Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink

Abstract

Satellite-based quantum key distribution (QKD) has lately received considerable attention due to its potential to establish a secure global network. Associated with its application is a turbulent atmosphere that sets a notable restriction to the transmission efficiency, which is especially challenging for ground-to-satellite uplink scenarios. Here, we propose a novel noiseless attenuation (NA) scheme involving a zero-photon catalysis operation for source preparation to improve the performance of continuous-variable (CV) QKD over uplink. Numerical analysis shows that the NA-based CV-QKD, under attenuation optimization, outperforms the traditional CV-QKD, which is embodied in extending the allowable zenith angle while improving the effective communication time. Attributing to characteristics of the attenuation optimization, we find that the NA-involved source preparation improves the security bound by relatively reducing the amount of information available to eavesdroppers. Taking the finite-size effect into account, we achieve a tighter bond of security, which is more practical compared with the asymptotic limit.

1. Introduction

In recent decades, a number of quantum cryptography protocols have been successfully realized, including quantum teleportation (QT) [1,2], quantum key distribution (QKD) [3], quantum secret sharing [4,5,6], and quantum coding [7,8]. QKD, which mainly relies on Heisenberg’s uncertainty principle, which provide us with an unconditional secure method for private information transmission over an insecure channel. Generally, it can be divided into two categories, referred to as discrete-variable (DV) QKD [9] and continuous-variable (CV) QKD [10]. In DV-QKD, the information is usually encoded in the polarization or time bin of single photons, using expensive and extreme cryo-cooling single-photon detectors [11]. Compared with DV-QKD, CV-QKD, which involves the coherent state, can achieve higher channel capacity by using homodyne or heterodyne detection, and has good compatibility with standard optical communications [12]. To date, CV-QKD has been successfully analyzed in fiber links both theoretical and experimental [13,14].

However, due to heavy losses in the practical fiber, the distance over which the coherent state of light can be transmitted is restricted within hundreds of kilometers. So far, the longest transmission distance of CV-QKD is only 202.81 km [3]. Since it is difficult to noiselessly amplify the quantum state at the receiver [15], exhibiting a global-scale network through a fiber link becomes a challenge. In the last few years, the research towards global-scale network has increased for a satellite-mediated scenario [16], where the channel losses are limited since the effective thickness of the atmosphere is only about 20 km [17]. The major works are being performed in downlink configuration, however, an uplink from ground to a quantum satellite is less explored because the atmospheric turbulence acts on early stages of optical signal transmission, leading to a larger diffraction-induced broadening and more serious beam wandering [18]. As has been pointed out, the secret key rate in uplink is less than one order of magnitude compared with the downlink scenario [19]. Despite the inferior performance compared with that of the downlink, the uplink CV-QKD is also of great value owing to the simple satellite structure, the variability of quantum sources, and the accessibility for maintenance repair [20,21].

To improve the performance of the uplink CV-QKD, we consider an enhanced source preparation scheme involving the noiseless attenuation (NA) operation [22]. The NA operation has been used for quantum key distribution, quantum teleportation, and other applications to reduce the detrimental effects of channel losses [23,24,25]. On the premise of the previous achievements, in this paper, we suggest an NA-based enhancement scheme for the Gaussian-modulated coherent-state protocol of uplink. Due to the limited space on the quantum satellite, the NA operation is applied to the ground station (the sender). We perform a suitable NA process by using a zero-photon catalysis (ZPC) operation, which happens to be a zero-energy-input operation that does not introduce the consumption of external energy [26]. We prove the security of the NA-based CV-QKD protocol against the Gaussian collective attack under both the asymptotic case and finite-size regime. Our results show that both the allowable zenith angle and effective communication time are improved compared with the traditional CV-QKD protocol. The research confirms the role of the NA operation on the uplink CV-QKD protocol and is conducive to making the CV-QKD protocol more practical.

The paper is organized as follows. In Section 2, we illustrate characteristics of the uplink CV-QKD system. In Section 3, we suggest the NA-based source preparation scheme, based on zero-photon catalysis, for improving the performance of the CV-QKD system. Section 4 shows the security of the NA-based protocol under both the asymptotic case and the finite-size regime. Finally, conclusions are drawn in Section 5. The motivation of this work is to improve the performance of the uplink CV-QKD to achieve a complete quantum network with more efficiency.

