Quantum key distribution using continuous-variable non-Gaussian states


In this work, we present a quantum key distribution protocol using continuous-variable non-Gaussian states, homodyne detection and post-selection. The employed signal states are the photon added then subtracted coherent states (PASCS) in which one photon is added and subsequently one photon is subtracted from the field. We analyze the performance of our protocol, compared with a coherent state-based protocol, for two different attacks that could be carried out by the eavesdropper (Eve). We calculate the secret key rate transmission in a lossy line for a superior channel (beam-splitter) attack, and we show that we may increase the secret key generation rate by using the non-Gaussian PASCS rather than coherent states. We also consider the simultaneous quadrature measurement (intercept-resend) attack, and we show that the efficiency of Eve’s attack is substantially reduced if PASCS are used as signal states.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    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, pp. 175 (1984)

  2. 2.

    Scarani, V., et al.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301 (2009)

    Article  ADS  Google Scholar 

  3. 3.

    Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441 (2000)

    Article  ADS  Google Scholar 

  4. 4.

    Stucki, D., et al.: High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres. New J. Phys. 11, 075003 (2009)

    Article  ADS  Google Scholar 

  5. 5.

    Ralph, T.C.: Continuous variable quantum cryptography. Phys. Rev. A 61, 010303(R) (1999)

    Article  MathSciNet  Google Scholar 

  6. 6.

    Hillery, M.: Quantum cryptography with squeezed states. Phys. Rev. A 61, 022309 (2000)

    Article  ADS  Google Scholar 

  7. 7.

    Cerf, N.J., Lévy, M., Van Assche, G.: Quantum distribution of Gaussian keys using squeezed states. Phys. Rev. A 63, 052311 (2001)

    Article  ADS  Google Scholar 

  8. 8.

    Horak, P.: The role of squeezing in quantum key distribution based on homodyne detection and post-selection. J. Mod. Opt. 51, 1249 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. 9.

    Grosshans, F., Grangier, P.: Continuous variable quantum cryptography using coherent states. Phys. Rev. Lett. 88, 057902 (2002)

    Article  ADS  Google Scholar 

  10. 10.

    Grosshans, F., et al.: Quantum key distribution using Gaussian-modulated coherent states. Nature 421, 238 (2003)

    Article  ADS  Google Scholar 

  11. 11.

    Vidiella-Barranco, A., Borelli, L.F.M.: Continuous variable quantum key distribution using polarized coherent states. Int. J. Mod. Phys. B 20, 1287 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. 12.

    Lorenz, S., Korolkova, N., Leuchs, G.: Continuous-variable quantum key distribution using polarization encoding and post selection. Appl. Phys. B 79, 273 (2004)

    Article  Google Scholar 

  13. 13.

    Namiki, R., Hirano, T.: Security of quantum cryptography using balanced homodyne detection. Phys. Rev. A 67, 022308 (2003)

    Article  ADS  Google Scholar 

  14. 14.

    Leverrier, A., Grangier, P.: Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation. Phys. Rev. 102, 180504 (2009)

    Google Scholar 

  15. 15.

    Peev, M., et al.: The SECOQC quantum key distribution network in Vienna. New J. Phys. 11, 075001 (2009)

    Article  ADS  Google Scholar 

  16. 16.

    Sasaki, M., et al.: Field test of quantum key distribution in the Tokyo QKD Network. Opt. Express 19, 10387 (2011)

    Article  ADS  Google Scholar 

  17. 17.

    Jouguet, Paul, et al.: Experimental demonstration of long-distance continuous-variable quantum key distribution. Nat. Photon. 7, 378 (2013)

    Article  ADS  Google Scholar 

  18. 18.

    Leverrier, A. et al.: Quantum communications with Gaussian and non-Gaussian states of light. In: International Conference on Quantum Information, OSA Technical Digest (CD) (Optical Society of America, 2011), paper QMF1. http://www.opticsinfobase.org/abstract.cfm?URI=ICQI-2011-QMF1

  19. 19.

    Pariggi, V., Zavatta, A., Kim, M., Bellini, M.: Probing quantum commutation rules by addition and subtraction of single photons to/from a light field. Science 317, 1890 (2007)

    Article  ADS  Google Scholar 

  20. 20.

    Silberhorn, C., Ralph, T.C., Lütkenhaus, N., Leuchs, G.: Continuous variable quantum cryptography: beating the 3 dB loss limit. Phys. Rev. Lett. 89, 167901 (2002)

    Article  ADS  Google Scholar 

  21. 21.

    Lütkenhaus, N.: Security against eavesdropping in quantum cryptography. Phys. Rev. A 54, 97 (1996)

    Article  ADS  Google Scholar 

  22. 22.

    Dakna, M., Knöll, L., Welsch, D.-G.: Quantum state engineering using conditional measurement on a beam splitter. Eur. Phys. J. D 3, 295 (1998)

    Article  ADS  Google Scholar 

  23. 23.

    Agarwal, G.S., Tara, K.: Nonclassical properties of states generated by the excitations on a coherent state. Phys. Rev. A 43, 492 (1991)

    Article  ADS  Google Scholar 

  24. 24.

    Zavatta, A., Viciani, S., Bellini, M.: Quantum-to-classical transition with single-photon-added coherent states of light. Science 306, 660 (2004)

    Article  ADS  Google Scholar 

  25. 25.

    Wang, Z., Yuan, H., Fan, H.: Nonclassicality of the photon addition-then-subtraction coherent state and its decoherence in the photon-loss channel. J. Opt. Soc. Am. B 28, 1964 (2011)

    Article  ADS  Google Scholar 

  26. 26.

    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)

    Article  ADS  Google Scholar 

  27. 27.

    Hillery, M., et al.: Distribution functions in physics: fundamentals. Phys. Rep. 106, 121 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  28. 28.

    Wu, J.W.: Violation of Bells inequalities and two-mode quantum-optical state measurement. Phys. Rev. A 61, 022111 (2000)

    Article  ADS  Google Scholar 

  29. 29.

    Ou, Z.Y., Hong, C.K., Mandel, L.: Relation between input and output states for a beam splitter. Opt. Commun. 63, 118 (1987)

    Article  ADS  Google Scholar 

  30. 30.

    Lütkenhaus, N.: Security against individual attacks for realistic quantum key distribution. Phys. Rev. A 61, 052304 (2000)

    Article  ADS  Google Scholar 

  31. 31.

    Shannon, C.E.: A Mathematical theory of communication. Bell Syst. Tech. J. 27, 379 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  32. 32.

    Becir, A., Wahiddin, M.R.: Phase coherent states for enhancing the performance of continuous variable quantum key distribution. J. Phys. Soc. Jpn. 81, 034005 (2012)

    Article  ADS  Google Scholar 

Download references


This work was partially supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico—INCT of Quantum Information), FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo—CePOF of Optics and Photonics) and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior), Brazil.

Author information



Corresponding author

Correspondence to A. Vidiella-Barranco.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Borelli, L.F.M., Aguiar, L.S., Roversi, J.A. et al. Quantum key distribution using continuous-variable non-Gaussian states. Quantum Inf Process 15, 893–904 (2016). https://doi.org/10.1007/s11128-015-1193-8

Download citation


  • Quantum cryptography
  • Continuous variables
  • Non-Gaussian states