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Mutual and coherent information for infinite-dimensional quantum channels

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Abstract

The paper is devoted to the study of quantum mutual information and coherent information, two important characteristics of a quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given, and their properties are studied in detail. Basic identities relating the quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of the quantum mutual information and coherent information, following from the above identities, is observed. An upper bound for the coherent information is obtained.

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References

  1. Nielsen, M.A., and Chuang, I.L., Quantum Computation and Quantum Information, Cambridge: Cambridge Univ. Press, 2000. Translated under the title Kvantovye vychisleniya i kvantovaya informatsiya, Moscow: Mir, 2006.

    MATH  Google Scholar 

  2. Holevo, A.S., Kvantovye sistemy, kanaly, informatsiya (Quantum Systems, Channels, and Information), Moscow: MCCME, 2010.

    Google Scholar 

  3. Wehrl, A., General Properties of Entropy, Rev. Modern Phys., 1978, vol. 50, no. 2, pp. 221–260.

    Article  MathSciNet  Google Scholar 

  4. Lindblad, G., Expectations and Entropy Inequalities for Finite Quantum Systems, Comm. Math. Phys., 1974, vol. 39, no. 2, pp. 111–119.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ohya, M. and Petz, D., Quantum Entropy and Its Use, Berlin: Springer, 2004, 2nd ed.

    Google Scholar 

  6. Holevo, A.S. and Shirokov, M.E., Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels, Teor. Veroyatnost. i Primenen., 2005, vol. 50, no. 1, pp. 98–114 [Theory Probab. Appl. (Engl. Transl.), 2006, vol. 50, no. 1, pp. 86–98].

    Google Scholar 

  7. Shirokov, M.E. and Holevo, A.S., On Approximation of Infinite-Dimensional Quantum Channels, Probl. Peredachi Inf., 2008, vol. 44, no. 2, pp. 3–22 [Probl. Inf. Trans. (Engl. Transl.), 2008, vol. 44, no. 2, pp. 73–90].

    Google Scholar 

  8. Adami, C. and Cerf, N.J., Von Neumann Capacity of Noisy Quantum Channels, Phys. Rev. A, 1997, vol. 56, no. 5, pp. 3470–3483.

    Article  MathSciNet  Google Scholar 

  9. Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V., Entanglement-Assisted Capacity of a Quantum Channel and the Reverse Shannon Theorem, IEEE Trans. Inform. Theory, 2002, vol. 48, no. 10, pp. 2637–2655.

    Article  MATH  MathSciNet  Google Scholar 

  10. Barnum, H., Nielsen, M.A., and Schumacher, B.W., Information Transmission through a Noisy Quantum Channel, Phys. Rev. A, 1998, vol. 57, no. 6, pp. 4153–4175.

    Article  Google Scholar 

  11. Devetak, I., The Private Classical Capacity and Quantum Capacity of a Quantum Channel, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 1, pp. 44–55.

    Article  MathSciNet  Google Scholar 

  12. Davies, E.B., Quantum Theory of Open Systems, London: Academic, 1976.

    MATH  Google Scholar 

  13. Holevo, A.S., Complementary Channels and the Additivity Problem, Teor. Veroyatnost. i Primenen., 2006, vol. 51, no. 1, pp. 133–143 [Theory Probab. Appl. (Engl. Transl.), 2007, vol. 51, no. 1, pp. 92–100].

    MathSciNet  Google Scholar 

  14. Shirokov, M.E., Continuity of the von Neumann Entropy, Commun. Math. Phys., 2010, vol. 296, no. 3, pp. 625–654.

    Article  MATH  MathSciNet  Google Scholar 

  15. Schumacher, B. and Westmoreland, M.D., Quantum Privacy and Quantum Coherence, Phys. Rev. Lett., 1998, vol. 80, no. 25, pp. 5695–5697.

    Article  Google Scholar 

  16. Ozawa, M., On Information Gain by Quantum Measurements of Continuous Observables, J. Math. Phys., 1986, vol. 27, no. 3, pp. 759–763.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    A. S. Holevo & M. E. Shirokov

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  1. A. S. Holevo
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  2. M. E. Shirokov
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Correspondence to A. S. Holevo.

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Original Russian Text © A.S. Holevo, M.E. Shirokov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 3, pp. 3–21.

Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00424a, and Scientific Program “Mathematical Control Theory” of the Russian Academy of Sciences.

Supported in part by the Analytical Departmental Target Program “Development of the Scientific Potential of the Higher School,” project no. 2.1.1/500, Federal Target Program “Research and Educational Personnel of Innovation Russia,” project no. 1.2.1, contract no. 938, and Russian Foundation for Basic Research, project no. 10-01-00139a.

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Holevo, A.S., Shirokov, M.E. Mutual and coherent information for infinite-dimensional quantum channels. Probl Inf Transm 46, 201–218 (2010). https://doi.org/10.1134/S0032946010030014

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  • DOI: https://doi.org/10.1134/S0032946010030014

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