Unidimensional Continuous-variable Quantum Key Distribution Based on Basis-encoding Coherent States Protocol

Abstract

We propose the unidimensional continuous-variable quantum key distribution(UDCVQKD) protocol based on the basis-encoding of Gaussian modulated coherent states. the UDCVQKD protocol disregards the necessity in one of the quadrature modulations in coherent states. On that basis, we propose our scheme by encoding the secret keys on the either randomly selected measurement bases: the phase quadrature (X) or the amplitude quadrature (P), in order to slightly weaken the effects of reconciliation efficiency and channel excess noise compare to the present UCVQKD protocol. The new protocol with a view to simplify the precedent unidimensional protocols in the decoding procedure, meanwhile ensure the security of the quantum communication.

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References

  1. 1.

    Huang, P., Huang, J., Wang, T., Li, H., Huang, D., Zeng, G.: Robust continuous-variable quantum key distribution against practical attacks. Phys. Rev. A 95, 052302 (2017)

    ADS  Article  Google Scholar 

  2. 2.

    Madsen, L.S., Usenko, V.C., Lassen, M., Filip, R., Andersen, U.L.: Continuous Variable Quantum key Distribution with Modulated Entangled States, p 1083. Nature Communications, London (2012)

    Google Scholar 

  3. 3.

    Jouguet, P., Kunz-Jacques, S., Diamanti, E.: Analysis of imperfections in practical continuous-variable quantum key distribution. Phys. Rev. A 86, 032309 (2012)

    ADS  Article  Google Scholar 

  4. 4.

    Weedbrook, C., Pirandola, S., Ralph, T.: Continuous-variable quantum key distribution using thermal states. Phys. Rev. A 86, 022318 (2012)

    ADS  Article  Google Scholar 

  5. 5.

    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)

    ADS  Article  Google Scholar 

  6. 6.

    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers Systems and Signal Processing, pp. 176–178 (1984)

  7. 7.

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

    ADS  Article  Google Scholar 

  8. 8.

    Waks, E., Zeevi, A., Yamamoto, Y.: Security of quantum key distribution with entangled photons against individual attacks. Phys. Rev. A 65, 052310 (2002)

    ADS  Article  Google Scholar 

  9. 9.

    Antonio, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-Independent Security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)

    ADS  Article  Google Scholar 

  10. 10.

    Grosshans, F.: Collective attacks and unconditional security in continuous variable quantum key distribution. Phys. Rev. Lett. 94, 020504 (2005)

    ADS  Article  Google Scholar 

  11. 11.

    Mertz, M., Kampermann, H., Bratzik, S., Dagmar, B.: Secret key rates for coherent attacks. Phys. Rev. A 87, 012315 (2013)

    ADS  Article  Google Scholar 

  12. 12.

    Furrer, F., Franz, T., Berta, M., Leverrier, A., Scholz, V.B., Tomamichel, M., Werner, R.F.: Continuous variable quantum key distribution: Finite-key analysis of composable security against coherent attacks. Phys. Rev. Lett. 109, 100502 (2014)

    Article  Google Scholar 

  13. 13.

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

    ADS  Article  Google Scholar 

  14. 14.

    Leverrier, A., Grosshans, F., Grangier, P.: Finite-size analysis of a continuous-variable quantum key distribution. Phys. Rev. A 81, 062343, 2 (2010)

    ADS  Google Scholar 

  15. 15.

    Jouguet, P., Kunz-Jacques, S., Leverrier, A.: Long-distance continuous-variable quantum key distribution with a Gaussian modulation. Phys. Rev. A 84, 062317 (2011)

    ADS  Article  Google Scholar 

  16. 16.

    Pirandola, S., Braunstein, S.L., Lloyd, S.: Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography. Phys. Rev. Lett. 101, 200504 (2008)

    ADS  Article  Google Scholar 

  17. 17.

    Navascues, M., Grosshans, F., Acin, A.: Optimalityof gaussian attacks in Continuous-Variable quantum cryptography. Phys. Rev. Lett. PRL 97, 190502 (2006)

    ADS  Article  Google Scholar 

  18. 18.

    Kumar, R., Qin, H., Alléaume, R.: Coexistence of continuous variable QKD with intense DWDM classical channels. New J. Phys. 17, 043027 (2015)

    ADS  Article  Google Scholar 

  19. 19.

    Grosshans, F., Assche, G.V., Wenger, J., Brouri, R., Cerf, N.J., Grangier, P.: Quantum key distribution using gaussian-modulated coherent states. Nature 421, 238 (2003)

    ADS  Article  Google Scholar 

  20. 20.

    Shen, Y., Peng, X., Yang, J., Guo, H.: Continuous-variable quantum key distribution with Gaussian source noise. Phys. Rev. A 83, 052304 (2011)

    ADS  Article  Google Scholar 

  21. 21.

    Huang, D., Huang, P., Li, H., Wang, T., Zhou, Y., Zeng, G.: Field demonstration of a continuous-variable quantum key distribution network. Opt. Lett. 41, 3511 (2016)

    ADS  Article  Google Scholar 

  22. 22.

