Optical channels, such as fibres or free-space links, are ubiquitous in today's telecommunication networks. They rely on the electromagnetic field associated with photons to carry information from one point to another in space. A complete physical model of these channels must necessarily take quantum effects into account to determine their ultimate performances. Single-mode, phase-insensitive bosonic Gaussian channels have been extensively studied over past decades, given their importance for practical applications. In spite of this, a long-standing unsolved conjecture on the optimality of Gaussian encodings has prevented finding their classical communication capacity. Here, this conjecture is solved by proving that the vacuum state achieves the minimum output entropy of these channels. This establishes the ultimate achievable bit rate under an energy constraint, as well as the long awaited proof that the single-letter classical capacity of these channels is additive.
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Gordon, J. P. Quantum effects in communications systems. Proc. IRE 1898–1908 (1962).
Caves, C. M. & Drummond, P. B. Quantum limits on bosonic communication rates. Rev. Mod. Phys. 66, 481–537 (1994).
Schumacher, B. & Westmoreland, W. D. Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997).
Holevo, A. S. The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998).
Bennett, C. H. & Shor, P. W. Quantum information theory. IEEE Trans. Inf. Theory 44, 2724–2742 (1998).
Holevo, A. S. & Werner, R. F. Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001).
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).
Giovannetti, V., Guha, S., Lloyd, S., Maccone, L. & Shapiro, J. H. Minimum bosonic channel output entropies. AIP Conf. Proc. 734, 21–24 (2004).
Giovannetti, V., Holevo, A. S., Lloyd, S. & Maccone, L. Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results. J. Phys. A 43, 415305 (2010).
García-Patrón, 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).
García-Patrón, R., Navarrete-Benlloch, C., Lloyd, S., Shapiro, J. H. & Cerf, N. J. The holy grail of quantum optical communication. in Proc. 11th International Conference on Quantum Communication, Measurement and Computing, 68 (2012).
Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012).
König, R. & Smith, G. Classical capacity of quantum thermal noise channels to within 1.45 bits. Phys. Rev. Lett. 110, 040501 (2013).
König, R. & Smith, G. Limits on classical communication from quantum entropy power inequalities. Nature Photon. 7, 142–146 (2013).
Giovannetti, V., Lloyd, S., Maccone, L. & Shapiro, J. H. Electromagnetic channel capacity for practical purposes. Nature Photon. 7, 834–838 (2013).
Giovannetti, V. & Lloyd, S. Additivity properties of a Gaussian channel. Phys. Rev. A 69, 062307 (2004).
Giovannetti, V., Lloyd, S., Maccone, L., Shapiro, J. H. & Yen, B. J. Minimum Rényi and Wehrl entropies at the output of bosonic channels. Phys. Rev. A 70, 022328 (2004).
Serafini, A., Eisert, J. & Wolf, M. M. Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs. Phys. Rev. A 71, 012320 (2005).
Guha, S., Shapiro, J. H. & Erkmen, B. I. Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture. Phys. Rev. A 76, 032303 (2007).
Giovannetti, V., Holevo, A. S. & García-Patrón, R. A solution of the Gaussian optimizer conjecture. Preprint at http://arxiv.org/abs/1312.2251 (2013).
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).
Modi, K., Brodutch, A., Cable, H., Paterek, T. & Vedral, V. The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012).
Pirandola, S., Spedalieri, G., Braunstein, S. L., Cerf, N. J. & Lloyd, S. Optimality of Gaussian discord. Preprint at http://arxiv.org/abs/1309.2215v2 (2014).
Cover, T. M. & Thomas, J. A. Elements of Information Theory (Wiley, 1968).
Holevo, A. S. Quantum Systems, Channels, Information. A Mathematical Introduction (De Gruyter, 2012).
Braunstein, S. L. & van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005).
Cerf, N. J., Leuchs, G. & Polzik, E. S. (eds) Quantum Information with Continuous Variables of Atoms and Light (Imperial College Press, 2007).