2. Uplink CV-QKD

For the convenience of security analysis, the uplink CV-QKD protocol is structured with the equivalent entanglement-based (EB) scheme [27]. As shown in Figure 1, Alice generates a two-mode squeezed vacuum (TMSV) state as an entangled resource, which can be represented by its covariance matrix

$${\gamma }_{AB}=\left[\begin{array}{cc}V\Pi & \sqrt{V^2-1}{\sigma }_z\\ \sqrt{V^2-1}{\sigma }_z& V\Pi \end{array}\right],$$

(1)

where ${\sigma }_z=\mathrm{diag}(1,-1)$, $\Pi =\mathrm{diag}(1,1)$ and $V=(1+{\lambda }^2)/(1-{\lambda }^2)$ with squeezing parameter $\lambda$. Then, Alice holds mode A for heterodyne detection, while sending another mode B to Bob over the insecure uplink, which is assumed to be controlled by a potential eavesdropper. Due to the random fading of the channel, the initial quantum state transfer to a mixture [28], whose composition depends on the random transmittance T together with channel excess noise $ϵ$. Here, we assume a fixed $ϵ$, and describe the random T by its probabilities distribution. Specifically, we divide the fading channel into K stable sub-channels of individuals, while each sub-channel has a fixed transmittance ${\left\{T_j\right\}}_{j\in \{1,\cdots ,K\}}$ with corresponding probability ${\left\{P_j\right\}}_{j\in \{1,\cdots ,K\}}$. Therefore, the arriving mixture shared by Alice and Bob is

$${\gamma }_{A{B}_2}=\left[\begin{array}{cc}V\Pi & \langle \sqrt{T}\rangle \sqrt{V^2-1}{\sigma }_z\\ \langle \sqrt{T}\rangle \sqrt{V^2-1}{\sigma }_z& \langle T\rangle (V+{\chi }_{\mathrm{line}})\Pi \end{array}\right],$$

(2)

where $\langle T\rangle ={\sum }_{j=1}^K{P}_j{T}_j$, $\langle \sqrt{T}\rangle ={\sum }_{j=1}^K{P}_j{\sqrt{T}}_j$, and ${\chi }_{\mathrm{line}}=1/\langle T\rangle -1+ϵ$. Finally, by implementing quadrature measurement, classical information reconciliation, and privacy amplification, Alice and Bob can extract the secret keys.

As shown in Equation (2), the statistical parameters $\langle T\rangle$ and $\langle \sqrt{T}\rangle$ represents the fading effects of uplink on quantum state. To describe these parameters, we apply the elliptic-beam model [29,30], which provides accurate agreement with the experimental data under weak and strong turbulence. In this model, the receiving beam is assumed to be an elliptic deformation, which can be expressed by a random vector

$$v=(x_0,y_0,W_1,W_2,\phi ),$$

(3)

where $(x_0,y_0)$ denotes the beam-centroid coordinates at telescope plane, $W_1$ and $W_2$ are the semi-axis of arriving elliptic beam, and $\phi$ is the orientation angle between semi-axis $W_1$ and the x axis. Based on the vector v, the probability distribution of transmittance (PDT) is given by

$$P\left(T\right)=\frac{2}{\pi }{\int }_{{\mathbb{R}}^4}{\mathrm{d}}^{4}v{\int }_0^{\frac{\pi }{2}}\mathrm{d}\phi {\rho }_G(v;\mu ,\Sigma )\delta (T-T\left(v\right)),$$

(4)

where $T\left(v\right)$ defines the relationship between T and v, and ${\rho }_G(v;\mu ,\Sigma )$ is a Gaussian probability density of v with the mean value $\mu$ and the covariance matrix $\Sigma$. The elements of $\mu$ and $\Sigma$ in the uplink case are listed in Appendix A. Based on $\mu$ and $\Sigma$, we can derive the PDT of the uplink channel through the Monte Carlo method [29], and then obtain the statistical parameters of T. Note that the simulation parameters of the work (as shown in Table 1) are the same as our previous work so that the numerical analysis of transmittance vs. zenith angle can refer to [21].

Subsequently, we demonstrate the performance of the uplink CV-QKD protocol. We consider the asymptotic regime and assume an ideal homodyne detection. The information reconciliation is the reverse reconciliation, since it is more robust against channel attenuation [31]. The secret key rate can be derived from Equation (2) with the support of Gaussian state extremality theorem [27], which is given by

$$K_A(\langle T\rangle ,\langle \sqrt{T}\rangle ,ϵ)={\beta }_r{I}_{AB}-{\chi }_E,$$

(5)

where ${\beta }_r$ denotes the reconciliation efficiency, $I_{AB}$ is the Shanon mutual information, and ${\chi }_E$ represents the Holevo bound. The calculations of $I_{AB}$ and ${\chi }_E$ are shown in Appendix B. Moreover, the angle-of-arrival fluctuation caused by wavefront distortion will introduce a coupling problem between spatial beam and fiber on receiver detection [32]. The relationship between the fiber coupling efficiency $\varsigma$ and angle-of-arrival ${\beta }_a$ can be expressed as [33]

$$\varsigma ={\int }_{-0.5d_{\mathrm{cor}}}^{0.5d_{\mathrm{cor}}}\frac{1}{\sqrt{2\pi \langle {\beta }_a^2\rangle }f}exp\left(\frac{-l^2}{2f^2\langle {\beta }_a^2\rangle }\right)\mathrm{d}l,$$

(6)

where f denotes the focal length, $d_{\mathrm{cor}}$ is the diameter of the fiber. $\langle {\beta }_a^2\rangle$ represents the variance of ${\beta }_a$, which is related to refraction index structure parameter $C_n^2$. Details of $\langle {\beta }_a^2\rangle$ in uplink case refer to Appendix C. Hence, the secret key rate in Equation (5) should be modified to

$$K_A(\langle T\rangle ,\langle \sqrt{T}\rangle ,ϵ)=\varsigma \times ({\beta }_r{I}_{AB}-{\chi }_E).$$

(7)

Table 1. The simulation parameters.

