Order structure and signalling in higher order quantum maps

We study the signalling structure of higher order quantum maps from an order-theoretic perspective, building on the combinatorial characterization of higher order types by Bisio and Perinotti. We have shown in a previous work arxiv:2411.09256 that types are represented by boolean functions called type functions, and that each such function is characterized by a related structure poset. We characterize the distributive lattice generated by all type functions with fixed indices of input and output systems - whose elements we call regular subtypes - by a monotonicity condition. Unlike the set of type functions, the lattice of regular subtypes is closed under the one-way signalling product, moreover, it is generated by a specific family of causally ordered types. We then study signalling relations for maps belonging to a regular subtype, showing that the no-signalling conditions between an input and an output system are determined by a single evaluation of the corresponding function. For higher order types specifically, we show that all signalling relations can be read off directly from the structure poset via a rank parity condition. Finally, we study relations between the structure poset of a type and its normal forms, that is, expressions of the type in terms of causally ordered types. We illustrate construction of normal forms on some examples, demonstrating the possibility that the normal form can be systematically derived from maximal chains of the poset and signalling relations between them.