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.
 E. N. Torgersen, Comparison of experiments when the parameter space is finite, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 16, 219 (1970).
 F. Buscemi, D. Sutter, and M. Tomamichel, An information-theoretic treatment of quantum dichotomies, arXiv:1907.08539.
 X. Wang and M. M. Wilde, ``Resource theory of asymmetric distinguishability'', arXiv:1905.11629 (2019).
 M. Dall'Arno, S. Brandsen, F. Buscemi, and V. Vedral, Device-independent tests of quantum measurements, Phys. Rev. Lett. 118, 250501 (2017).
 M. Dall'Arno, Device-independent tests of quantum states, Phys. Rev. A 99, 052353 (2019).
 S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10, 165 (1976).
 F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, 187–204, (Interscience Publishers, New York, 1948).
 Frederik vom Ende, "Strict positivity and D-majorization", Linear and Multilinear Algebra 1 (2020).
 Sagnik Chakraborty, Dariusz Chruściński, Gniewomir Sarbicki, and Frederik vom Ende, "On the Alberti-Uhlmann Condition for Unital Channels", Quantum 4, 360 (2020).
 Katarzyna Siudzińska, Sagnik Chakraborty, and Dariusz Chruściński, "Interpolating between Positive and Completely Positive Maps: A New Hierarchy of Entangled States", Entropy 23 5, 625 (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2021-08-01 14:43:04). The list may be incomplete as not all publishers provide suitable and complete citation data.
On SAO/NASA ADS no data on citing works was found (last attempt 2021-08-01 14:43:04).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.