Performance analysis of the satellite-to-ground continuous-variable quantum key distribution with orthogonal frequency division multiplexed modulation


Continuous-variable quantum key distribution (CVQKD) is aiming at widespread application and adoption in different scenarios, and thus, the application of satellite-to-ground link may play a vital role in establishing the global secure quantum communications. In this paper, we propose an improved tunable CVQKD scheme for the satellite-to-ground free space optical (FSO) link in an orthogonal frequency division multiplexing (OFDM) system. The OFDM-based CVQKD can effectively suppress the random fading effect that resulted from the atmospheric channel since it divides the initial transmission channel into multi-subcarriers working in parallel so as to compensate the weakness of single-channel transmission of the signals. Moreover, the influence of the intensity scintillation, atmospheric transmittance and phase noise caused by atmospheric turbulence is involved in security analysis. The results reveal that satellite-to-ground CVQKD system can reduce the atmosphere influence in an OFDM system. Compared with the single-channel CVQKD in the FSO link, the secret key rate is improved with the OFDM technique within a certain range of subcarrier numbers. Nevertheless, the reduction of symbols per channel cannot make the asymptotic assumption due to the limitation of technology and devices.

This is a preview of subscription content, log in to check access.

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


  1. 1.

    Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H.: Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012)

    ADS  Google Scholar 

  2. 2.

    Gisin, N., Thew, R.: Quantum communication. Nat. Photonics 55, 298–303 (2011)

    Google Scholar 

  3. 3.

    Pirandola, S., Mancini, S., Lloyd, S., Braunstein, S.L.: Continuous variable quantum cryptography using two-way quantum communication. Nat. Phys. 5, 726–730 (2006)

    Google Scholar 

  4. 4.

    Huang, D., Huang, P., Lin, D., Zeng, G.: Long-distance continuous-variable quantum key distribution by controlling excess noise. Sci. Rep. 6, 19201 (2015)

    ADS  Google Scholar 

  5. 5.

    Shor, P.W., Preskill, J.: Simple proof of security of the bb84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441–444 (2000)

    ADS  Google Scholar 

  6. 6.

    Pomerene, A., Starbuck, A.L., Lentine, A.L., Long, C.M., Derose, C.T., Trotter, D.C.: Silicon photonic transceiver circuit for high-speed polarization-based discrete variable quantum key distribution. Opt. Express 25, 12282 (2017)

    ADS  Google Scholar 

  7. 7.

    Huang, P., He, G., Fang, J., Zeng, G.: Performance improvement of continuous-variable quantum key distribution via photon subtraction. Phys. Rev. A 87, 530–537 (2013)

    Google Scholar 

  8. 8.

    Fang, J., Huang, P., Zeng, G.: Multichannel parallel continuous-variable quantum key distribution with gaussian modulation. Phys. Rev. A 89, 022315 (2014)

    ADS  Google Scholar 

  9. 9.

    Liao, Q., Guo, Y., Huang, D., Huang, P., Zeng, G.: Long-distance continuous-variable quantum key distribution using non-gaussian state-discrimination detection. New J. Phys. 20, 2 (2017)

    Google Scholar 

  10. 10.

    Jouguet, P., Kunzjacques, S., Leverrier, A., Grangier, P., Diamanti, E.: Experimental demonstration of continuous-variable quantum key distribution over 80 km of standard telecom fiber. Nat. Photonics 7, 378–381 (2013)

    ADS  Google Scholar 

  11. 11.

    Huang, D., Lin, D., Wang, C., Liu, W., Fang, S., Peng, J.: Continuous-variable quantum key distribution with 1 mbps secure key rate. Opt. Express 23, 17511–17519 (2015)

    ADS  Google Scholar 

  12. 12.

    Vasylyev, D., Semenov, A.A., Vogel, W., Günthner, K., Thurn, A., Bayraktar, O., Marquardt, C.: Free-space quantum links under diverse weather conditions. Phys. Rev. A 96, 043856 (2017)

    ADS  Google Scholar 

  13. 13.

    Liao, S.K., Cai, W.Q., Liu, W.Y., Zhang, L., Li, Y., Ren, J.G.: Satellite-to-ground quantum key distribution. Nature 549, 43 (2017)

    ADS  Google Scholar 

  14. 14.

    Zhang, H., Mao, Y., Huang, D., Li, J.W., Zhang, L., Guo, Y.: Security analysis of orthogonal-frequency-division-multiplexing-based continuous-variable quantum key distribution with imperfect modulation. Phys. Rev. A 97, 052328 (2018)

    ADS  Google Scholar 

  15. 15.

