Extension of the Alberti-Ulhmann criterion beyond qubit dichotomies

Michele Dall'Arno1,2, Francesco Buscemi3, and Valerio Scarani1,4

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
2Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan
3Graduate School of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan
4Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore

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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.

As soon as entanglement was recognised as a resource, theorists started studying the interconversions properties of this resource. The most famous such question is: given N copies of a state rho, how many copies N' of the state rho' can one obtain with local operations and classical communication? This question led to the definition of entanglement of formation (rho is the maximally entangled state), of distillation (rho' is the maximally entangled state), to the discovery of inequivalent entanglement classes for multipartite systems... The amount of literature on this question is enormous.

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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[1] J. M. Renes, Relative submajorization and its use in quantum resource theories, J. Math. Phys. 57, 122202 (2016).

[2] D. Blackwell, Equivalent Comparisons of Experiments, Ann. Math. Statist. 24, 265 (1953).

[3] E. N. Torgersen, Comparison of statistical experiments, (Cambridge University Press, 1991).

[4] E. N. Torgersen, Comparison of experiments when the parameter space is finite, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 16, 219 (1970).

[5] K. Matsumoto, An example of a quantum statistical model which cannot be mapped to a less informative one by any trace preserving positive map, arXiv:1409.5658.

[6] K. Matsumoto, On the condition of conversion of classical probability distribution families into quantum families, arXiv:1412.3680 (2014).

[7] F. Buscemi and G. Gour, Quantum Relative Lorenz Curves, Phys. Rev. A 95, 012110 (2017).

[8] D. Reeb, M. J. Kastoryano, and M. M. Wolf, Hilbert's projective metric in quantum information theory, J. Math. Phys. 52, 082201 (2011).

[9] A. Jenčová, Comparison of Quantum Binary Experiments, Reports on Mathematical Physics 70, 237 (2012).

[10] F. Buscemi, Comparison of Quantum Statistical Models: Equivalent Conditions for Sufficiency, Communications in Mathematical Physics 310, 625 (2012).

[11] K. Matsumoto, A quantum version of randomization criterion, arXiv: 1012.2650 (2010).

[12] A. Jenčová, Comparison of quantum channels and statistical experiments, in 2016 IEEE International Symposium on Information Theory (ISIT), 2249 (2016).

[13] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications (Springer, 2011).

[14] K. Matsumoto, Reverse Test and Characterization of Quantum Relative Entropy, arXiv:1010.1030.

[15] F. Buscemi, D. Sutter, and M. Tomamichel, An information-theoretic treatment of quantum dichotomies, arXiv:1907.08539.

[16] X. Wang and M. M. Wilde, ``Resource theory of asymmetric distinguishability'', arXiv:1905.11629 (2019).

[17] P. M. Alberti and A. Uhlmann, A problem relating to positive linear maps on matrix algebras, Reports on Mathematical Physics 18, 163 (1980).

[18] M. Dall'Arno, S. Brandsen, F. Buscemi, and V. Vedral, Device-independent tests of quantum measurements, Phys. Rev. Lett. 118, 250501 (2017).

[19] M. Dall'Arno, Device-independent tests of quantum states, Phys. Rev. A 99, 052353 (2019).

[20] M. Dall'Arno, F. Buscemi, A. Bisio, and A. Tosini, Data-driven inference, reconstruction, and observational completeness of quantum devices, arXiv:1812.08470.

[21] F. Buscemi and M. Dall'Arno, Data-driven Inference of Physical Devices: Theory and Implementation, New J. Phys. 21, 113029 (2019).

[22] M. Dall'Arno, A. Ho, F. Buscemi, and V. Scarani, Data-driven inference and observational completeness of quantum devices, arXiv:1905.04895.

[23] S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10, 165 (1976).

[24] M. M. Wilde, Quantum Information Theory, (Cambridge University Press, 2017).

[25] F. Buscemi, G. M. D'Ariano, M. Keyl, P. Perinotti, and R. Werner, Clean Positive Operator Valued Measures, J. Math. Phys. 46, 082109 (2005).

[26] 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).

[27] K. M. Ball, Ellipsoids of maximal volume in convex bodies, Geom. Dedicata. 41, 241 (1992).

[28] Michael J. Todd, Minimum-Volume Ellipsoids: Theory and Algorithms, (Cornell University, 2016).

[29] S. Boyd and L. Vandenberghe, Convex Optimization, (Cambridge University Press, 2004).

[30] G. M. D'Ariano, G. Chiribella, and P. Perinotti, Quantum Theory from First Principles: An Informational Approach (Cambridge University Press, 2017).

Cited by

[1] Frederik vom Ende, "Strict positivity and D-majorization", Linear and Multilinear Algebra 1 (2020).

[2] Sagnik Chakraborty, Dariusz Chruściński, Gniewomir Sarbicki, and Frederik vom Ende, "On the Alberti-Uhlmann Condition for Unital Channels", Quantum 4, 360 (2020).

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