Communications in Mathematical Physics

, Volume 334, Issue 3, pp 1553–1571 | Cite as

A Solution of Gaussian Optimizer Conjecture for Quantum Channels

  • V. Giovannetti
  • A. S. Holevo
  • R. García-Patrón


The long-standing conjectures of the optimality of Gaussian inputs and additivity are solved for a broad class of gauge-covariant or contravariant bosonic Gaussian channels (which includes in particular thermal, additive classical noise, and amplifier channels) restricting to the class of states with finite second moments. We show that the vacuum is the input state which minimizes the entropy at the output of such channels. This allows us to show also that the classical capacity of these channels (under the input energy constraint) is additive and is achieved by Gaussian encodings.


Coherent State Quantum Channel Partial Isometry Gaussian Channel Classical Capacity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cover T.M., Thomas J.A.: Elements of Information Theory. Wiley, New York (1968)Google Scholar
  2. 2.
    Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379 (1948)Google Scholar
  3. 3.
    Bennett C.H., Shor P.W.: Quantum information theory. IEEE Trans. Inform. Theory 44, 2724 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Holevo A.S., Giovannetti V.: Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75, 046001 (2012)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inform Theory 44, 269 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Schumacher B., Westmoreland W.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131 (1997)CrossRefADSGoogle Scholar
  7. 7.
    Holevo A.S., Werner R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001)CrossRefADSGoogle Scholar
  8. 8.
    Caves C.M., Drummond P.B.: Quantum limits on bosonic communication rates. Rev. Mod. Phys. 66, 481 (1994)CrossRefADSGoogle Scholar
  9. 9.
    Holevo A.S.: Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Holevo A.S., Shirokov M.E.: Continuous ensembles and the χ-capacity of infinite-dimensional channels. Probab. Theory Appl. 50, 86–98 (2005)CrossRefGoogle Scholar
  11. 11.
    Holevo A.S.: Quantum systems, channels, information. A mathematical introduction. De Gruyter, Berlin (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    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)CrossRefADSGoogle Scholar
  13. 13.
    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)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Giovannetti V., Guha S., Lloyd S., Maccone L., Shapiro J.H.: Minimum bosonic channel output entropies. AIP Conf. Proc. 734, 21 (2004)CrossRefADSGoogle Scholar
  15. 15.
    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). arXiv:1111.1986 [quant-ph]
  16. 16.
    König R., Smith G.: Classical capacity of quantum thermal noise channels to within 1.45 bits. Phys. Rev. Lett. 110, 040501 (2013)CrossRefGoogle Scholar
  17. 17.
    König R., Smith G.: Limits on classical communication from quantum entropy power inequalities. Nat. Photon. 7, 142 (2013)CrossRefADSGoogle Scholar
  18. 18.
    Giovannetti, V., Lloyd, S., Maccone, L., Shapiro, J.H.: Electromagnetic channel capacity for practical purposes. Nat. Photon. 7, 834–838 (2013)Google Scholar
  19. 19.
    Giovannetti, V., Guha, S., Lloyd, S., Maccone, L., Shapiro, J.H., Yuen, H.P.: Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92, 027902 (2004). arXiv:quant-ph/0308012
  20. 20.
    Mari A., Giovannetti V., Holevo A.S.: Quantum state majorization at the output of bosonic Gaussian channels. Nat. Comms. 5, 3826 (2014)CrossRefADSGoogle Scholar
  21. 21.
    Walls G.J., Milburn D.F.: Quantum Optics. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Heinosaari, T., Holevo, A.S., Wolf, M.M.: The semigroup structure of Gaussian channels. Quantum Inf. Comp. 10, 0619–0635 (2010). arXiv:0909.0408
  23. 23.
    Caruso, F. Giovannetti, V., Holevo, A.S.: One-mode bosonic Gaussian channels: a full weak-degradability classification, New J. Phys. 8, 310 (2006). arXiv:quant-ph/0609013
  24. 24.
    Shor, P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43, 4334–4340 (2002). arXiv:quant-ph/0201149
  25. 25.
    Shirokov, M.E.: The convex closure of the output entropy of infinite dimensional channels and the additivity problem. Russ. Math. Surv. 61, 1186–1188 (2006). arXiv:quant-ph/0608090
  26. 26.
    Holevo, A.S., Shirokov, M.E.: On Shor’s channel extension and constrained channels. Commun. Math. Phys. 249, 417–430 (2004). arXiv:quant-ph/0306196
  27. 27.
    Holevo, A.S.: On complementary channels and the additivity problem. Probab. Theory Appl. 51, 133–134 (2006). arXiv:quant-ph/0509101
  28. 28.
    King, C., Matsumoto, K., Nathanson, M., Ruskai, M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. Markov Process Relat. Fields 13, 391–423 (2007). arXiv:quant-ph/0509126
  29. 29.
    Garcia-Patron, R., Navarrete-Benlloch, C., Lloyd, S., Shapiro, J.H., Cerf, N.J.: The Holy Grail of Quantum Optical Communication. In: Schmiedmayer, J. (ed.) Proceedings of the 11th International Conference on Quantum Communication Measurement and Computing, AIP, New York (to be published)Google Scholar
  30. 30.
    Holevo A.S., Sohma M., Hirota O.: Capacity of quantum Gaussian channels. Phys. Rev. A 59, 1820 (1999)CrossRefADSGoogle Scholar
  31. 31.
    Giovannetti V., Lloyd S., Maccone L., Shor P.W.: Broadband channel capacities. Phys. Rev. A 68, 062323 (2003)CrossRefADSGoogle Scholar
  32. 32.
    Schäfer J., Daems D., Karpov E., Cerf N.J.: Capacity of a bosonic memory channel with Gauss–Markov noise. Phys. Rev. A 80, 062313 (2009)CrossRefADSGoogle Scholar
  33. 33.
    Pilyavets O.V., Lupo C., Mancini S.: Methods for estimating capacities and rates of Gaussian quantum channels. IEEE Trans. Inf. Theory 58, 6126–6164 (2012)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Wehrl A.: General properties of entropy. Rev. Mod. Phys. 50, 221–260 (1978)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • V. Giovannetti
    • 1
  • A. S. Holevo
    • 2
    • 3
  • R. García-Patrón
    • 4
    • 5
  1. 1.NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNRPisaItaly
  2. 2.Steklov Mathematical Institute, RASMoscowRussia
  3. 3.National Research University Higher School of Economics (HSE)MoscowRussia
  4. 4.Center for Quantum Information and Communication, Ecole Polytechnique de Bruxelles, CP 165Universite Libre de BruxellesBruxellesBelgium
  5. 5.Max-Planck Institut für QuantenoptikGarchingGermany

Personalised recommendations