Matthias Salzger

The process matrix framework models quantum protocols without assuming a definite acyclic causal order between the operations of parties in the protocol. This leads to so-called indefinite causal order processes which have been shown to provide advantages for quantum information processing. However, there have been longstanding open questions regarding the subset of practically realisable process matrices, as well as challenges in formulating their composition. We make progress on addressing such questions by connecting an important subset of process matrices, namely quantum circuits with quantum control of causal order (QC-QC) with so-called causal boxes which describe practical quantum information protocols in spacetime. Causal boxes are fully closed under composition and incorporate physical principles such as relativistic causality in spacetime. We first identify a notion of operational equivalence between QC-QCs and causal boxes by connecting state spaces and operations in the two formalisms. We then explicitly construct for each QC-QC an operationally equivalent causal box that satisfies certain special mathematical properties that allows the causal box to be interpreted as a process, this corresponds to a subset of causal boxes previously known as process boxes. This allows us to define composition of QC-QCs in terms of composition of causal boxes which is well-defined. We conjecture with a proof sketch that the spatiotemporal labels involved in their description can be simplified to a certain totally ordered form. Based on this, we establish through a constructive proof that every process box can be mapped to an operationally equivalent QCQC. This indicates that the subset of indefinite causal order processes realisable in a background spacetime correspond to controlled superpositions of acyclic orders, which in particular rules out processes violating so-called causal inequalities. Our results shed light on the practicality and composability questions for indefinite causal structures while introducing new physically-motivated tools for studying their applications for quantum information processing in spacetime.