Undecidability in resource theory: can you tell theories apart?

Matteo Scandi, Jacopo Surace

Published 2021-05-19Version 1

A central question in resource theory is whether one can find a set of monotones that completely characterise the allowed transitions dictated by a set of free operations. This is part of the more general problem about the possibility of certifying the equivalence of two different characterisations of the same resource theory. A similar question is whether two distinct sets of free operations generate the same class of transitions. In the present letter we prove that in the context of quantum resource theories this class of problems is undecidable in general. This is done by proving the undecidability of the reachability problem for CPTP maps, which subsumes all the other results.