Amosov, G. G.; Zhdanovskii, I. Yu. Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity. (English. Russian original) Zbl 1348.81129 Math. Notes 99, No. 6, 924-927 (2016); translation from Mat. Zametki 99, No. 6, 929-932 (2016). From the introduction: M. E. Shirokov [Quantum Inf. Process. 14, No. 8, 3057–3074 (2015; Zbl 1327.81098)] recently suggested a construction of a noncommutative operator graph, depending on a complex parameter \(\theta\), which enables one to construct channels with positive quantum capacity for which the \(n\)-shot capacity is zero. We study the algebraic structure of this graph. Relations for the algebra generated by the graph are derived. In the limit case \(\theta = \pm 1\), the graph becomes commutative and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein group. Cited in 3 Documents MSC: 81P45 Quantum information, communication, networks (quantum-theoretic aspects) 94A40 Channel models (including quantum) in information and communication theory 46L07 Operator spaces and completely bounded maps 46L10 General theory of von Neumann algebras 46L30 States of selfadjoint operator algebras 46L60 Applications of selfadjoint operator algebras to physics 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) Keywords:von Neumann algebra; noncommutative operator graph; superactivation phenomenon; quantum channel; quantum state; Kraus operator Citations:Zbl 1327.81098 PDFBibTeX XMLCite \textit{G. G. Amosov} and \textit{I. Yu. Zhdanovskii}, Math. Notes 99, No. 6, 924--927 (2016; Zbl 1348.81129); translation from Mat. Zametki 99, No. 6, 929--932 (2016) Full Text: DOI References: [1] Shirokov, M. E., No article title, Quantum Inf. Process., 14, 3057 (2015) · Zbl 1327.81098 · doi:10.1007/s11128-015-1014-0 [2] Smith, G.; Yard, J., No article title, Science, 321, 1812 (2008) · Zbl 1226.94011 · doi:10.1126/science.1162242 [3] R. Duan, Super-Activation of Zero-Error Capacity of Noisy Quantum Channels, arXiv: 0906.2527 (2009). [4] T. S. Cubitt, J. Chen, and A. W. Harrow, Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel, arXiv: 0906.2547 (2009). [5] Duan, R.; Severini, S.; Winter, A., No article title, IEEE Trans. Inform. Theory, 59, 1164 (2013) · Zbl 1364.81059 · doi:10.1109/TIT.2012.2221677 [6] Shirokov, M. E.; Shulman, T., No article title, Comm. Math. Phys., 335, 1159 (2015) · Zbl 1310.81043 · doi:10.1007/s00220-015-2345-5 [7] Shirokov, M. E.; Shulman, T. V., No article title, Problemy Peredachi Informatsii, 50, 35 (2014) [8] A. S. Holevo, Quantum Systems, Channels, Information: AMathematical Introduction, in De Gruyter Studies in Math. Phys. (De Gruyter, Berlin, 2012), vol. 16. · Zbl 1332.81003 [9] Kholevo, A. S., No article title, Teor. Veroyatn. Primen., 51, 133 (2006) · doi:10.4213/tvp151 [10] T. S. Cubitt, M. B. Ruskai, and G. Smith, J. Math. Phys. 49 (10) (2008). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.