Undecidability in Resource Theory: Can You Tell Resource Theories Apart?

A central question in resource theory is whether one can construct a set of monotones that completely characterize the allowed transitions dictated by a set of free operations. A similar question is whether two distinct sets of free operations generate the same class of transitions. These questions are part of the more general problem of whether it is possible to pass from one characterization of a resource theory to another. In the present Letter, we prove that in the context of quantum resource theories this class of problem is undecidable in general. This is done by proving the undecidability of the membership problem for completely positive trace preserving maps, which subsumes all the other results.