    Wang, Y., Wang, D., Ma, J.: On the performance of coherent ofdm systems in free-space optical communications. IEEE. Photon. J. 7, 1–10 (2015)

    Google Scholar 

  16. 16.

    Armstrong, J.: OFDM for optical communications. J. Lightw. Technol. 27, 189–204 (2009)

    ADS  Google Scholar 

  17. 17.

    Bahrani, S., Razavi, M., Salehi, J.A.: Orthogonal frequency-division multiplexed quantum key distribution. J. Lightw. Technol. 33, 4687–4698 (2015)

    ADS  Google Scholar 

  18. 18.

    Jin, X.M., Ren, J.G., Yang, B., Yi, Z.H., Zhou, F., Xu, X.F.: Experimental free-space quantum teleportation. Nat. Photonics 4, 376–381 (2010)

    ADS  Google Scholar 

  19. 19.

    Alhabash, A., Andrews, L.C.: New mathematical model for the intensity pdf of a laser beam propagating through turbulent media. Opt. Eng. 40, 1554–1562 (2001)

    ADS  Google Scholar 

  20. 20.

    Bekkali, A., Naila, C.B., Kazaura, K., Wakamori, K., Matsumoto, M.: Transmission analysis of ofdm-based wireless services over turbulent radio-on-fso links modeled by Gamma–Gamma distribution. IEEE. Photon. J. 2, 510–520 (2010)

    ADS  Google Scholar 

  21. 21.

    Bai, F., Su, Y., Sato, T.: Performance analysis of polarization modulated directdetection optical cdma systems over turbulent fso linksmodeled by the gamma-gamma distribution. Photonics 2, 139–155 (2015)

    Google Scholar 

  22. 22.

    Wang, Y., Wang, D., Ma, J.: Performance analysis of multihop coherent ofdm free-space optical communication systems. Opt. Commun. 376, 35–40 (2016)

    ADS  Google Scholar 

  23. 23.

    Qing, C., Wu, X., Li, X., Zhu, W., Qiao, C., Rao, R.: Use of weather research and forecasting model outputs to obtain near-surface refractive index structure constant over the ocean. Opt. Express 24, 13303 (2016)

    ADS  Google Scholar 

  24. 24.

    Paterson, C.: Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Phys. Rev. Lett. 94, 153901 (2005)

    ADS  Google Scholar 

  25. 25.

    Vasylyev, D.Y., Semenov, A.A., Vogel, W.: Toward global quantum communication: beam wandering preserves nonclassicality. Phys. Rev. Lett. 108, 220501 (2012)

    ADS  Google Scholar 

  26. 26.

    Vasylyev, D., Semenov, A.A., Vogel, W.: Atmospheric quantum channels with weak and strong turbulence. Phys. Rev. Lett. 117, 090501 (2016)

    ADS  Google Scholar 

  27. 27.

    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963)

    ADS  MathSciNet  MATH  Google Scholar 

  28. 28.

    Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys Rev. Lett. 10, 277–279 (1963)

    ADS  MathSciNet  MATH  Google Scholar 

  29. 29.

    Usenko, V.C., Heim, B., Peuntinger, C., Wittmann, C., Marquardt, C., Leuchs, G.: Entanglement of gaussian states and the applicability to quantum key distribution over fading channels. New J. Phys. 14, 93048 (2012)

    Google Scholar 

  30. 30.

    Gappmair, W., Flohberger, M.: Error performance of coded fso links in turbulent atmosphere modeled by gamma-gamma distributions. IEEE Trans. Wirel. Commun. 8, 2209–2213 (2009)

    Google Scholar 

  31. 31.

    Xie, G., Wang, F., Dang, A., Guo, H.: A novel polarization-multiplexing system for free-space optical links. IEEE Photon. Technol. Lett. 23, 1484–1486 (2011)

    ADS  Google Scholar 

  32. 32.

    Gyongyosi, L.: Subcarrier domain of multicarrier continuous-variable quantum key distribution. Mathematics 1, 1–25 (2013)

    Google Scholar 

  33. 33.

    Gyongyosi, L.: Diversity extraction for multicarrier continuous-variable quantum key distribution. In: Signal Processing Conference, pp. 478–482 (2016)

  34. 34.

    Gyongyosi, L.: Adaptive multicarrier quadrature division modulation for continuous-variable quantum key distribution. In: Proceedings of SPIE, p. 912307 (2014)

  35. 35.

    García-Patrón, R., Cerf, N.J.: Unconditional optimality of gaussian attacks against continuous-variable quantum key distribution. Phys. Rev. Lett. 97, 190503 (2006)

    ADS  Google Scholar 

  36. 36.

