Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Electromagnetic channel capacity for practical purposes

Nature Photonics volume 7, pages 834–838 (2013)Cite this article

Subjects

Abstract

What is the maximum rate at which digital information can be communicated without error using electromagnetic signals, such as radio communication? According to Shannon theory this rate is the capacity of the communication channel, which is obtained by maximizing the mutual information between the channel's input and output. Shannon theory, however, has been developed within classical physics, whereas electromagnetic signals are, ultimately, quantum-mechanical entities. To account for this fact, the capacity must be expressed in terms of a complicated optimization of the Holevo information, but explicit solutions are still unknown for arguably the most elementary electromagnetic channel, the one degraded by additive thermal noise. We place bounds on the thermal channel's Holevo information that determine the capacity up to corrections that are insignificant for practical scenarios such as those with high noise or low transmissivity. Our results apply to any bosonic thermal-noise channel, including electromagnetic signalling at any frequency.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Plots of bounds for the thermal bosonic channel and the amplifier in the practically relevant regimes of high thermal noise (large N).
Figure 2: Plots of bounds in regimes that emphasize the gap between the lower and upper bounds.

Similar content being viewed by others

References

  1. Shannon, C. E. A mathematical theory of communication. Bell Syst. Technol. J. 27, 9–423, 623–656 (1948).

    Article  MathSciNet  Google Scholar 

  2. Caves, C. M. & Drummond, P. D. Quantum limits on bosonic communication rates. Rev. Mod. Phys. 66, 481–537 (1994).

    Article  ADS  Google Scholar 

  3. Holevo, A. S. & Giovannetti, V. Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75, 046001 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  4. König, R. & Smith, G. Classical capacity of quantum thermal noise channels to within 1.45 Bits. Phys. Rev. Lett. 110, 040501 (2013).

    Article  ADS  Google Scholar 

  5. König, R. & Smith, G. Limits on classical communication from quantum entropy power inequalities. Nature Photon. 7, 142–146 (2013).

    Article  ADS  Google Scholar 

  6. Holevo, A. S. & Werner, R. F. Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001).

    Article  ADS  Google Scholar 

  7. Giovannetti, V. et al. Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92, 027902 (2004).

    Article  ADS  Google Scholar 

  8. Giovannetti, V., Guha, S., Lloyd, S., Maccone, L. & Shapiro, J. H. Minimum output entropy of bosonic channels: a conjecture. Phys. Rev. A 70, 032315 (2004).

    Article  ADS  Google Scholar 

  9. Shirokov, M. E. The Holevo capacity of infinite dimensional channels and the additivity problem. Commun. Math. Phys. 262, 137–159 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  10. Shirokov, M. E. On entropic quantities related to the classical capacity of infinite dimensional quantum channels. Proc. SPIE 5833, 222–228 (2005).

    Article  ADS  Google Scholar 

  11. Holevo, A. S. & Shirokov, M. E. Continuous ensembles and the χ-capacity of infinite-dimensional channels. Theory Probab. Appl. 50, 86–96 (2006).

    Article  MathSciNet  Google Scholar 

  12. Hiroshima, T. Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs. Phys. Rev. A 73, 012330 (2006).

    Article  ADS  Google Scholar 

  13. Giovannetti, V., Holevo, A. S., Lloyd, S. & Maccone, L. Generalized minimal output entropy conjecture for Gaussian channels: definitions and some exact results. J. Phys. A 43, 415305 (2010).

    Article  MathSciNet  Google Scholar 

  14. Shapiro, J. H., Guha, S. & Erkmen, B. I. Ultimate channel capacity of free-space optical communications. J. Opt. Commun. Netw. 4, 501–516 (2005).

    Article  Google Scholar 

  15. Giovannetti, V., Lloyd, S., Maccone, L. & Shor, P. W. Entanglement assisted capacity of the broadband lossy channel. Phys. Rev. Lett. 91, 047901 (2003).

    Article  ADS  Google Scholar 

  16. Hastings, M. B. A counterexample to additivity of minimum output entropy. Nature Phys. 5, 255–257 (2009).

    Article  ADS  Google Scholar 

  17. Holevo, A. S. The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998).

    Article  MathSciNet  Google Scholar 

  18. Schumacher, B. & Westmoreland, M. D. Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997).

    Article  ADS  Google Scholar 

  19. Horodecki, M., Shor, P. W. & Ruskai, M. B. Entanglement breaking channels. Rev. Math. Phys. 15, 629–641 (2003).

    Article  MathSciNet  Google Scholar 

  20. Holevo, A. S. Entanglement-breaking channels in infinite dimensions. Probl. Inf. Transm. 44, 171–184 (2008).

    Article  MathSciNet  Google Scholar 

  21. Shor, P. W. Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43, 4334–4340 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  22. Giovannetti, V. et al. Minimum bosonic channel output entropies. AIP Conf. Proc. 734, 21–24 (2004).

    Article  ADS  Google Scholar 

  23. Garcia-Patron, R., Navarrete-Benlloch, C., Lloyd, S., Shapiro, J. H. & Cerf, N. J. Majorization theory approach to the Gaussian channel minimum entropy conjecture. Phys. Rev. Lett. 108, 110505 (2012).

    Article  ADS  Google Scholar 

  24. Zy˙czkowski, K. Rényi extrapolation of Shannon entropy. Open Syst. Inf. Dyn. 10, 297–310 (2003).

    Article  MathSciNet  Google Scholar 

  25. Beck, C. & Schlögl, F. Thermodynamics of Chaotic Systems (Cambridge Univ. Press, 1993).

  26. Giovannetti, V. & Lloyd, S. Additivity properties of a Gaussian bosonic channel. Phys. Rev. A 69, 062307 (2004).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

V.G. acknowledges support from the Italian Ministry of Instruction, University and Research (MIUR) through FIRB-IDEAS project no. RBID08B3FM, and L.M. from European project COQUIT (Collective Quantum Operations for Information Technologies). S.L. and J.H.S. acknowledge support from a US Office of Naval Research (ONR) Basic Research Challenge grant.

Author information

Authors and Affiliations

  1. NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, piazza dei Cavalieri 7, Pisa, I-56126, Italy

    Vittorio Giovannetti

  2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139, Massachusetts, USA

    Seth Lloyd

  3. Dipartimento Fisica ‘A. Volta’, University of Pavia, via Bassi 6, Pavia, I-27100, Italy

    Lorenzo Maccone

  4. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, 02139, Massachusetts, USA

    Jeffrey H. Shapiro

Authors
  1. Vittorio Giovannetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seth Lloyd
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lorenzo Maccone
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jeffrey H. Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the derivation and writing of the manuscript.

Corresponding author

Correspondence to Vittorio Giovannetti.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 427 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giovannetti, V., Lloyd, S., Maccone, L. et al. Electromagnetic channel capacity for practical purposes. Nature Photon 7, 834–838 (2013). https://doi.org/10.1038/nphoton.2013.193

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphoton.2013.193

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing