Skip to main content
SpringerLink
Log in

A Simple Heuristic Derivation of Shannon’s Formula for a Communication Channel with Continuous Variables in the Quantum Case

  • ATOMS, MOLECULES, OPTICS
  • Published:
Journal of Experimental and Theoretical Physics Aims and scope Submit manuscript

Abstract

The aim of the paper is the derivation of formulas for the capacities of a classical–quantum communication channel with Gaussian noise, which are analogs of the classical capacities and arise in many problems of quantum information theory, on a simple, heuristic and physically intuitively clear level. The classical capacity of a classical communication channel with limited frequency band and Gaussian information states diverges as noise tends to zero. In a consistent quantum consideration, the divergence is removed, which is a consequence of the fundamental physical principle of indistinguishability of particles—bosons in information states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

Similar content being viewed by others

Notes

  1. Note that the condition |ω| ≤ W is used for the bandwidth ω. Since negative frequencies are impossible for a physically stable system, we use the condition 0 ≤ ω ≤ Ω for the bandwidth (the position of the interval on the frequency axis is unimportant); therefore, the bandwidth in our notation is, for example, twice the bandwidth in [3].

REFERENCES

  1. C. E. Shannon, Bell Syst. Technol. J. 27, 379 (1948).

    Article  Google Scholar 

  2. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

    Book  Google Scholar 

  3. R. G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968).

    MATH  Google Scholar 

  4. A. S. Holevo, Probl. Inf. Transm. 9, 177 (1973).

    Google Scholar 

  5. A. S. Kholevo, Usp. Mat. Nauk 53, 193 (1998).

    Article  Google Scholar 

  6. A. S. Holevo, Quantum Systems, Channels, Information (MTsNMO, Moscow, 2010; De Gruyter, Berlin, 2013).

  7. H. J. Landau and H. O. Pollak, Bell Syst. Technol. J. 40, 65 (1961).

    Article  Google Scholar 

  8. D. Slepian and H. O. Pollak, Bell Syst. Technol. J. 40, 43 (1961).

    Article  Google Scholar 

  9. W. H. J. Fuchs, J. Math. Anal. Appl. 9, 317 (1964).

    Article  MathSciNet  Google Scholar 

  10. F. A. Berezin, The Method of Second Quantization, Vol. 24 of Monographs and Textbooks in Pure and Applied Physics (IO NFMI, Novokuznetsk, 2000; Academic, New York, 1966).

Download references

ACKNOWLEDGMENTS

We are grateful to our colleagues from the Academy of Cryptography of the Russian Federation for discussions and support. We are grateful to K.A. Balygin, A.N. Klimov, and S.P. Kulik for numerous discussions and comments.

Funding

This work was supported by the Russian Science Foundation, project no. 21-12-00005.

Author information

Authors and Affiliations

  1. Academy of Cryptography of the Russian Federation, 121552, Moscow, Russia

    I. M. Arbekov & S. N. Molotkov

  2. Institute of Solid State Physics, Russian Academy of Sciences, 142432, Chernogolovka, Moscow oblast, Russia

    S. N. Molotkov

  3. Quantum Technologies Center, Moscow State University, 119899, Moscow, Russia

    S. N. Molotkov & I. V. Sinil’shchikov

Authors
  1. I. M. Arbekov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. N. Molotkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. V. Sinil’shchikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Molotkov.

Additional information

Translated by I. Nikitin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arbekov, I.M., Molotkov, S.N. & Sinil’shchikov, I.V. A Simple Heuristic Derivation of Shannon’s Formula for a Communication Channel with Continuous Variables in the Quantum Case. J. Exp. Theor. Phys. 132, 362–373 (2021). https://doi.org/10.1134/S1063776121030110

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063776121030110

Search

Navigation