    Navascués, M., Acín, A.: Security bounds for continuous variables quantum key distribution. Phys. Rev. Lett. 94, 020505 (2005)

    ADS  Google Scholar 

  37. 37.

    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  Google Scholar 

  38. 38.

    Guo, Y., Liao, Q., Wang, Y., Huang, D., Huang, P., Zeng, G.: Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction. Phys. Rev. A 95, 032304 (2017)

    ADS  Google Scholar 

  39. 39.

    Gyongyosi, L.: Distribution statistics and random matrix formalism of multicarrier continuous-variable quantum key distribution. Mathematics 1, 1–47 (2014)

    MathSciNet  Google Scholar 

  40. 40.

    Gyongyosi, L.: Security thresholds of multicarrier continuous-variable quantum key distribution. Mathematics 1, 1–58 (2014)

    MathSciNet  Google Scholar 

  41. 41.

    Gyongyosi, L.: Adaptive multicarrier quadrature division modulation for continuous-variable quantum key distribution. In: Proceedings of SPIE, vol. 9123, pp. 1–37 (2013)

  42. 42.

    Beland, R.R., Brown, J.H.: A deterministic temperature model for stratospheric optical turbulence. Phys. Scripta. 37, 419–423 (1988)

    ADS  Google Scholar 

  43. 43.

    Berrisford, P., Kallberg, P., Kobayashi, S., Dee, D., Uppala, S., Simmons, A.J.: Atmospheric conservation properties in ERA-Interim. Q. J. R. Meteorol. Soc. 137, 1381–1399 (2011)

    ADS  Google Scholar 

  44. 44.

    Gatto, M.A., Seery, J.B.: Numerical evaluation of the modified bessel functions i, and k. Comput. Math. Appl. 7, 203–209 (1981)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the lambert w function. Adv. Comput. Math. 5, 329–359 (1996)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Newman, M.E.J., Barkema, G.T.: Monte carlo methods in statistical physics. Top. Curr. Phys. 46, 252–253 (1999)

    MATH  Google Scholar 

Download references


This work is supported by the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2018zzts539) and the National Natural Science Foundation of China (Grant Nos. 61379153, 61572529).

Author information



Corresponding author

Correspondence to Qin Liao.


A The scintillation index analysis

In this appendix, we illustrate the Rytov variance \({\delta }^2_{R}\) in Sect. 2.1. Rytov variance, which is used to estimate the optical fluctuation intensity, can be defined as

$$\begin{aligned} {\delta }^2_{R}=2.25k^{7/6}sec^{11/6}(\zeta ) \int _{h_0}^{H} C_n^2 (h)(h-h_0)^{5/6}\mathrm {d}h, \end{aligned}$$

where \(k=2 \pi / \lambda \) is the optical wave number, \(H=h_0+Lcos \zeta \) is the satellite altitude, \(h_0\) is the height above ground level, L is the propagation distance, \(\zeta \) is the zenith angle, and \(C_n^2(h)\) is the refraction index structure constant parameter. When \({\delta }^2_{R} < 1\), it indicates the weak turbulence, when \({\delta }^2_{R} \cong 1\), it indicates the medium turbulence, and when \({\delta }^2_{R} > 1\), it indicates the strong turbulence. As for \(C_n^2(h)\), one of the most widely used models is the Hufnagel–Valley (H–V) model described by [42]

$$\begin{aligned} \begin{aligned} C_n^2(h) = 0.00594(v/27)^2(10^{-5}h)^{10} e^{-\frac{h}{1000}} +2.7 \times 10^{-16} e^{-\frac{h}{1500}}+Ae^{-\frac{h}{100}}, \end{aligned} \end{aligned}$$

where v is the pseudowind in meters per second (m/s) and A is the nominal value of \(C_n^2(0)\) at the ground in \(m^{-2/3}\).

Figure 10 shows \(C_n^2\) in the satellite altitude of 0 km, 5 km, 15 km and 20 km based on the varying wind speed v and constant \(C_n^2(0)\). In the satellite-to-ground scenario, \(C_n^2(0)\) expresses the atmospheric refraction structure above the ocean near-surface. From the weather research and forecasting (WRF) model outputs [23], the near-surface refractive index structure constant is mainly varying in the range of \(10^{-15}{-} 10^{-13}\,\mathrm {m}^{-2/3}\). The wind speed v can be extracted from the ECMWF Re-Analysis (ERA-Interim) data set [43], which ranges roughly from 0 to 25 m/s. We can see clearly that the ocean near-surface turbulence level has little effect above 5 km and the wind speed governs the profile behavior primarily in the vicinity of 10 km. Accordingly, the atmospheric turbulence could be negligible when the altitude is higher than 10 km.

