Extension of the Alberti-Ulhmann criterion beyond qubit dichotomies

The Alberti-Ulhmann criterion states that any given qubit dichotomy can be transformed into any other given qubit dichotomy by a quantum channel if and only if the testing region of the former dichotomy includes the testing region of the latter dichotomy. Here, we generalize the Alberti-Ulhmann criterion to the case of arbitrary number of qubit or qutrit states. We also derive an analogous result for the case of qubit or qutrit measurements with arbitrary number of elements. We demonstrate the possibility of applying our criterion in a semi-device independent way.