Towards a relational interpretation of indefinite causality

Quantum indefinite causality has become a central topic in quantum foundations. One can employ a particular extension of quantum mechanics, called the process matrix formalism (PMF), to study these processes. The PMF allows one to represent correlations between quantum events among multiple parties without specifying, a priori, their spatiotemporal locations. Crucially, since the PMF makes no assumption about the global causal structure among quantum events, it allows for the existence of causally non-separable quantum processes. Such processes are said to have an indefinite causal structure. This work aims to investigate some conceptual lessons from causal non-separability. A preliminary discussion reviews the higher-order quantum operation (HOQO) formalism and shows that process matrices are a class of HOQOs. Then two notions of indefinite causality—namely, non-causality and causal non-separability—are introduced and compared. Finally, a particular implementation of a causally non-separable process, the quantum switch, is presented in both its acyclic and circuital representations. The study lays the groundwork for interpreting the circuital representation of the quantum switch, analyzing the roles of agents, quantum systems, and events within it. A relational interpretation is then proposed.