Parity erasure: a foundational principle for indefinite causal order

Processes with indefinite causal order can arise when quantum theory is locally valid and they allow accomplishing new informational tasks. Despite recent progress, the correlations allowed in such processes have not been clearly understood. Here, we propose to study the constraints on information exchange through such processes in a paradigm of locally sequential operations. In this paradigm, we identify an information-theoretic principle constraining the correlations, termed parity erasure, which follows from the local validity of causality. We show that this principle completely characterizes the local-tomography representation of higher-order processes with indefinite causal order: among all multipartite channels, the ones describing valid higher-order processes are those and only those whose input-output correlations respect parity erasure. This approach reveals a fundamental property of information exchange in scenarios with indefinite causal structure and opens a new arena for exploring their potential applications.