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On inner geometry of noncommutative operator graphs

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Abstract

Operator systems (noncommutative operator graphs in other terminology) play a major role in the theory of quantum error correcting codes. Any operator graph is associated with a number of quantum channels. The possibility to transmit quantum information through a quantum channel with zero error is determined by the geometrical properties of the corresponding graph. Noncommutative operator graphs are known to be generated by positive operator-valued measures (POVMs). In turn, many principal POVMs consist of multiple of projections. We construct the model in which the graph is a linear envelope of two projection-valued resolutions of identities in a Hilbert space. Conditions for the existence of quantum anticliques (error-correcting codes) for the graph are investigated. The connection with Shirokov’s example of quantum superactivation (Shirokov in Probl Inform Transm 51(2):87–102, 2015; Shirokov and Shulman in Commun Math Phys 335:1159, 2015) is revealed.

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Acknowledgements

This work was funded by the Ministry of Science and Higher Education of the Russian Federation (grant number 075-15-2020-788).

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, Russia, 119991

    G. G. Amosov

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  1. G. G. Amosov
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Correspondence to G. G. Amosov.

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Amosov, G.G. On inner geometry of noncommutative operator graphs. Eur. Phys. J. Plus 135, 865 (2020). https://doi.org/10.1140/epjp/s13360-020-00871-1

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  • DOI: https://doi.org/10.1140/epjp/s13360-020-00871-1

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