  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.

Quantum Physics