Institute for Physics & FDM, University of Freiburg, 79104 Freiburg, Germany and Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany.
Phys Rev Lett. 2016 Jan 8;116(1):010402. doi: 10.1103/PhysRevLett.116.010402. Epub 2016 Jan 7.
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
最近人们意识到，机器学习中的许多因果发现概念和工具与量子信息，尤其是量子非定域性问题密切相关。连接这两个领域的关键因素是因果关系的数学理论，它允许表示任意的因果结构，并为推理概率因果关系提供了严格的工具。实际上，贝尔定理涉及一种非常特殊的因果结构，贝尔不等式是这种模型所产生的线性约束的特殊情况。因此，寻找涉及更复杂贝尔场景的推广是很自然的。然而，这个问题依赖于这样一个事实，即这种广义场景的特点是多项式贝尔不等式，而目前除了非常简单的情况之外，没有方法可以推导出这些不等式。在这项工作中，我们朝着这个方向迈出了重要的一步，为广泛的场景中多项式贝尔不等式的推导提供了一种新的、通用的、概念上清晰的方法。我们还展示了如何利用我们的构造来放宽因果约束，并自然地引出广义贝尔网络中的非信号概念。
