Embedding cyclic information-theoretic structures in acyclic spacetimes: no-go results for indefinite causality

Abstract: The notions of causality adopted within the quantum information and spacetime physics communities are distinct. Although both notions play a role in physical experiments, their general interplay is little understood in theory. We develop a theoretical framework that connects the two causality notions, while also clearly distinguishing them. The framework describes a composition of quantum operations through feedback loops, and the embedding of the resulting, possibly cyclic information-theoretic structure in an acyclic spacetime structure. Relativistic causality (which forbids superluminal communication) follows as a graph-theoretic compatibility condition between the two structures. Formulating indefinite causal order (ICO) processes in our framework, we shed light on the links between indefinite and cyclic causality, and on questions regarding their physicality. In particular, there are several experiments that claim to implement ICO processes in Minkowski spacetime, presenting an apparent paradox: how can an indefinite information-theoretic causal structure be consistent with a definite spacetime causal structure? We address this through no-go theorems, showing that as a consequence of relativistic causality, (a) realisations of ICO processes necessarily involve the non-localisation of systems in spacetime and (b) will nevertheless admit an explanation in terms of a definite and acyclic causal order process, at a fine-grained level. This fully resolves the apparent paradox and bears implications for the physical interpretation of ICO experiments, and is achieved by introducing the concept of fine-graining that allows causal structures to be analysed at different levels of detail. Our work also sheds light on the limits of quantum information processing in spacetime and on the operational meaning of indefinite causality, within and beyond the context of a fixed spacetime.