Quantum communication with Gaussian channels of zero quantum capacity


As with classical information1,2, error-correcting codes enable reliable transmission of quantum information through noisy or lossy channels3,4,5. In contrast to classical theory, imperfect quantum channels exhibit a strong kind of synergy: pairs of discrete memoryless quantum channels exist, each of zero quantum capacity, which acquire positive quantum capacity when used together6. Here, we show that this ‘superactivation’ phenomenon also occurs in the more realistic setting of optical channels with attenuation and Gaussian noise7,8. This paves the way for its experimental realization and application in real-world communications systems.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Symplectic extensions of Gaussian channels.
Figure 2: A family of examples.
Figure 3: Superactivation for a wide range of parameters.
Figure 4: Generating the Hamiltonian.


  1. 1

    Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

  2. 2

    Cover, T. M. & Thomas, J. A. Elements of Information Theory (Wiley, 1991).

  3. 3

    Devetak, I. The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51, 44–55 (2005).

  4. 4

    Shor, P. W. The quantum channel capacity and coherent information, in MSRI Workshop on Quantum Computation (Berkeley, 2002), available at http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/.

  5. 5

    Lloyd, S. Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613–1622 (1997).

  6. 6

    Smith, G. & Yard, J. Quantum communication with zero-capacity channels. Science 321, 1812–1815 (2008).

  7. 7

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

  8. 8

    Harrington, J. & Preskill, J. Achievable rates for the Gaussian quantum channel. Phys. Rev. A 64, 062301 (2001).

  9. 9

    Bennett, C. H. & Brassard, G. Quantum cryptography: public key distribution and coin tossing, in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, 175–179 (IEEE Press, 1984).

  10. 10

    Eisert, J. & Wolf, M. M. Gaussian quantum channels, in Quantum Information with Continuous Variables of Atoms and Light, 23–42 (Imperial College Press, 2007).

  11. 11

    DiGuglielmo, J. et al. Preparing the bound instance of quantum entanglement. Preprint at http://www.arxiv.org/abs/1006.4651 (2010).

  12. 12

    Wolf, M. M., Perez-Garcia, D. & Giedke, G. Quantum capacities of bosonic channels. Phys. Rev. Lett. 98, 130501 (2007).

  13. 13

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

  14. 14

    Horodecki, P. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333–339 (1997).

  15. 15

    Horodecki, P., Horodecki, M. & Horodecki, R. Bound entanglement can be activated. Phys. Rev. Lett. 82, 1056–1059 (1999).

  16. 16

    Shor, P. W., Smolin, J. A. & Terhal, B. M. Nonadditivity of bipartite distillable entanglement follows from a conjecture on bound entangled Werner states. Phys. Rev. Lett. 86, 2681–2684 (2000).

  17. 17

    Pirandola, S., Mancini, S., Lloyd, S. & Braunstein, S. L. Continuous-variable quantum cryptography using two-way quantum communication. Nature Phys. 4, 726–730 (2008).

  18. 18

    Hastings, M. Superadditivity of communication capacity using entangled inputs. Nature Phys. 5, 255–257 (2009).

  19. 19

    Smith, G. & Smolin, J. A. Extensive nonadditivity of privacy. Phys. Rev. Lett. 103, 120503 (2009).

  20. 20

    Li, K., Winter, A., Zou, X. & Guo, G-C. Private capacity of quantum channels is not additive. Phys. Rev. Lett. 103, 120501 (2009).

  21. 21

    Czekaj, L. & Horodecki, P. Purely quantum superadditivity of classical capacities of quantum multiple access channels. Phys. Rev. Lett. 102, 110505 (2009).

  22. 22

    Czekaj, L., Korbicz, J. K., Chhajlany, R. W. & Horodecki, P. Quantum superadditivity in linear optics networks: sending bits via multiple-access Gaussian channels. Phys. Rev. A 82, 020302(R)10.1103/PhysRevA.82.020302 (2010).

  23. 23

    Devetak, I. & Shor, P. W. The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256, 287–303 (2005).

  24. 24

    Horodecki, M., Horodecki, P. & Horodecki, R. Mixed-state entanglement and distillation: is there a ‘bound’ entanglement in nature? Phys. Rev. Lett. 80, 5239–5242 (1998).

  25. 25

    Peres, A. Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996).

  26. 26

    Horodecki, M., Horodecki, P. & Horodecki, R. Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996).

  27. 27

    Werner, R. F. & Wolf, M. M. Bound entangled Gaussian states. Phys. Rev. Lett. 86, 3658–3661 (2001).

  28. 28

    Vahlbruch, H. et al. Observation of squeezed light with 10-dB quantum-noise reduction. Phys. Rev. Lett. 100, 033602 (2008).

  29. 29

    Giedke, G., Duan, R., Cirac, J. I. & Zoller, P. Distillability criterion for all bipartite Gaussian states. Quant. Inf. Comp. 1, 79–86 (2001).

  30. 30

    Horodecki, K., Horodecki, M., Horodecki, P. & Oppenheim, J. Secure key from bound entanglement. Phys. Rev. Lett. 94, 160502 (2005).

Download references


The authors thank J. Eisert and M.M. Wolf for helpful advice in the early stages of this work and C.H. Bennett for many useful suggestions. G.S. and J.Y. are especially grateful to the Institut Mittag-Leffler, where some of this work was performed, for their hospitality. J.Y.'s research was supported by grants through the Laboratory Directed Research and Development programme of the US Department of Energy. G.S. and J.A.S. were supported by the Defense Advanced Research Projects Agency (QUEST contract HR0011-09-C-0047).

Author information




G.S., J.A.S. and J.Y. designed the research, carried out the research, and wrote the manuscript.

Corresponding author

Correspondence to Graeme Smith.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 300 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Smith, G., Smolin, J. & Yard, J. Quantum communication with Gaussian channels of zero quantum capacity. Nature Photon 5, 624–627 (2011). https://doi.org/10.1038/nphoton.2011.203

Download citation

Further reading