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.
Very little however is known about a different problem, the one we consider here. The question is whether a pair of states (rho,sigma) can be converted into another pair of states (rho',sigma'). This question does not need to refer to entanglement: in fact, here we don't consider composite systems, and consequently we don't restrict the possible operations. A very simple answer would be the one that holds for classical probability distributions: Pair 1 can be converted into Pair 2, if all the statistics that can be observed with Pair 2 can also be observed with Pair 1. This conveys the idea that Pair 1 can do all that Pair 2 can do, and possibly more. This answer holds for two states of qubits (Alberti and Uhlmann, 1980), but counter-examples are known already when Pair 1 comprises qutrit states. In this paper, we prove that the classical-like characterisation still holds when Pair 1 is generalized to any family of qubit states, as soon as they can all be expressed with real coefficients, and Pair 2 is generalized to any family of qubit or, under certain hypotheses, qutrit, states. We also exploit a duality between states and measurements to present a similar characterisation of measurement devices.
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