  We consider the question of characterising the incompatibility of sets of
high-dimensional quantum measurements. We introduce the concept of measurement
incompatibility in subspaces. That is, starting from a set of measurements that
is incompatible, one considers the set of measurements obtained by projection
onto any strict subspace of fixed dimension. We identify three possible forms
of incompatibility in subspaces: (i) incompressible incompatibility:
measurements that become compatible in every subspace, (ii) fully compressible
incompatibility: measurements that remain incompatible in every subspace, and
(iii) partly compressible incompatibility: measurements that are compatible in
some subspace and incompatible in another. For each class we discuss explicit
examples. Finally, we present some applications of these ideas. First we show
that joint measurability and coexistence are two inequivalent notions of
incompatibility in the simplest case of qubit systems. Second we highlight the
implications of our results for tests of quantum steering.