    Martinez-Mateo, J., Elkouss, D., Martin, V.: Key reconciliation for high performance quantum key distribution. Sci. Rep. 3, 1576 (2013)

    ADS  Article  Google Scholar 

  23. 23.

    Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information. Rev. Mod. Phys. 84(2), 621 (2012)

    ADS  Article  Google Scholar 

  24. 24.

    Usenko, V.C., Grosshans, F.: Unidimensional continuous-variable quantum key distribution. Phys Rev. A 92, 062337 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Zhang, H., Ruan, X., Wu, X.: Plug-and-play unidimensional continuous-variable quantum key distribution. Quantum Inf Process 18(5) (2019)

  26. 26.

    Usenko, V.C.: Unidimensional continuous-variable quantum key distribution using squeezed states. Phys. Rev. A 98(3), 032321 (2018)

    ADS  Article  Google Scholar 

  27. 27.

    liao, Q., Guo, Y., Xie, C., Huang, D., Huang, P., Zeng, G.: Composable security of unidimensional continuous-variable quantum key distribution. Quantum Inf. Process 17(5), 113 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Huang, P., Huang, J., Zhang, Z., Zeng, G.: Quantum key distribution using basis encoding of Gaussian-modulated coherent states. Phys. Rev. A 97, 042311 (2018)

    ADS  Article  Google Scholar 

  29. 29.

    Gessner, M., Pezze, L., Smerzi, A.: Efficient entanglement criteria for discrete, continuous, and hybrid variables. Phys. Rev. A 94(2), 020101 (2016)

    ADS  Article  Google Scholar 

  30. 30.

    Huang, D., Huang, P., Lin, D., Zeng, G.: Long-distance continuous-variable quantum key distribution by controlling excess noise. Sci. Rep. 6, 19201 (2016)

    ADS  Article  Google Scholar 

  31. 31.

    Brádler, K., Weedbrook, C.: Security proof of continuous-variable quantum key distribution using three coherent states. Phys. Rev. A97, 022310 (2018)

    ADS  Article  Google Scholar 

  32. 32.

    Navascués, M., Acín, A.: Security bounds for continuous variables quantum key distribution. Phys. Rev. Lett. 94, 020505 (2008)

    Article  Google Scholar 

  33. 33.

    Lodewyck, J., Bloch, M.: Quantum key distribution over 25 km with an all-fiber continuous-variable system. Phy. Rev. A 76, 042305 (2007)

    ADS  Article  Google Scholar 

  34. 34.

    Tobias, G., Vitus, H., Jrg, D., Fabian, F., Torsten, F., Christoph, P., Werner, R.F., Roman, S.: Implementation of continuous-variable quantum key distribution with composable and one-sided-device-independent security against coherent attacks. Nat. Commun. 6, 8795 (2015)

    Article  Google Scholar 

  35. 35.

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  36. 36.

    Grosshans, F., Cerf, N.J.: Continuous-Variable Quantum cryptography is secure against Non-Gaussian attacks. Phys. Rev. Lett. 92, 047905 (2004)

    ADS  Article  Google Scholar 

  37. 37.

    Scarani, V., Bechmann-Pasquinucci, H., Nicolas, J.C., Dušek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81 (2009)

  38. 38.

    Thearle, O., Assad, S.M., Symul, T.: Estimation of output-channel noise for continuous-variable quantum key distribution. Phys. Rev. A 93(4), 042343 (2016)

    ADS  Article  Google Scholar 

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Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant No.61801522), and National Nature Science Foundation of Hunan Province, China (Grant No. 2019JJ40352).

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Correspondence to Duan Huang.

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Appendix

Appendix

In the BEUD CVQKD protocols, Alice sends the single modulated key information through the lossy and noisy channel to Bob. We focus on the Gaussian attacks in this paper, in which we assume the parameters εx = 𝜖p = ε as the channel excess noise, we suppose that Eve would replace the quantum channel with a perfect displacement with no noise and lose, then she entangle ρb0 with her own states ∣ψE〉 and resends to Bob as camouflage, so that the entangled ρb0 would be the same as the state that passed through the noisy and lossy channel. Suppose that Bob uses homodyne detection with the electronics noise v and efficiency η, then the quadratures received by Bob can be calculated as

$$ \begin{array}{@{}rcl@{}} X_{B} &=& \sqrt{\eta T}X_{A}+\sqrt{\eta T}X_{ex}+\delta X_{v}+\delta X_{el},\\ P_{B} &=& \sqrt{\eta T}P_{A}+\sqrt{\eta T}P_{ex}+\delta P_{v}+\delta P_{el}. \end{array} $$
(18)

Where Xex(Pex), δXv(δPv), δXel(δPel) are generated from homodyne detection and denoted to the channel excess noise, shot noise and electronic noise, respectively. where\( \left \langle {X_{ex}^{2}} \right \rangle \) = \( \left \langle P_{ex}^{2} \right \rangle \) = εc,\( \left \langle {X_{v}^{2}} \right \rangle \) = \( \left \langle {P_{v}^{2}} \right \rangle \) = 1, and\( \left \langle \delta X_{el}^{2} \right \rangle \) = \( \left \langle \delta P_{el}^{2} \right \rangle \) = v in shot noise units, respectively.

