  While there is general consensus on the definition of incompatible POVMs,
moving up to the level of instruments one finds a much less clear situation,
with mathematically different and logically independent definitions of
incompatibility. Here we close this gap by introducing the notion of
q-compatibility, which unifies different notions of POVMs, channels, and
instruments incompatibility into one hierarchy of resource theories of
communication between separated parties. The resource theories that we obtain
are complete, in the sense that they contain complete families of free
operations and monotones providing necessary and sufficient conditions for the
existence of a transformation. Furthermore, our framework is fully operational,
in the sense that free transformations are characterized explicitly, in terms
of local operations aided by causally-constrained directed classical
communication, and all monotones possess a game-theoretic interpretation making
them experimentally measurable in principle. We are thus able to pinpoint
exactly what each notion of incompatibility consists of, in terms of
information-theoretic resources.
