Skip to main content
Springer Nature Link
Log in

Strong Converse for the Feedback-Assisted Classical Capacity of Entanglement-Breaking Channels

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

Abstract

Quantum entanglement can be used in a communication scheme to establish a correlation between successive channel inputs that is impossible by classical means. It is known that the classical capacity of quantum channels can be enhanced by such entangled encoding schemes, but this is not always the case. In this paper, we prove that a strong converse theorem holds for the classical capacity of an entanglement-breaking channel even when it is assisted by a classical feedback link from the receiver to the transmitter. In doing so, we identify a bound on the strong converse exponent, which determines the exponentially decaying rate at which the success probability tends to zero, for a sequence of codes with communication rate exceeding capacity. Proving a strong converse, along with an achievability theorem, shows that the classical capacity is a sharp boundary between reliable and unreliable communication regimes. One of the main tools in our proof is the sandwiched Rényi relative entropy. The same method of proof is used to derive an exponential bound on the success probability when communicating over an arbitrary quantum channel assisted by classical feedback, provided that the transmitter does not use entangled encoding schemes.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Book  MATH  Google Scholar 

  2. El Gamal, A. and Kim, Y.-H., Network Information Theory, Cambridge: Cambridge Univ. Press, 2011.

    Book  MATH  Google Scholar 

  3. 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  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Hastings, M.B., Superadditivity of Communication Capacity Using Entangled Inputs, Nat. Phys., 2009, vol. 5, no. 4, pp. 255–257.

    Article  MathSciNet  Google Scholar 

  6. Horodecki, M., Shor, P.W., and Ruskai, M.B., Entanglement Breaking Channels, Rev. Math. Phys., 2003, vol. 15, no. 6, pp. 629–641.

    Article  MathSciNet  MATH  Google Scholar 

  7. Shor, P.W., Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels, J. Math. Phys., 2002, vol. 43, no. 9, pp. 4334–4340.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bowen, G. and Nagarajan, R., On Feedback and the Classical Capacity of a Noisy Quantum Channel, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 1, pp. 320–324.

    Article  MathSciNet  MATH  Google Scholar 

  9. 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, p. 062301.

    Article  Google Scholar 

  10. Wolfowitz, J., Coding Theorems of Information Theory, Berlin: Springer, 1978, 3rd ed.

    Book  MATH  Google Scholar 

  11. Arimoto, S., On the Converse to the Coding Theorem for Discrete Memoryless Channels, IEEE Trans. Inform. Theory, 1973, vol. 19, no. 3, pp. 357–359.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ogawa, T. and Nagaoka, H., Strong Converse to the Quantum Channel Coding Theorem, IEEE Trans. Inform. Theory, 1999, vol. 45, no. 7, pp. 2486–2489.

    Article  MathSciNet  MATH  Google Scholar 

  13. Winter, A., Coding Theorem and Strong Converse for Quantum Channels, IEEE Trans. Inform. Theory, 1999, vol. 45, no. 7, pp. 2481–2485.

    Article  MathSciNet  MATH  Google Scholar 

  14. Koenig, R. and Wehner, S., A Strong Converse for Classical Channel Coding Using Entangled Inputs, Phys. Rev. Lett., 2009, vol. 103, no. 7, p. 070504.

    Article  Google Scholar 

  15. Wilde, M.M., Winter, A., and Yang, D., Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy, Comm. Math. Phys., 2014, vol. 331, no. 2, pp. 593–622.

    Article  MathSciNet  MATH  Google Scholar 

  16. Bardhan, B.R., García-Patrón, R., Wilde, M.M., and Winter, A., Strong Converse for the Classical Capacity of All Phase-Insensitive Bosonic Gaussian Channels, IEEE Trans. Inform. Theory, 2015, vol. 61, no. 4, pp. 1842–1850.

    Article  MathSciNet  MATH  Google Scholar 

  17. Nagaoka, H., Strong Converse Theorems in Quantum Information Theory, Asymptotic Theory of Quantum Statistical Inference: Selected Papers, Hayashi, M., Ed., Singapore: World Sci., 2005, pp. 64–65.

    Chapter  Google Scholar 

  18. Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., and Tomamichel, M., On Quantum Rényi Entropies: A New Generalization and Some Properties, J. Math. Phys., 2003, vol. 54, no. 12, p. 122203.

    Article  MATH  Google Scholar 

  19. King, C., Maximal p-Norms of Entanglement Breaking Channels, Quantum Inf. Comput., 2003, vol. 3, no. 2, pp. 186–190.

    MathSciNet  MATH  Google Scholar 

  20. Mosonyi, M. and Hiai, F., On the Quantum Rényi Relative Entropies and Related Capacity Formulas, IEEE Trans. Inform. Theory, 2011, vol. 57, no. 4, pp. 2474–2487.

    Article  MathSciNet  MATH  Google Scholar 

  21. Cooney, T., Mosonyi,M., and Wilde, M.M., Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication, Comm. Math. Phys., 2016, vol. 344, no. 3, pp. 797–829.

    Article  MathSciNet  MATH  Google Scholar 

  22. Umegaki, H., Conditional Expectation in an Operator Algebra, IV (Entropy and Information), Kōdai Math. Sem. Rep., 1962, vol. 14, no. 2, pp. 59–85.

    Article  MathSciNet  MATH  Google Scholar 

  23. Frank, R.L. and Lieb, E.H., Monotonicity of a Relative Rényi Entropy, J. Math. Phys., 2013, vol. 54, no. 12, p. 122201.

    Article  MathSciNet  MATH  Google Scholar 

  24. Beigi, S., Sandwiched Rényi Divergence Satisfies Data Processing Inequality, J. Math. Phys., 2013, vol. 54, no. 12, p. 122202.

