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 Zap. Nauchn. Sem. POMI, 2015, Volume 436, Pages 49–75 (Mi znsl6159)

On the noncommutative deformation of the operator graph corresponding to the Klein group

G. G. Amosova, I. Yu. Zhdanovskiybc

a Steklov Mathematical Institute, Moscow, Russia
b Moscow Institute of Physics and Technology, Moscow, Russia
c Higher School of Economics, Moscow, Russia

Abstract: We study the noncommutative operator graph $\mathcal L_\theta$ depending on a complex parameter $\theta$ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing $n$-shot capacity. We define a noncommutative group $G$ and an algebra $\mathcal A_\theta$ which is a quotient of $\mathbb CG$ with respect to a special algebraic relation depending on $\theta$ such that the matrix representation $\phi$ of $\mathcal A_\theta$ results in the algebra $\mathcal M_\theta$ generated by $\mathcal L_\theta$. In the case of $\theta=\pm1$, the representation $\phi$ degenerates into an faithful representation of $\mathbb CK_4$, where $K_4$ is the Klein group. Thus, $\mathcal L_\theta$ can be regarded as a noncommutative deformation of the graph associated with the Klein group.

Key words and phrases: quantum channel, noncommutative operator graph, noncommutative deformation of the ring generated by the Klein group.

 Funding Agency Grant Number Russian Science Foundation 14-21-00162 Russian Foundation for Basic Research 13-01-0023414-01-00416 The first part of the work (Secs. 1, 2, 3, and 4) was fulfilled by G.G. Amosov. The second part of the work (Secs. 5, 6, Appendix A, and Appendix B) was fulfilled by I.Yu. Zhdanovskiy. The work of G.G. Amosov is supported by the Russian Science Foundation under the grant No. 14-21-00162 and performed in the Steklov Mathematical Institute of the Russian Academy of Sciences. The work of I.Yu. Zhdanovskiy is supported by the RFBR, research projects 13-01-00234 and 14-01-00416, and was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

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English version:
Journal of Mathematical Sciences (New York), 2016, 215:6, 659–676

Bibliographic databases:

UDC: 512.547+512.553+512.7+519.72
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Citation: G. G. Amosov, I. Yu. Zhdanovskiy, “On the noncommutative deformation of the operator graph corresponding to the Klein group”, Representation theory, dynamical systems, combinatorial methods. Part XXV, Zap. Nauchn. Sem. POMI, 436, POMI, St. Petersburg, 2015, 49–75; J. Math. Sci. (N. Y.), 215:6 (2016), 659–676

Citation in format AMSBIB
\Bibitem{AmoZhd15} \by G.~G.~Amosov, I.~Yu.~Zhdanovskiy \paper On the noncommutative deformation of the operator graph corresponding to the Klein group \inbook Representation theory, dynamical systems, combinatorial methods. Part~XXV \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 436 \pages 49--75 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6159} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3498185} \elib{http://elibrary.ru/item.asp?id=27159446} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 215 \issue 6 \pages 659--676 \crossref{https://doi.org/10.1007/s10958-016-2872-4} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84966642469}