We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. We define causaltopes, our chosen portmanteau of "causal polytopes", for arbitrary spaces of input histories and arbitrary choices of input contexts. We show that causaltopes are obtained by slicing simpler polytopes of conditional probability distributions with a set of causality equations, which we fully characterise. We provide efficient linear programs to compute the maximal component of an empirical model supported by any given sub-causaltope, as well as the associated causal fraction. We introduce a notion of causal separability relative to arbitrary causal constraints. We provide efficient linear programs to compute the maximal causally separable component of an empirical model, and hence its causally separable fraction, as the component jointly supported by certain sub-causaltopes. We study causal fractions and causal separability for several novel examples, including a selection of quantum switches with entangled or contextual control. In the process, we demonstrate the existence of "causal contextuality", a phenomenon where causal inseparability is clearly correlated to, or even directly implied by, non-locality and contextuality.

翻译：我们提供了一个统一的操作框架，用于在完全设备无关和理论无关的背景下研究因果关系、非局域性和上下文相关性。我们定义了“因果拓扑”（causaltopes），即我们选用的“因果多面体”的合成词，适用于任意输入历史空间和任意输入上下文选择。我们证明，因果拓扑是通过用一组我们完全刻画了的因果方程对更简单的条件概率分布多面体进行切割而得到的。我们提供了高效的线性规划方法，用于计算由任意给定子因果拓扑支持的实证模型的最大分量，以及相应的因果分数。我们引入了相对于任意因果约束的因果可分性概念。我们提供了高效的线性规划方法，用于计算实证模型的最大因果可分分量，从而得到其因果可分分数，作为由某些子因果拓扑共同支持的分量。我们通过若干新颖的示例研究了因果分数和因果可分性，包括一系列带有纠缠或上下文控制的量子开关。在此过程中，我们证明了“因果上下文性”的存在，这是一种因果不可分性与非局域性和上下文性明显相关甚至直接蕴含的现象。