Abstract
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.
Highlights
When quantum states are looked at as resources, it is natural to study which states can be transformed into which others by means of an allowed set of operations
We show that any family of n qubit states which can all become simultaneously real under a single unitary transformation can be transformed into any other family of n qubit states by a completely positive trace preserving (CPTP) map if the testing region of the former includes the testing region of the latter
We demonstrate the possibility of witnessing statistical sufficiency in a semi-device independent way, that is, without any assumption on the devices except their Hilbert space dimension
Summary
When quantum states are looked at as resources, it is natural to study which states can be transformed into which others by means of an allowed set of operations. As a consequence of a well-known result by Alberti and Uhlmann, there exists a CPTP map transforming (ρ0, ρ1) into (σ0, σ1) if and only if the testing region of the former contains the testing region of the latter [17]. This is the perfect analog of Blackwell’s theorem; but counterexamples are known as soon as (ρ0, ρ1) is a qutrit dichotomy [5].
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