On Noncommutative Operator Graphs Generated by Resolutions of Identity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 14-22.

Voir la notice de l'article provenant de la source Math-Net.Ru

Noncommutative operator graphs play an important role in the theory of quantum error correction. In this paper, we briefly review recent results devoted to the graphs generated by resolutions of identity for which there exists a quantum error-correcting code. We discuss examples of such graphs and touch upon the problem of describing quantum noise within this theory.
@article{TM_2021_313_a1,
     author = {G. G. Amosov and A. S. Mokeev},
     title = {On {Noncommutative} {Operator} {Graphs} {Generated} by {Resolutions} of {Identity}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {14--22},
     publisher = {mathdoc},
     volume = {313},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_313_a1/}
}
TY  - JOUR
AU  - G. G. Amosov
AU  - A. S. Mokeev
TI  - On Noncommutative Operator Graphs Generated by Resolutions of Identity
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2021
SP  - 14
EP  - 22
VL  - 313
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2021_313_a1/
LA  - ru
ID  - TM_2021_313_a1
ER  - 
%0 Journal Article
%A G. G. Amosov
%A A. S. Mokeev
%T On Noncommutative Operator Graphs Generated by Resolutions of Identity
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2021
%P 14-22
%V 313
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2021_313_a1/
%G ru
%F TM_2021_313_a1
G. G. Amosov; A. S. Mokeev. On Noncommutative Operator Graphs Generated by Resolutions of Identity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 14-22. http://geodesic.mathdoc.fr/item/TM_2021_313_a1/

[1] Amosov G.G., “On general properties of non-commutative operator graphs”, Lobachevskii J. Math., 39:3 (2018), 304–308 | DOI | MR | Zbl

[2] G. G. Amosov and A. S. Mokeev, “On construction of anticliques for noncommutative operator graphs”, J. Math. Sci., 234:3 (2018), 269–275 | DOI | MR | MR | Zbl

[3] Amosov G.G., Mokeev A.S., “On non-commutative operator graphs generated by covariant resolutions of identity”, Quantum Inf. Process., 17:12 (2018), 325 | DOI | MR | Zbl

[4] Amosov G.G., Mokeev A.S., “On linear structure of non-commutative operator graphs”, Lobachevskii J. Math., 40:10 (2019), 1440–1443 | DOI | MR | Zbl

[5] Amosov G.G., Mokeev A.S., “Non-commutative graphs in the Fock space over one-particle Hilbert space”, Lobachevskii J. Math., 41:4 (2020), 592–596 | DOI | MR | Zbl

[6] Amosov G.G., Mokeev A.S., “On non-commutative operator graphs generated by reducible unitary representation of the Heisenberg–Weyl group”, Int. J. Theor. Phys., 60:2 (2021), 457–463 | DOI | MR

[7] Amosov G.G., Mokeev A.S., Pechen A.N., “Non-commutative graphs and quantum error correction for a two-mode quantum oscillator”, Quantum Inf. Process., 19:3 (2020), 95 | DOI | MR

[8] G. G. Amosov and I. Yu. Zhdanovskii, “Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity”, Math. Notes, 99:5–6 (2016), 924–927 | DOI | MR | Zbl

[9] Choi M.-D., Effros E.G., “Injectivity and operator spaces”, J. Funct. Anal., 24:2 (1977), 156–209 | DOI | MR | Zbl

[10] Duan R., Super-activation of zero-error capacity of noisy quantum channels, E-print, 2009, arXiv: 0906.2527 [quant-ph]

[11] Duan R., Severini S., Winter A., “Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number”, IEEE Trans. Inf. Theory, 59:2 (2013), 1164–1174 | DOI | MR | Zbl

[12] Glauber R.J., “Coherent and incoherent states of the radiation field”, Phys. Rev., 131:6 (1963), 2766–2788 | DOI | MR | Zbl

[13] Gottesman D., “Theory of fault-tolerant quantum computation”, Phys. Rev. A, 57:1 (1998), 127–137 | DOI

[14] Knill E., Laflamme R., “Theory of quantum error-correcting codes”, Phys. Rev. A, 55:2 (1997), 900–911 | DOI | MR

[15] Knill E., Laflamme R., Viola L., “Theory of quantum error correction for general noise”, Phys. Rev. Lett., 84:11 (2000), 2525–2528 | DOI | MR | Zbl

[16] Laflamme R., Miquel C., Paz J.P., Zurek W.H., “Perfect quantum error correcting code”, Phys. Rev. Lett., 77:1 (1996), 198–201 | DOI

[17] Shirokov M.E., Shulman T., “On superactivation of zero-error capacities and reversibility of a quantum channel”, Commun. Math. Phys., 335:3 (2015), 1159–1179 | DOI | MR | Zbl

[18] Shor P.W., “Scheme for reducing decoherence in quantum computer memory”, Phys. Rev. A, 52:4 (1995), R2493–R2496 | DOI

[19] Steane A., “Multiple-particle interference and quantum error correction”, Proc. R. Soc. London A, 452:1954 (1996), 2551–2577 | DOI | MR | Zbl

[20] Sudarshan E.C.G., “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams”, Phys. Rev. Lett., 10:7 (1963), 277–279 | DOI | MR | Zbl

[21] Weaver N., “A “quantum” Ramsey theorem for operator systems”, Proc. Amer. Math. Soc., 145:11 (2017), 4595–4605 | DOI | MR | Zbl

[22] Weaver N., “The “quantum” Turán problem for operator systems”, Pac. J. Math., 301:1 (2019), 335–349 | DOI | MR | Zbl

[23] Yashin V.I., “Properties of operator systems, corresponding to channels”, Quantum Inf. Process., 19:7 (2020), 195 | DOI | MR