Skip to main content
Log in

A note on random coding bounds for classical-quantum channels

  • Information Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

A modified derivation of achievability results in classical-quantum channel coding theory is proposed, which has, in our opinion, two main benefits over previously used methods: it allows to (i) follow in a simple and clear way how binary hypothesis testing relates to channel coding achievability results, and (ii) derive in a unified way all previously known random coding achievability bounds on error exponents for classical and classical-quantum channels.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Shannon, C.E., A Mathematical Theory of Communication, Bell Syst. Tech. J., 1948, vol. 27, no. 3, pp. 379–423; no. 4, pp. 623–656.

    Article  MATH  MathSciNet  Google Scholar 

  2. Fano, R.M., Transmission of Information: A Statistical Theory of Communication, New York: Wiley, 1961.

    Google Scholar 

  3. Gallager, R.G., A Simple Derivation of the Coding Theorem and Some Applications, IEEE Trans. Inform. Theory, 1965, vol. 11, no. 1, pp. 3–18.

    Article  MATH  MathSciNet  Google Scholar 

  4. Hausladen, P., Jozsa, R., Schumacher, B., Westmoreland, M., and Wootters, W.K., Classical Information Capacity of a Quantum Channel, Phys. Rev. A, 1996, vol. 54, no. 3, pp. 1869–1876.

    Article  MathSciNet  Google Scholar 

  5. Schumacher, B. and Westmoreland, M.D., Sending Classical Information via Noisy Quantum Channels, Phys. Rev. A, 1997, vol. 56, no. 1, pp. 131–138.

    Article  Google Scholar 

  6. Holevo, A.S., The Capacity of the Quantum Channel with General Signal States, IEEE Trans. Inform. Theory, 1998, vol. 44, no. 1, pp. 269–273.

    Article  MATH  MathSciNet  Google Scholar 

  7. Hayashi, M. and Nagaoka, H., General Formulas for Capacity of Classical-Quantum Channels, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 7, pp. 1753–1768.

    Article  MATH  MathSciNet  Google Scholar 

  8. Burnashev, M.V. and Holevo, A.S., On the Reliability Function for a Quantum Communication Channel, Probl. Peredachi Inf., 1998, vol. 34, no. 2, pp. 3–15 [Probl. Inf. Trans. (Engl. Transl.), 1998, vol. 34, no. 2, pp. 97–107].

    MATH  MathSciNet  Google Scholar 

  9. Holevo, A.S., Reliability Function of General Classical-Quantum Channel, IEEE Trans. Inform. Theory, 2000, vol. 46, no. 6, pp. 2256–2261.

    Article  MATH  MathSciNet  Google Scholar 

  10. Hayashi, M., Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel Coding, Phys. Rev. A, 2007, vol. 76, no. 6, pp. 062301.

    Article  Google Scholar 

  11. Dalai, M., Sphere Packing Bound for Quantum Channels, in Proc. 201. IEEE Int. Sympos. on Information Theory (ISIT’2012), Cambridge, MA, USA, July 1–6, 2012, pp. 160–164.

    Chapter  Google Scholar 

  12. Dalai, M., Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels, IEEE Trans. Inform. Theory, 2013, vol. 59, no. 12, pp. 8027–8056.

    Article  MATH  MathSciNet  Google Scholar 

  13. Cover, T.M. and Thomas, J.A., Elements of Information Theory, New York: Wiley, 1991.

    Book  MATH  Google Scholar 

  14. Vazquez-Vilar, G., Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds, in Proc. 201. IEEE Int. Sympos. on Information Theory (ISIT’2016), Barcelona, Spain, July 10–15, 2016, pp. 2854–2857.

    Google Scholar 

  15. Shannon, C.E., Certain Results in Coding Theory for Noisy Channels, Inform. Control, 1957, vol. 1, pp. 6–25.

    Article  MATH  MathSciNet  Google Scholar 

  16. Holevo, A.S., Coding Theorems for Quantum Channels, arXiv:quant-ph/9809023v1, 1998.

    MATH  Google Scholar 

  17. Wilde, M.M., Quantum Information Theory, Cambridge, UK: Cambridge Univ. Press, 2013.

    Book  Google Scholar 

  18. Shamai, S. and Sason, I., Variations on the Gallager Bounds, Connections, and Applications, IEEE Trans. Inform. Theory, 2002, vol. 48. no. 12, pp. 3029–3051.

    Article  MATH  MathSciNet  Google Scholar 

  19. Audenaert, K.M.R., Calsamiglia, J., Mu˜noz-Tapia, R., Bagan, E., Masanes, L., Acin, A., and Verstraete, F., Discriminating States: The Quantum Chernoff Bound, Phys. Rev. Lett., 2007, vol. 98, no. 16, pp. 160501.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Dalai.

Additional information

Partially supported by the Italian Ministry of Education under grant PRIN 2015 D72F1600079000.

Original Russian Text © M. Dalai, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 3, pp. 23–29.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dalai, M. A note on random coding bounds for classical-quantum channels. Probl Inf Transm 53, 222–228 (2017). https://doi.org/10.1134/S0032946017030036

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946017030036

Navigation