Skip to main content
SpringerLink
Log in

Entanglement-breaking channels in infinite dimensions

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

Abstract

In the first part of the paper we give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the “Kraus decomposition with rank-one operators” and use it to describe complementary channels. We also note that coherent information for antidegradable channel is always nonpositive. In the second part, we give necessary and sufficient condition for entanglement breaking for the general quantum Gaussian channel. Application of this condition to one-mode channels provides several new cases where the additivity conjecture holds in the strongest form.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Holevo, A.S., Quantum Coding Theorems, Uspekhi Mat. Nauk, 1998, vol. 53, no. 6, pp. 193–230 [Russian Math. Surveys (Engl. Transl.), 1998, vol. 53, no. 6, pp. 1295–1331].

    MathSciNet  Google Scholar 

  3. Werner, R.F., Holevo, A.S., and Shirokov, M.E., On the Concept of Entanglement in Hilbert Spaces, Uspekhi Mat. Nauk, 2005, vol. 60, no. 2, pp. 153–154 [Russian Math. Surveys (Engl. Transl.), 2005, vol. 60, no. 2, pp. 359–360].

    MathSciNet  Google Scholar 

  4. Holevo, A.S., Veroyatnostnye i statisticheskie aspekty kvantovoi teorii, Moscow: Nauka, 1980. Translated under the title Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam: North-Holland, 1982.

    Google Scholar 

  5. 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  MATH  MathSciNet  Google Scholar 

  6. 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].

    Google Scholar 

  7. 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 

  8. Shirokov, M.E., The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem, Comm. Math. Phys., 2006, vol. 262, no. 1, pp. 137–159.

    Article  MATH  MathSciNet  Google Scholar 

  9. Holevo, A.S. and Shirokov, M.E., On Shor’s Channel Extension and Constrained Channels, Comm. Math. Phys., 2004, vol. 249, no. 2, pp. 417–430.

    Article  MATH  MathSciNet  Google Scholar 

  10. Holevo, A.S., Radon-Nikodym Derivatives of Quantum Instruments, J. Math. Phys., 1998, vol. 39, no. 3, pp. 1373–1387.

    Article  MATH  MathSciNet  Google Scholar 

  11. Devetak, I. and Shor, P.W., The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information, Comm. Math. Phys., 2005, vol. 256, no. 2, pp. 287–303.

    Article  MATH  MathSciNet  Google Scholar 

  12. Caruso, F. and Giovannetti, V., Degradability of Bosonic Gaussian Channels, Phys. Rev. A, 2006, vol. 74, no. 6, pp. 062307(6).

  13. 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 

  14. Holevo, A.S., Covariant Measurements and Imprimitivity Systems, Quantum Probability and Applications to the Quantum Theory of Irreversible Processes (Proc. Int. Workshop, Villa Mondragone, Italy, 1982), Accardi, L., Frigerio, A., and Gorini, V., Eds., Lect. Notes Math., vol. 1055, Berlin: Springer, 1984, pp. 153–172.

    Chapter  Google Scholar 

  15. Cubitt, T., Ruskai, M.-B., and Smith, G., The Structure of Degradable Quantum Channels, arXiv e-print quant-ph/0802.1360, 2008.

  16. Holevo, A.S., Statistical Decision Theory for Quantum Systems, J. Multivariate Anal., 1973, vol. 3, pp. 337–394.

    Article  MATH  MathSciNet  Google Scholar 

  17. Demoen, B., Vanheuverzwijn, P., and Verbeure, A., Completely Positive Quasi-free Maps on the CCR Algebra, Rep. Math. Phys., 1979, vol. 15, no. 1, pp. 27–39.

    Article  MATH  MathSciNet  Google Scholar 

  18. Holevo, A.S. and Werner, R.F., Evaluating Capacities of Bosonic Gaussian Channels, Phys. Rev. A, 2001, vol. 63, no. 3, pp. 032312(14).

  19. Eisert, J. and Wolf, M.M., Gaussian Quantum Channels, Quantum Information with Continuous Variables of Atoms and Light, Cerf, N.J., Leuchs, G., and Polzik, E.S., Eds., London: Imperial College Press, 2007, pp. 23–42.

    Google Scholar 

  20. Werner, R.F. and Wolf, M.M., Bound Entangled Gaussian States, Phys. Rev. Lett., 2001, vol. 86, no. 16, pp. 3658–3661.

    Article  Google Scholar 

  21. Holevo, A.S., One-Mode Quantum Gaussian Channels: Structure and Quantum Capacity, Probl. Peredachi Inf., 2007, vol. 43, no. 1, pp. 3–14 [Probl. Inf. Trans. (Engl. Transl.), 2007, vol. 43, no. 1, pp. 1–11].

    MathSciNet  Google Scholar 

  22. Caruso, F., Giovannetti, V., and Holevo, A.S., One-Mode Bosonic Gaussian Channels: A Full Weak-Degradability Classification, New J. Physics, 2006, vol. 8, no. 12, pp. 310(18).

    Google Scholar 

  23. Giovannetti, V., Guha, S., Lloyd, S., Maccone, L., Shapiro, J.H., and Yuen, H.P., Classical Capacity of the Lossy Bosonic Channel: The Exact Solution, Phys. Rev. Lett., 2004, vol. 92, no. 2, pp. 027902(4).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, RAS, Moscow, Russia

    A. S. Holevo

Authors
  1. A. S. Holevo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. S. Holevo.

Additional information

Original Russian Text © A.S. Holevo, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 3, pp. 3–18.

Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00164a, and the program “Modern Problems of Theoretical Mathematics” of the Russian Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holevo, A.S. Entanglement-breaking channels in infinite dimensions. Probl Inf Transm 44, 171–184 (2008). https://doi.org/10.1134/S0032946008030010

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946008030010

Keywords

Search

Navigation