On Errors Generated by Unitary Dynamics of Bipartite Quantum Systems

Abstract

Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group \({\mathbb{Z}}_{n}\) and the additive group \(\mathbb{R}\) are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.

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REFERENCES

  1. 1

    P. Shor, ‘‘Scheme for reducing decoherence in quantum memory,’’ Phys. Rev. A 52, 2493 (1995).

    Article  Google Scholar 

  2. 2

    D. Gottesman, ‘‘Stabilizer codes and quantum error correction,’’ Ph. D. Thesis (Caltech, 1997); arXiv: quant-ph/9705052.

  3. 3

    E. Knill and R. Laflamme, ‘‘Theory of error-correction codes,’’ Phys. Rev. A 55, 900 (1997).

    MathSciNet  Article  Google Scholar 

  4. 4

    R. Duan, S. Severini, and A. Winter, ‘‘Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function,’’ IEEE Trans. Inform. Theory 59, 1164-1174 (2013).

    MathSciNet  Article  Google Scholar 

  5. 5

    R. Duan, ‘‘Superactivation of zero-error capacity of noisy quantum channels,’’ arXiv:0906.2527 (2009).

  6. 6

    M. E. Shirokov and T. Shulman, ‘‘On superactivation of zero-error capacities and reversibility of a quantum channel,’’ Commun. Math. Phys. 335, 1159–1179 (2015).

    MathSciNet  Article  Google Scholar 

  7. 7

    V. I. Yashin, ‘‘Properties of operator systems, corresponding to channels (2020),’’ arXiv: 2004.13661 (2020).

  8. 8

    G. G. Amosov, A. S. Mokeev, and A. N. Pechen, ‘‘Non-commutative graphs and quantum error correction for a two-mode quantum oscillator,’’ Quantum Inform. Process. 19, 95 (2020).

    MathSciNet  Article  Google Scholar 

  9. 9

    G. G. Amosov and A. S. Mokeev, ‘‘On non-commutative operator graphs generated by reducible unitary representation of the Heisenberg–Weyl group,’’ Int. J. Theor. Phys. (2018). https://doi.org/10.1007/s10773-018-3963-4

  10. 10

    G. G. Amosov and A. S. Mokeev, ‘‘On non-commutative operator graphs generated by covariant resolutions of identity,’’ Quantum Inform. Process. 17, 325 (2018).

    MathSciNet  Article  Google Scholar 

  11. 11

    G. G. Amosov and A. S. Mokeev, ‘‘On linear structure of non-commutative operator graphs,’’ Lobachevskii J. Math. 40 (10), 1440–1443 (2019).

    MathSciNet  Article  Google Scholar 

  12. 12

    M. D. Choi and E. G. Effros, ‘‘Injectivity and operator spaces,’’ J. Funct. Anal. 24, 156–209 (1977).

    MathSciNet  Article  Google Scholar 

  13. 13

    N. Weaver, ‘‘A ‘‘quantum’’ Ramsey theorem for operator systems,’’ Proc. Am. Math. Soc. 145, 4595-4605 (2017).

    MathSciNet  Article  Google Scholar 

  14. 14

    A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011).

    Google Scholar 

  15. 15

    A. S. Holevo, Quantum System, Channels, Information (De Gruyter, Berlin, 2012).

    Google Scholar 

  16. 16

    G. G. Amosov, ‘‘On operator systems generated by reducible projective unitary representations of compact groups,’’ Turk. J. Math. 43, 2366–2370 (2019).

    MathSciNet  Article  Google Scholar 

  17. 17

    C. H. Bennett, G. Brassard, R. Jozsa, C. Crepeau, A. Peres, and W. K. Wootters, ‘‘Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,’’ Phys. Rev. Lett. 70, 1895 (1993).

    MathSciNet  Article  Google Scholar 

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Funding

The work is supported by Russian Science Foundation under the grant no. 19-11-00086 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.

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Correspondence to G. G. Amosov or A. S. Mokeev.

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(Submitted by S. A. Grigoryan)

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Amosov, G.G., Mokeev, A.S. On Errors Generated by Unitary Dynamics of Bipartite Quantum Systems. Lobachevskii J Math 41, 2310–2315 (2020). https://doi.org/10.1134/S1995080220120069

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Keywords:

  • non-commutative operator graphs
  • covariant resolution of identity
  • symmetric Fock space
  • quantum anticliques