We study the relation of causal influence between input systems of a reversible evolution and its output systems, in the context of operational probabilistic theories. We analyse two different definitions that are borrowed from the literature on quantum theory—where they are equivalent. One is the notion based on signalling, and the other one is the notion used to define the neighbourhood of a cell in a quantum cellular automaton. The latter definition, that we adopt in the general scenario, turns out to be strictly weaker than the former: it is possible for a system to have causal influence on another one without signalling to it. Remarkably, the counterexample comes from classical theory, where the proposed notion of causal influence determines a redefinition of the neighbourhood of a cell in cellular automata. We stress that, according to our definition, it is impossible anyway to have causal influence in the absence of an interaction, e.g. in a Bell-like scenario. We study various conditions for causal influence, and introduce the feature that we call $\textit{no interaction without disturbance}$, under which we prove that signalling and causal influence coincide. The proposed definition has interesting consequences on the analysis of causal networks, and leads to a revision of the notion of neighbourhood for classical cellular automata, clarifying a puzzle regarding their quantisation that apparently makes the neighbourhood larger than the original one.

中文翻译：

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操作概率理论中的因果影响

我们在操作概率理论的背景下研究可逆进化的输入系统与其输出系统之间的因果关系。我们分析了从量子理论文献中借用的两个不同定义——它们是等价的。一种是基于信令的概念，另一种是用于定义量子元胞自动机中元胞邻域的概念。我们在一般情况下采用的后一种定义实际上比前一种更弱：一个系统可能会对另一个系统产生因果影响，而无需向它发出信号。值得注意的是，反例来自经典理论，其中提出的因果影响概念决定了元胞自动机中细胞邻域的重新定义。我们强调，根据我们的定义，在没有交互作用的情况下，无论如何都不可能产生因果影响，例如在类似贝尔的场景中。我们研究了因果影响的各种条件，并引入了我们称之为 $\textit{无干扰无干扰}$ 的特征，在此基础上我们证明了信号和因果影响是一致的。提议的定义对因果网络的分析产生了有趣的影响，并导致了经典元胞自动机邻域概念的修订，澄清了一个关于它们的量化的难题，这显然使邻域大于原始邻域。并引入我们称之为 $\textit{无干扰无交互}$ 的特征，在该特征下我们证明信号和因果影响是一致的。提议的定义对因果网络的分析产生了有趣的影响，并导致了经典元胞自动机邻域概念的修订，澄清了一个关于它们的量化的难题，这显然使邻域大于原始邻域。并引入我们称之为 $\textit{无干扰无交互}$ 的特征，在该特征下我们证明信号和因果影响是一致的。提议的定义对因果网络的分析产生了有趣的影响，并导致了经典元胞自动机邻域概念的修订，澄清了一个关于它们的量化的难题，这显然使邻域大于原始邻域。