    Article  MATH  Google Scholar 

  25. Ohya, M., Petz, D., and Watanabe, N., On Capacities of Quantum Channels, Probab. Math. Statist., 1997, vol. 17, no. 1, pp. 179–196.

    MathSciNet  MATH  Google Scholar 

  26. Schumacher, B. and Westmoreland, M.D., Optimal Signal Ensembles, Phys. Rev. A, 2001, vol. 63, no. 2, p. 022308.

    Article  Google Scholar 

  27. Schumacher, B. and Westmoreland, M.D., Relative Entropy in Quantum Information Theory, Quantum Computation and Information (Proc. AMS Special Session, Jan. 19–21, 2000, Washington, DC), Lomonaco, S.J., Jr. and Brandt, H.E., Eds., Providence, RI: Amer. Math. Soc., 2002, pp. 265–290.

    Google Scholar 

  28. Holevo, A.S., Multiplicativity of p-Norms of Completely Positive Maps and the Additivity Problem in Quantum Information Theory, Uspekhi Mat. Nauk, 2006, vol. 61, no. 2 (368), pp. 113–152 [Russian Math. Surveys (Engl. Transl.), 2006, vol. 61, no. 2, pp. 301–339].

    Article  MathSciNet  MATH  Google Scholar 

  29. Sion, M., On General Minimax Theorems, Pacific J. Math., 1958, vol. 8, no. 1, pp. 171–176.

    Article  MathSciNet  MATH  Google Scholar 

  30. Audenaert, K.M.R., A Note on the p → q Norms of 2-PositiveMaps, Linear Algebra Appl., 2009, vol. 430, no. 4, pp. 1436–1440.

    Article  MathSciNet  MATH  Google Scholar 

  31. Bennett, C.H., Devetak, I., Shor, P.W., and Smolin, J.A., Inequalities and Separations among Assisted Capacities of Quantum Channels, Phys. Rev. Lett., 2006, vol. 96, no. 15, p. 150502.

    Article  MathSciNet  MATH  Google Scholar 

  32. Smith, G. and Smolin, J.A., Extensive Nonadditivity of Privacy, Phys. Rev. Lett., 2009, vol. 103, no. 12, p. 120503.

    Article  Google Scholar 

  33. Korbicz, J.K., Horodecki, P., and Horodecki, R., Quantum-Correlation Breaking Channels, Broadcasting Scenarios, and Finite Markov Chains, Phys. Rev. A, 2012, vol. 86, no. 4, p. 042319.

    Article  Google Scholar 

  34. Polyanskiy, Y. and Verdú, S., Arimoto Channel Coding Converse and Rényi Divergence, in Proc. 48th Annual Allerton Conf. on Communication, Control, and Computation, Sept. 29–Oct. 1, 2010, Allerton, IL, USA, pp. 1327–1333.

  35. Mosonyi, M. and Ogawa, T., Strong Converse Exponent for Classical-Quantum Channel Coding, Comm. Math. Phys., 2017, vol. 355, no. 1, pp. 373–426.

    Article  MathSciNet  MATH  Google Scholar 

  36. Augustin, U., Noisy Channels, Habilitation Thesis, Univ. Erlangen-Nürnberg, Germany, 1978.

    Google Scholar 

  37. Dueck, G. and Körner, J., Reliability Function of a Discrete Memoryless Channel at Rates above Capacity, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 1, pp. 82–85.

    Article  MathSciNet  MATH  Google Scholar 

  38. Csiszár, I. and Körner, J., Feedback Does Not Affect the Reliability Function of a DMC at Rates above Capacity, IEEE Trans. Inform. Theory, 1982, vol. 28, no. 1, pp. 92–93.

    Article  MathSciNet  MATH  Google Scholar 

  39. Brádler, K., Hayden, P., Touchette, D., and Wilde, M.M., Trade-off Capacities of the Quantum Hadamard Channels, Phys. Rev. A, 2010, vol. 81, no. 6, p. 062312.

    Article  Google Scholar 

  40. Wilde, M.M., Quantum Information Theory, Cambridge: Cambridge Univ. Press, 2017, 2nd ed.

    Book  MATH  Google Scholar 

  41. Fannes, M., A Continuity Property of the Entropy Density for Spin Lattice Systems, Comm. Math. Phys., 1973, vol. 31, no. 4, pp. 291–294.

    Article  MathSciNet  MATH  Google Scholar 

  42. Audenaert, K.M.R., A Sharp Continuity Estimate for the von Neumann Entropy, J. Phys. A, 2007, vol. 40, no. 28, pp. 8127–8136.

    Article  MathSciNet  MATH  Google Scholar 

  43. Tomamichel, M., Wilde, M.M., and Winter, A., Strong Converse Rates for Quantum Communication, IEEE Trans. Inform. Theory, 2017, vol. 63, no. 1, pp. 715–727.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Physics, Stanford University, Stanford, USA

    D. Ding

  2. Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, USA

    M. M. Wilde

Authors
  1. D. Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Wilde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Ding.

Additional information

Original Russian Text © D. Ding, M.M. Wilde, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 1, pp. 3–23.

Supported from a Stanford Graduate Fellowship.

Supported from startup funds from the Department of Physics and Astronomy at LSU, the NSF under Award No. CCF-1350397, and the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ding, D., Wilde, M.M. Strong Converse for the Feedback-Assisted Classical Capacity of Entanglement-Breaking Channels. Probl Inf Transm 54, 1–19 (2018). https://doi.org/10.1134/S0032946018010015

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946018010015

Search

Navigation