Walls, D. F. & Milburn, G. J. Quantum Optics (Springer, 1994).
Schäfer, J., Karpov, E., García-Patrón, R., Pilyavets, O. V. & Cerf, N. J. Equivalence relations for the classical capacity of single-mode Gaussian quantum channels. Phys. Rev. Lett. 111, 030503 (2013).
Caruso, F. & Giovannetti, V. Degradability of bosonic Gaussian channels. Phys. Rev. A 74, 062307 (2006).
Holevo, A. S. One-mode quantum Gaussian channels: structure and quantum capacity. Probl. Inf. Transm. 43, 1–11 (2007).
Caruso, F., Giovannetti, V. & Holevo, A. S. One-mode bosonic Gaussian channels: a full weak-degradability classification. New J. Phys. 8, 310 (2006).
Giovannetti, V. et al. Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92, 027902 (2004).
Holevo, A. S. Quantum coding theorems. Russ. Math. Surveys 53, 1295–1331 (1998).
Holevo, A. S. & Shirokov, M. E. Continuous ensembles and the χ-capacity of infinite-dimensional channels. Theory Prob. Appl. 50, 86–98 (2006).
Wehrl, A. General properties of entropy. Rev. Mod. Phys. 50, 221–260 (1978).
Holevo, A. S. Complementary channels and the additivity problem. Theory Prob. Appl. 51, 92–100 (2007).
King, C., Matsumoto, K., Natanson, M. & Ruskai, M. B. Properties of conjugate channels with applications to additivity and multiplicativity. Markov Process. Relat. Fields 13, 391–423 (2007).
Holevo, A. S. Entanglement-breaking channels in infinite dimensions. Probl. Inf. Transm. 44, 171–184 (2008).
Lupo, C., Pilyavets, O. V. & Mancini, S. Capacities of lossy bosonic channel with correlated noise. New J. Phys. 11, 063023 (2009).
Giedke, G., Wolf, M. M., Krüger, O., Werner, R. F. & Cirac, J. I. Entanglement of formation for symmetric Gaussian states. Phys. Rev. Lett. 91, 107901 (2003).
Wolf, M. M., Giedke, G., Krüger, O., Werner, R. F. & Cirac, J. I. Gaussian entanglement of formation. Phys. Rev. A 69, 052320 (2004).
Matsumoto, K., Shimono, T. & Winter, A. Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. Commun. Math. Phys. 246, 427–442 (2004).
The authors thank L. Ambrosio, A. Mari, C. Navarrete-Benloch, J. Oppenheim, M.E. Shirokov, R.F. Werner and A. Winter for comments and discussions. The authors acknowledge support and the catalysing role of the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; an important part of this work was conducted when attending the Newton Institute programme ‘Mathematical Challenges in Quantum Information’. A.H. acknowledges the Rothschild Distinguished Visiting Fellowship, which enabled him to participate in the programme, and partial support from RAS Fundamental Research Programs (the Russian Quantum Center). N.J.C. and R.G.-P. acknowledge financial support from the Belgian Fonds de la Recherche Scientifique (F.R.S.–FNRS) under projects T.0199.13 and HIPERCOM (High-Performance Coherent Quantum Communications), as well as from the ‘Interuniversity Attraction Poles’ programme of the Belgian Science Policy Office (grant no. IAP P7-35 Photonics@be). R.G.-P. acknowledges financial support from the Alexander von Humboldt Foundation, the F.R.S.–FNRS and Back-to-Belgium grant from the Belgian Federal Science Policy.
The authors declare no competing financial interests.
About this article
Cite this article
Giovannetti, V., García-Patrón, R., Cerf, N. et al. Ultimate classical communication rates of quantum optical channels. Nature Photon 8, 796–800 (2014). https://doi.org/10.1038/nphoton.2014.216
npj Quantum Information (2021)
Communications in Mathematical Physics (2021)
npj Quantum Information (2019)
Scientific Reports (2019)
npj Quantum Information (2019)