After identify the quadrature of the output mode in proposed scheme, we will calculate the QBER of Alice’s decoding according to Bob’s measurement results. Because of the symmetry of the jugement rule for either X or P quadratures, we neglect the unmodulated quadrature and just focus on the case of XA > PA for simplicity. Accroding to (18), the inequality \({{\upbeta }_{A}^{x}}X_{B}<C_{A}\) and \({{\upbeta }_{A}^{p}}P_{B}>C_{A}\) can be expanded to [28].

$$ \begin{array}{@{}rcl@{}} X_{A}-P_{A} &=& -2\delta X_{ex}-\frac{2}{\sqrt{\eta T}}(\delta X_{v}+\delta X_{el}),\\ X_{A}-P_{A} &=& -2\delta P_{ex}-\frac{2}{\sqrt{\eta T}}(\delta P_{v}+\delta P_{el}). \end{array} $$
(19)

We set M = XAPA, \(\text {N}_{\text {x}}=-2\delta X_{ex}-\frac {2}{\sqrt {\eta T}}(\delta X_{v}+\delta X_{el})\) and \(\text {N}_{\text {p}}=-2\delta \text {P}_{\text {ex}}-\frac {2}{\sqrt {\eta T}}(\delta P_{v}+\delta P_{el})\), the variables follow the normal distribution as

$$ M\sim(0,{\sigma_{m}^{2}}),N_{x},N_{p}\sim(0,{\sigma_{n}^{2}}). $$
(20)

where \({\sigma _{m}^{2}}=2V_{M}\) and \({\sigma _{n}^{2}}=4\left (\frac {\varepsilon _{c}\eta T+1+v}{\eta T}\right )\), The QBER between Alice and Bob can be derived as

$$ \begin{array}{@{}rcl@{}} P_{A B}& =&\frac{1}{2}[P(M<N_{x}\mid M>0)+P(M<N_{p}\mid M>0)]\\ & =&\frac{1}{2}[P(0<M<N_{x})/P(M>0) +P(0<M<N_{p})/P(M>0)]\\ & =&P(0<M<N_{x})+P(0<M<N_{p})\\ & =&\iint_{0<m<n_{x}} \frac{e^{\frac{m^{2}}{2{\sigma_{m}^{2}}}-\frac{{n_{x}^{2}}}{2\sigma_{n_{1}}^{2}}}}{2\pi\sigma_{m}\sigma_{n_{1}}} \text{dm} \text{dn}_{\text{x}}+ \iint_{0<m<n_{x}} \frac{e^{\frac{m^{2}}{2{\sigma_{m}^{2}}}-\frac{{n_{p}^{2}}}{2\sigma_{n_{1}}^{2}}}}{2\pi\sigma_{m}\sigma_{n_{1}}} \text{dm} \text{dn}_{\text{p}}\\ & =&\frac{1}{2\pi\sigma_{n_{1}}\sigma_{m}}{\int}_{n_{x}}^{0} e^{\frac{m^{2}}{2{\sigma_{m}^{2}}}} \text{dm}{\int}_{-\infty}^{0} e^{\frac{{n_{x}^{2}}}{2\sigma_{n_{1}}^{2}} } \text{dn}_{\text{x}}+ \quad\ \frac{1}{2\pi\sigma_{n_{1}}\sigma_{m}}{\int}_{n_{p}}^{0} e^{\frac{m^{2}}{2{\sigma_{m}^{2}}}} dm{\int}_{-\infty}^{0} e^{\frac{{n_{p}^{2}}}{2\sigma_{n_{1}}^{2}}} \text{dn}_{\text{p}}\\ & =&{\int}_{0}^{+\infty} \frac{1}{\sqrt{2\pi}\sigma_{n_{1}}}erf\left( \frac{n}{\sqrt{2}\sigma_{m}}\right)e^{-\frac{n^{2}}{2\sigma_{n_{1}}^{2}}} \text{dn}\\ & =&\text{arctan}\left( \frac{\sigma_{n_{1}}}{\sigma_{m}}\right)/\pi. \end{array} $$
(21)

where \(\text {erf}(x)={{\int \limits }_{0}^{x}}e^{-t^{2}}dt\) is the error function.

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Guo, L., Ran, Q., Zhao, W. et al. Unidimensional Continuous-variable Quantum Key Distribution Based on Basis-encoding Coherent States Protocol. Int J Theor Phys 59, 1730–1741 (2020). https://doi.org/10.1007/s10773-020-04439-8

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Keywords

  • Quantum bite error rate
  • Single quadrature modulation
  • Basis-encoding