Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland.
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Institute of Informatics, University of Gdańsk, 80-952 Gdańsk, Poland.
Phys Rev Lett. 2014 Mar 28;112(12):120401. doi: 10.1103/PhysRevLett.112.120401. Epub 2014 Mar 26.
Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Despite some recent proposals, it still faces a fundamental problem: how to quantify its presence? In this work, we provide a universal framework for quantifying contextuality. We conduct two complementary approaches: (i) the bottom-up approach, where we introduce a communication game, which grasps the phenomenon of contextuality in a quantitative manner; (ii) the top-down approach, where we just postulate two measures, relative entropy of contextuality and contextuality cost, analogous to existent measures of nonlocality (a special case of contextuality). We then match the two approaches by showing that the measure emerging from the communication scenario turns out to be equal to the relative entropy of contextuality. Our framework allows for the quantitative, resource-type comparison of completely different games. We give analytical formulas for the proposed measures for some contextual systems, showing in particular that the Peres-Mermin game is by order of magnitude more contextual than that of Klyachko et al. Furthermore, we explore properties of these measures such as monotonicity or additivity.
语境相关性是量子理论基础和新型信息处理任务的核心。尽管最近有一些提议，但它仍然面临着一个基本问题：如何量化其存在？在这项工作中，我们提供了一个通用的框架来量化语境相关性。我们进行了两种互补的方法：（i）自下而上的方法，我们引入了一个通信博弈，以定量的方式抓住语境相关性的现象；（ii）自上而下的方法，我们只是假设了两个度量，相对语境熵和语境代价，类似于非局域性的现有度量（语境相关性的一个特例）。然后，我们通过表明从通信场景中出现的度量与语境相关性的相对熵相等来匹配这两种方法。我们的框架允许对完全不同的博弈进行定量的、资源类型的比较。我们给出了一些语境系统的拟议度量的解析公式，特别表明 Peres-Mermin 游戏比 Klyachko 等人的游戏在数量级上更具语境相关性。此外，我们还探索了这些度量的性质，如单调性或可加性。
