Causality in higher order process theories

Quantum supermaps provide a framework in which higher order quantum processes can act on lower order quantum processes. In doing so, they enable the definition and analysis of new quantum protocols and causal structures. Recently, the key features of quantum supermaps were captured through a general categorical framework, which when iterated yields the framework of higher order process theories (HOPT). The HOPT framework combines lower and higher order transformations in a single unified theory, with its mathematical structure shown to coincide with the notion of a closed monoidal category. Here we provide an equivalent construction of the HOPT framework from four simple axioms of process-theoretic nature. We then use the HOPT framework to establish connections between foundational features such as causality, determinism, and signalling, and to explore their interaction with the mathematical structure of *-autonomy.