Skip to main content
SpringerLink
Log in

On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory

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

Abstract

We consider a family of energy-constrained diamond norms on the set of Hermitian- preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional channels than the diamond norm topology). We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained infinite-dimensional quantum channels (as functions of a channel) with respect to the energy-constrained diamond norms, which imply uniform continuity of these characteristics with respect to the strong convergence topology.

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

Similar content being viewed by others

References

  1. Holevo, A.S., Kvantovye sistemy, kanaly, informatsiya, Moscow: MCCME, 2010. Translated under the title Quantum Systems, Channels, Information: A Mathematical Introduction, Berlin: De Gruyter, 2012.

    Google Scholar 

  2. Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., and Lloyd, S., Gaussian Quantum Information, Rev. Mod. Phys., 2012, vol. 84, no. 2, pp. 621–669.

    Article  Google Scholar 

  3. Paulsen, V.I., Completely Bounded Maps and Operators Algebras, Cambridge: Cambridge Univ. Press, 2002.

    MATH  Google Scholar 

  4. Aharonov, D., Kitaev, A., and Nisan, N., Quantum Circuits with Mixed States, in Proc. 30th Annual ACM Sympos. on Theory of Computing (STOC’98), May 23–26, 1998, Dallas, TX, USA. New York: ACM, 1999, pp. 20–30.

    Google Scholar 

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

    Book  MATH  Google Scholar 

  6. Leung, D. and Smith, G., Continuity of Quantum Channel Capacities, Comm. Math. Phys., 2009, vol. 292, no. 1, pp. 201–215.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kretschmann, D., Schlingemann, D., and Werner, R.F., A Continuity Theorem for Stinespring’s Dilation, arXiv:0710.2495 [quant-ph], 2007.

    MATH  Google Scholar 

  8. 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].

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Lindblad, G., Entropy, Information and Quantum Measurements, Comm. Math. Phys., 1973, vol. 33, no. 4, pp. 305–322.

    Article  MathSciNet  Google Scholar 

  12. Reed, M. and Simon, B., Methods of Modern Mathematical Physics, vol. 1: Functional Analysis, New York: Academic, 1972. Translated under the title Metody sovremennoi matematicheskoi fiziki, vol. 1: Funktsional’nyi analiz, Moscow: Mir, 1978.

    MATH  Google Scholar 

  13. Pirandola, S., Laurenza, R., Ottaviani, C., and Banchi, L., Fundamental Limits of Repeaterless Quantum Communications, Nat. Commun., 2017, vol. 8, Article no. 15043.

  14. Shirokov, M.E., Entropy Characteristics of Subsets of States. I, Izv. Ross. Akad. Nauk, Ser. Mat., 2006, vol. 70, no. 6, pp. 193–222 [Izv. Math. (Engl. Transl.), 2006, vol. 70, no. 6, pp. 1265–1292].

    Article  MathSciNet  MATH  Google Scholar 

  15. Shirokov, M.E., Tight Uniform Continuity Bounds for the Quantum Conditional Mutual Information, for the Holevo Quantity, and for Capacities of Quantum Channels, J. Math. Phys., 2017, vol. 58, no. 10, p. 102202.

    Article  MathSciNet  MATH  Google Scholar 

  16. Holevo, A.S., Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel, Probl. Peredachi Inf., 1973, vol. 9, no. 3, pp. 3–11 [Probl. Inf. Trans. (Engl. Transl.), 1973, vol. 9, no. 3, pp. 177–183].

    MATH  Google Scholar 

  17. 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].

    Article  MathSciNet  MATH  Google Scholar 

  18. Billingsley, P., Convergence of Probability Measures, New York: Wiley, 1968. Translated under the title Skhodimost’ veroyatnostnykh mer, Moscow: Nauka, 1977.

    MATH  Google Scholar 

  19. Winter, A., Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints, Comm. Math. Phys., 2016, vol. 347, no. 1, pp. 291–313.

    Article  MathSciNet  MATH  Google Scholar 

  20. Holevo, A.S., Classical Capacities of a Quantum Channel with a Restriction at the Input, Teor. Veroyatnost. i Primenen., 2003, vol. 48, no. 2, pp. 359–374 [Theory Probab. Appl. (Engl. Transl.), 2004, vol. 48, no. 2, pp. 243–255].

    Article  MathSciNet  Google Scholar 

  21. Wilde, M.M. and Qi, H., Energy-Constrained Private and Quantum Capacities of Quantum Channels, arXiv:1609.01997 [quant-ph], 2016.

    Google Scholar 

  22. Giovannetti, V., Holevo, A.S., and García-Patrón, R., A Solution of Gaussian Optimizer Conjecture for Quantum Channels, Comm. Math. Phys., 2015, vol. 334, no. 3, pp. 1553–1571.

    Article  MathSciNet  MATH  Google Scholar 

  23. Holevo, A.S., On the Constrained Classical Capacity of Infinite-Dimensional Covariant Quantum Channels, J. Math. Phys., 2016, vol. 57, no. 1, p. 015203.

    Article  MATH  Google Scholar 

  24. Holevo, A.S. and Shirokov, M.E., On Classical Capacities of Infinite-Dimensional Quantum Channels, Probl. Peredachi Inf., 2013, vol. 49, no. 1, pp. 19–36 [Probl. Inf. Trans. (Engl. Transl.), 2013, vol. 49, no. 1, pp. 15–31].

    MathSciNet  MATH  Google Scholar 

  25. Winter, A., Energy-Constrained Diamond Norm with Applications to the Uniform Continuity of Continuous Variable Channel Capacities, arXiv:1712.10267 [quant-ph], 2017.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    M. E. Shirokov

Authors
  1. M. E. Shirokov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. Shirokov.

Additional information

Original Russian Text © M.E. Shirokov, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 1, pp. 24–38.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shirokov, M.E. On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory. Probl Inf Transm 54, 20–33 (2018). https://doi.org/10.1134/S0032946018010027

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946018010027

Search

Navigation