Quantum marginal problem and incompatibility

Erkka Haapasalo1,2, Tristan Kraft3, Nikolai Miklin4, and Roope Uola5

1Centre for Quantum Technologies, National University of Singapore, Science Drive 2, Block S15-03-18, Singapore 117543
2Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China
3Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany
4Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdańsk, Poland
5Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

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Abstract

One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.

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Cited by

[1] Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469.

[2] Mark Girard, Martin Plávala, and Jamie Sikora, "Jordan products of quantum channels and their compatibility", Nature Communications 12, 2129 (2021).

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