Quantum measurement incompatibility in subspaces

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