系统是具有单维希尔伯特空间的平凡量子系统时，我们的结果恢复了已知的基本结果，即格莱森定理，这表明量子理论中的本生规则是唯一能够以非上下文化的方式为量子系统测量结果分配概率的方式。因此，我们的结果建立了格莱森定理到可能非空间状分离系统上测量的直接扩展，从而将量子相关性的空间数学理论扩展到空间和时间上的量子相关性。
英文摘要：
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.
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