  Existing work on quantum causal structure assumes that one can perform
arbitrary operations on the systems of interest. But this condition is often
not met. Here, we extend the framework for quantum causal modelling to
situations where a system can suffer sectorial constraints, that is,
restrictions on the orthogonal subspaces of its Hilbert space that may be
mapped to one another. Our framework (a) proves that a number of different
intuitions about causal relations turn out to be equivalent; (b) shows that
quantum causal structures in the presence of sectorial constraints can be
represented with a directed graph; and (c) defines a fine-graining of the
causal structure in which the individual sectors of a system bear causal
relations. As an example, we apply our framework to purported photonic
implementations of the quantum switch to show that while their coarse-grained
causal structure is cyclic, their fine-grained causal structure is acyclic. We
therefore conclude that these experiments realize indefinite causal order only
in a weak sense. Notably, this is the first argument to this effect that is not
rooted in the assumption that the causal relata must be localized in spacetime.