Fig. 10

The refraction structure \(C_n^2\) at altitudes 0 km, 5 km, 10 km and 20 km

B The atmospheric transmittance analysis

In this appendix, we illustrate the expressions of the above parameters in Sect. 2.2. The transmittance \(T_0\) can be estimated by [12]

$$\begin{aligned} T_0= & {} 1 -I_0 \left( a^2 \left[ \frac{1}{W_1^2}-\frac{1}{w_2^2} \right] \right) {\mathrm {exp}} \left[ -a^2 \left( \frac{1}{W_1^2} + \frac{1}{W_2^2} \right) \right] -2 \left\{ 1-{\mathrm {exp}} \left[ -\frac{a^2}{2} \left( \frac{1}{W_1} - \frac{1}{W_2} \right) ^2 \right] \right\} \nonumber \\&\times {\mathrm {exp}} \left\{ - \left[ \frac{ \frac{(W_1+W_2)^2}{|W_1^2-W_2^2|}}{R\left( \frac{1}{W_1}-\frac{1}{W_2}\right) } \right] ^{ \lambda \left( \frac{1}{W_1}-\frac{1}{W_2}\right) } \right\} . \end{aligned}$$

\(R(\cdot )\) and \(\lambda (\cdot )\), which are scale and shape functions, respectively,

$$\begin{aligned} R(\xi )= & {} \left[ {\mathrm {ln}} \left( 2 \frac{1-{\mathrm {exp}}[-\frac{1}{2}a^2\xi ^2]}{1-{\mathrm {exp}}[-a^2\xi ^2 ]I_0 (a^2\xi ^2) } \right) \right] ^{ -\frac{1}{\lambda (\xi )}} , \end{aligned}$$
$$\begin{aligned} \lambda (\xi )= & {} 2a^2 \xi ^2 \frac{{\mathrm {exp}}(-a^2\xi ^2) I_1(a^2\xi ^2)}{1-{\mathrm {exp}}(-a^2\xi ^2)I_0(a^2\xi ^2)} \times \left[ {\mathrm {ln}} \left( 2 \frac{1-{\mathrm {exp}}[-\frac{1}{2}a^2\xi ^2]}{1-{\mathrm {exp}}[-a^2\xi ^2 ]I_0 (a^2\xi ^2) } \right) \right] ^{-1}\!,\nonumber \\ \end{aligned}$$

where \(I_i(\cdot )\) is the modified Bessel function of i-th order [44]. For the given angle \(\chi =\phi -\varphi _0\), the effective spot radius \(W_{\mathrm {eff}}(\cdot )\) can be approximated by

$$\begin{aligned} \begin{aligned} W_{\mathrm {eff}}^2(\chi ) = 4a^2 \left\{ {\mathscr {W}} \left( \frac{4a^2}{W_1W_2} {\mathrm {exp}}\left[ \frac{a^2}{W_1^2} (1+2{\mathrm {cos}}^2\chi ) \right] \times {\mathrm {exp}} \left[ \frac{a^2}{W_2^2}(1+2\mathrm {sin}^2 \chi ) \right] \right) \right\} , \end{aligned} \end{aligned}$$

Here, \({\mathscr {W}}(\cdot )\) is the Lambert W function [45].

Based on Eqs. (6), (28)–(31), the probability distribution of the atmospheric transmittance can be evaluated using the Monte Carlo method [46]. In the above model, there are five parameters \((x_0,y_0,W_1,W_2,\phi )\). As shown in Fig. 11, parameter \(\phi \) is uniformly distributed and independent of others. Besides, parameters \((W_1,W_2)\) can be obtained by multiplying large number of small random contributions, which gives a good argument for assuming that \((W_1,W_2)\) is log-normally distributed. Therefore, random parameters \(\varTheta _1={\mathrm {ln}} \frac{W_1}{W_0}\) and \(\varTheta _2={\mathrm {ln}} \frac{W_2}{W_0}\) are introduced to calculate the probability distribution of the atmospheric transmittance, where \(W_0\) is the transmitter beam. The correlation of \((x_0,y_0,\varTheta _1,\varTheta _2)\) could be defined by its covariance matrix, which reads [12, 26]


