Embedding cyclic information-theoretic structures in acyclic space-times: No-go results for indefinite causality
Résumé
The notions of causality adopted within the quantum information and space-time physics communities are distinct. Although experience tells us that these notions play together in a compatible manner in physical experiments, their general interplay is little understood in theory. Therefore 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 space-time structure. Relativistic causality (which forbids superluminal communication) then follows as a graph-theoretic compatibility condition between the two structures. Demonstrating that indefinite causal order (ICO) processes, widely studied in the quantum information community, can be formulated within 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 space-time, presenting an apparent theoretical paradox: how can an indefinite information-theoretic causal structure be consistent with a definite space-time structure? We address this through no-go theorems, showing that as a consequence of relativistic causality, (a) realizations of ICO processes necessarily involve the nonlocalization of systems in space-time and (b) will nevertheless admit an explanation in terms of a definite and acyclic causal order process, at a more fine-grained level. These results are made possible by introducing the concept of fine-graining that allows causal structures to be analyzed at different levels of detail. This fully resolves the apparent paradox and bears implications for the physical interpretation of ICO experiments. Our work also sheds light on the limits of quantum information processing in space-time and offers concrete insights on the operational meaning of indefinite causality, both within and beyond the context of a fixed space-time.
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