Axiomatic framework for higher-order quantum maps

Sequential models of classical or quantum computation, such as Turing machines and circuits, describe computation as state changes over time. In contrast, higher-order models like the Lambda calculus operate on functions themselves—an idea that extends naturally to Higher-Order Quantum Theory (HOQT), where quantum channels become inputs to “second-order” transformations, which extend recursively to form a hierarchy of higher-order maps (HOMs).

I will present an axiomatic framework for HOQT, analogous to Kraus’ axiomatization of quantum operations, characterizing admissible HOMs via their Choi-Jamiołkowski operators. Some HOMs correspond to circuits with open slots, while others (e.g., the quantum switch) feature indefinite causal order. I will show how to relate each map’s functional description to its causal structure through no-signaling relations. Finally I will present a how to generalize the higher-order framework to Operational Probabilistic Theories (OPTs).