Causality is pivotal to our understanding of the world, presenting itself in different forms: information-theoretic and relativistic, the former linked to the flow of information, the latter to the structure of space-time. Leveraging a framework introduced in PRA, 106, 032204 (2022), which formally connects these two notions in general physical theories, we study their interplay. Here, information-theoretic causality is defined through a causal modelling approach. First, we improve the characterization of information-theoretic signalling as defined through so-called affects relations. Specifically, we provide conditions for identifying redundancies in different parts of such a relation, introducing techniques for causal inference in unfaithful causal models (where the observable data does not "faithfully" reflect the causal dependences). In particular, this demonstrates the possibility of causal inference using the absence of signalling between certain nodes. Second, we define an order-theoretic property called conicality, showing that it is satisfied for light cones in Minkowski space-times with $d>1$ spatial dimensions but violated for $d=1$. Finally, we study the embedding of information-theoretic causal models in space-time without violating relativistic principles such as no superluminal signalling (NSS). In general, we observe that constraints imposed by NSS in a space-time and those imposed by purely information-theoretic causal inference behave differently. We then prove a correspondence between conical space-times and faithful causal models: in both cases, there emerges a parallel between these two types of constraints. This indicates a connection between informational and geometric notions of causality, and offers new insights for studying the relations between the principles of NSS and no causal loops in different space-time geometries and theories of information processing.

翻译：因果关系是理解世界的关键，它以不同形式呈现：信息论因果关系与相对论因果关系，前者关联信息流动，后者涉及时空结构。我们利用《物理评论A》106卷032204号（2022）引入的框架（该框架在一般物理理论中正式连接这两种概念），研究其相互作用。此处，信息论因果关系通过因果建模方法定义。首先，我们改进了通过所谓影响关系定义的信息论信号的刻画。具体而言，我们提供了识别此类关系不同部分冗余性的条件，引入了在非忠实因果模型（其中观测数据不能“忠实地”反映因果依赖关系）中进行因果推断的技术。这尤其展示了利用某些节点间信号缺失进行因果推断的可能性。其次，我们定义了一种称为锥性的序理论性质，证明其在空间维度$d>1$的闵可夫斯基时空的光锥中成立，但在$d=1$时被违反。最后，我们研究了在不违反相对论原理（如无超光速信号传输，NSS）的前提下，信息论因果模型在时空中的嵌入。总体而言，我们观察到由NSS在时空中施加的约束与纯信息论因果推断施加的约束表现不同。随后，我们证明了锥性时空与忠实因果模型之间的对应关系：在这两种情形下，这两类约束之间出现了平行关系。这揭示了因果的信息论与几何概念之间的关联，并为研究不同时空几何与信息处理理论中NSS原理与无因果循环原理之间的关系提供了新见解。