  Several central problems in quantum information theory (such as measurement
compatibility and quantum steering) can be rephrased as membership in the
minimal matrix convex set corresponding to special polytopes (such as the
hypercube or its dual). In this article, we generalize this idea and introduce
the notion of polytope compatibility, by considering arbitrary polytopes. We
find that semiclassical magic squares correspond to Birkhoff polytope
compatibility. In general, we prove that polytope compatibility is in
one-to-one correspondence with measurement compatibility, when the measurements
have some elements in common and the post-processing of the joint measurement
is restricted. Finally, we consider how much tuples of operators with
appropriate joint numerical range have to be scaled in the worst case in order
to become polytope compatible and give both analytical sufficient conditions
and numerical ones based on linear programming.
