Skip to main content
SpringerLink
Log in

The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The Holevo capacity of an arbitrarily constrained infinite dimensional quantum channel is considered and its properties are discussed. The notion of optimal average state is introduced. The continuity properties of the Holevo capacity with respect to constraint and to the channel are explored.

The main result of this paper is the statement that additivity of the Holevo capacity for all finite dimensional channels implies its additivity for all infinite dimensional channels with arbitrary constraints.

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. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. vol.I, New York-Heidelberg-Berlin, Springer Verlag, 1979

  2. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed State Entanglement and Quantum Error Correction. Phys. Rev. A 54, 3824–3851 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  3. Dell'Antonio, G.F.: On the limits of sequences of normal states. Commun. Pure Appl. Math. 20, 413–430 (1967)

    Article  MathSciNet  Google Scholar 

  4. Davies, E.B.: Information and Quantum Measurements. IEEE Trans.Inf.Theory 24, 596–599 (1978)

    Article  MATH  Google Scholar 

  5. Davies, E.B.: Quantum theory of open systems. London, Academic Press, 1976

  6. Donald, M.J.: Further results on the relative entropy. Math. Proc. Cam. Phil. Soc. 101, 363–373 (1987)

    Article  MathSciNet  Google Scholar 

  7. Eisert, J., Simon, C., Plenio, M.B.: The quantification of entanglement in infinite-dimensional quantum systems. J. Phys. A 35, 3911 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Holevo, A.S.: Quantum coding theorems. Russ. Math. Surv. 53, N6, 1295–1331 (1998)

    Google Scholar 

  9. Holevo, A.S.: On quantum communication channels with constrained inputs. http://arxiv.org/list/ quant-ph/9705054, 1997

  10. Holevo, A.S.: Classical capacities of quantum channels with constrained inputs. Probability Theory and Applications. 48, N.2, 359–374 (2003)

  11. Holevo, A.S., Shirokov, M.E.: On Shor's channel extension and constrained channels. Commun. Math. Phys. 249, 417–430 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Holevo, A.S., Shirokov, M.E.: Continuous ensembles and the χ-capacity of infinite dimensional channels. Probability Theory and Applications 50, N.1, 98–114 (2005)

    Google Scholar 

  13. Holevo, A.S., Werner, R.F.: Evaluating capacities of Bosonic Gaussian channels. http://arxiv.org/list/ quant-ph/9912067, 1999

  14. Holevo, A.S., Shirokov, M.E., Werner, R.F.: On the notion of entanglement in Hilbert space. Russ. Math. Surv. 60, N.2, 359–360 (2005)

    Google Scholar 

  15. Horodecki, M., Shor, P.W., Ruskai, M.B.: General Entanglement Breaking Channels. Rev. Math. Phys. 15, 629–641 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lindblad, G.: Entropy, Information and Quantum Measurements. Commun. Math. Phys. 33, N.4, 305–322 (1973)

  17. Lindblad, G.: Expectation and Entropy Inequalities for Finite Quantum Systems. Commun. Math. Phys. 39, N.2, 111–119 (1974)

  18. Lindblad, G.: Completely Positive Maps and Entropy Inequalities. Commun. Math. Phys. 40, N.2, 147–151 (1975)

  19. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1993

  20. Sarymsakov, T.A.: Introduction to Quantum Probability Theory. Tashkent, FAN: 1985, (In Russian)

  21. Schumacher, B., Westmoreland, M.D.: Sending Classical Information via Noisy Quantum Channels. Phys. Rev. A 56, 131–138 (1997)

    Article  ADS  Google Scholar 

  22. Schumacher, B., Westmoreland, M.D.: Optimal signal ensemble. Phys. Rev. A 63, 022308 (2001)

    Article  ADS  Google Scholar 

  23. Shirokov, M.E.: On the additivity conjecture for channels with arbitrary constraints. http://arxiv.org/list/quant-ph/0308168, 2003

  24. Shirokov, M.E.: On entropic quantities related to the classical capacity of infinite dimensional quantum channels. http://arxiv.org/list/quant-ph/0411091, 2004

  25. Shirokov, M.E.: The Holevo capacity of infinite dimensional channels and the additivity problem. http://arxiv.org/list/quant-ph/0408009, 2004

  26. Shor, P.W.: Additivity of the classical capacity of entanglement breaking quantum channel. J.Math.Phys, 43, 4334–4340 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Shor, P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, N.3, 453–472 (2004)

    Google Scholar 

  28. Simon, B.: Convergence theorem for entropy. Appendix in Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum mechanical entropy. J.Math.Phys. 14, 1938–1941 (1973)

    Google Scholar 

  29. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50, 221–250 (1978)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, 119991, 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

Communicated by M.B. Ruskai

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shirokov, M. The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem. Commun. Math. Phys. 262, 137–159 (2006). https://doi.org/10.1007/s00220-005-1457-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1457-8

Keywords

Search

Navigation