Combing the results of the scintillation index in Eqs. (26)–(27) with the turbulent regimes, the satellite-to-ground link can be regarded to be affected by weak-to-moderate turbulence (\(\delta _R^2 < 1\)). The elements of \(\varXi \) is expressed by [26]

$$\begin{aligned} \begin{aligned} \langle x_0^2\rangle&= \langle y_0^2\rangle =0.33W_0^2 \delta _R^2 \varOmega ^{-\frac{7}{6}} , \\ \langle \varTheta _1^2 \rangle&= \langle \varTheta _2^2 \rangle ={\mathrm {ln}} \left[ 1+\frac{1.2\delta _R^2 \varOmega ^{\frac{5}{6}}}{(1+2.96\delta _R^2 \varOmega ^{\frac{5}{6}} )^2} \right] ,\\ \langle \varTheta _1 \varTheta _2 \rangle&= \langle \varTheta _2^2 \rangle ={\mathrm {ln}} \left[ 1-\frac{0.8\delta _R^2 \varOmega ^{\frac{5}{6}}}{(1+2.96\delta _R^2 \varOmega ^{\frac{5}{6}} )^2} \right] , \end{aligned} \end{aligned}$$

where \(\varOmega =\frac{kW_0^2}{2L}\) is the Fresnel parameter of the beam and their expectations read

$$\begin{aligned} \begin{aligned} \langle x_0\rangle&= \langle y_0\rangle =0 ,\\ \langle \varTheta _1 \rangle&= \langle \varTheta _2 \rangle = {\mathrm {ln}} \left[ \frac{ \left( 1+2.96\delta _R^2 \varOmega ^{\frac{5}{6}} \right) ^2 }{ \varOmega ^2 \sqrt{ \left( 1+2.96\delta _R^2 \varOmega ^{\frac{5}{6}} \right) ^2 +1.2\delta _R^2 \varOmega ^{\frac{5}{6}} } } \right] . \end{aligned} \end{aligned}$$
Fig. 11

The aperture of radius a and the elliptical beam profile with the half axis \(W_1\) rotated on the angle \(\phi \) relative to the x-axis and on the angle \(\chi \) relative to the \(r_0\)-associated axis are shown

C Crosstalk noise on the subchannels

In this appendix, we illustrate crosstalk noise in Sect. 4. The effect of interchannel crosstalk in the modulation is depicted in Fig. 12. The crosstalk information \(\gamma _i (N_i)\) on the i-th subchannel \(N_i\) is expressed as [34]

$$\begin{aligned} \gamma _i (N_i) = \sum _{j\ne i} x_j \chi (A_j:B_j) , \end{aligned}$$

where \(0 \le x_j \le 1\), \(\chi (A_j:B_j)=H(B_j)-H(B_j|A_j)\) is the Holevo information of Alice and Bob conveyed by the neighboring subchannel \(N_j\). Since the crosstalk noise \( \gamma _i (N_i)\) on the subchannel \(N_i\) acts for Bob as Gaussian noise, it can be modeled by \( \gamma _i (N_i) \in \mathscr {C} {\mathbb {N}} (0,\sigma ^2_{\gamma _i})\) with variance \(\sigma ^2_{\gamma _i} {\mathbb {E}} [|\gamma _i|^2]\). This additional noise does not change the rate formulas of the system, since this additional noise is already contained in the subchannel’s noise variance \(\sigma _{N_i}^2\) (see Ref. [41], Theorem 2). In other words, the mutual information between Alice and Bob is completely characterized for a given \(N_i\) by the noise variance \(\sigma _{N_i}^2\), which can be decomposed as

$$\begin{aligned} \sigma _{N_i}^2 = \sigma _{E_i}^2 + \sigma _{\gamma _i}^2. \end{aligned}$$

where \( \sigma _{E_i}^2 \) is the noise variance of Eve’s optimal Gaussian collective attack. Accordingly, the \(\chi (B:E)\) Holevo information of Eve changes is as follows [34]

$$\begin{aligned} \chi (B:E)_{\gamma _i} = \chi (B:E) +\sum _{n} \sum _{j\ne i} |T_{E,i}|^2 x_j \chi (A_j:B_j) . \end{aligned}$$

Furthermore, crosstalk noise has no effect on the security of the modulation, since it allows no more leaking of information to an eavesdropper than single-carrier CVQKD protocols (see Ref. [34], Theorem 4). Moreover, the influence of excess noise has been analyzed in Ref. [40].

Fig. 12

The effect of crosstalk noise between subchannels. Due to the imperfections of practical devices, some information from neighboring subchannels can be leaked to a given subchannel, which acts as noise to the receiver

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, W., Liao, Q., Huang, D. et al. Performance analysis of the satellite-to-ground continuous-variable quantum key distribution with orthogonal frequency division multiplexed modulation. Quantum Inf Process 18, 39 (2019).

Download citation


  • Continuous-variable quantum key distribution
  • Satellite-to-ground link
  • Orthogonal frequency division multiplexing technique