Variable	Value	Description	Reference
$\zeta$	0–60${}^{\circ }$	Zenith angle
a	1 m	Receiver telescope radius	[21]
f/2a	1	Focal ratio	[33]
$d_{\mathrm{cor}}$	10 $\mathsf{\mu }\mathrm{m}$	Fiber core diameter	[33]
$W_0$	80 mm	Intial beam radius	[33]
$V-1$	4 SNU	Modulation variance	[34]
${\lambda }_0$	800 nm	Wavelength	[21]
${\beta }_r$	$95\%$	Reconciliation efficiency	[21]
$ϵ$	0.01 SNU	Excess noise	[35]
$C_n^2$	$1.12$×${10}^{-16}{\mathrm{m}}^{-2/3}$	Refraction index structure parameter	[17]
$n_0$	0.61 ${\mathrm{m}}^{-3}$	Mean number of scatterer particles	[17]

Table 1. The simulation parameters.

Variable	Value	Description	Reference
$\zeta$	0–60${}^{\circ }$	Zenith angle
a	1 m	Receiver telescope radius	[21]
f/2a	1	Focal ratio	[33]
$d_{\mathrm{cor}}$	10 $\mathsf{\mu }\mathrm{m}$	Fiber core diameter	[33]
$W_0$	80 mm	Intial beam radius	[33]
$V-1$	4 SNU	Modulation variance	[34]
${\lambda }_0$	800 nm	Wavelength	[21]
${\beta }_r$	$95\%$	Reconciliation efficiency	[21]
$ϵ$	0.01 SNU	Excess noise	[35]
$C_n^2$	$1.12$×${10}^{-16}{\mathrm{m}}^{-2/3}$	Refraction index structure parameter	[17]
$n_0$	0.61 ${\mathrm{m}}^{-3}$	Mean number of scatterer particles	[17]

The secret key rates $K_A$ as a function of the zenith angle $\zeta$ for the given satellite altitude H is shown in Figure 2, where the simulation parameters refer to Table 1 in Section 4. In addition, the performance of the downlink scenario, analyzing with the same parameters, is also plotted for comparison. Numerical analysis reveals the favorable of downlink for communication, while illustrating the urgency of an enhancement scheme for the uplink case. Note that the maximum secret key rate ($\zeta =0^{\circ }$) of the downlink at 100 km is lower than that of the uplink because the fiber coupling problem is more severe in the downlink scenario [21].

3. Modified Protocol with Noiseless Attenuation

Due to the limited size of the quantum satellite, an enhanced source preparation scheme at the ground station becomes a feasible solution. Motivated by characteristics of the source preparation [23,24,25], we apply a suitable NA operation before transmission for performance improvement (see Figure 1). In this section, the NA-based source preparation is realized via the ZPC operation, which can be implemented by mixing zero photon at an asymmetrical beam splitter (ABS) and then post-selecting the ABS output by conditional detection of zero photon (see Figure 3). For the TMSV state, the catalyzed mode (mode B) is sent to one of the input ports of ABS with transmittance $T_A$, while a zero-photon ancillary Fock state incidents on another port for interference. In an ideal case, the auxiliary zero-photon state will not be destroyed at all and the photon number resolving detector (PNRD) at the corresponding output ports can register zero photon. Therefore, the ZPC-involved attenuation happens to be a zero-energy-input attenuation that does not cause the consumption of external energy. Mathematically, the ZPC process can be regarded as an equivalent operation ${\widehat{O}}_0$ given by

$${\widehat{O}}_0=\frac{{\partial }^m}{m!\partial {\chi }^m}{\left\{\frac{T_A^m}{1-\chi }{\left(\frac{{\sqrt{T}}_A-\chi /{\sqrt{T}}_A}{1-\chi }\right)}^{b^{†}b}\right\}}_{m=0,\chi =0},$$

(8)

where $b^{†}\left(b\right)$ denotes the addition (annihilation) photon operation of input mode, m is the number of photons detected by PNRD when the ZPC process is successful. Obviously, $m=0$ in the ZPC process as the description above.

When Alice performs heterodyne detection on mode A, the transmission mode B will project to a coherent state [27]. It means that the ZPC operation on TMSV is equivalent to that acts on a coherent state, which can be written as

$${\widehat{O}}_0\mid \alpha \rangle ={\left(\sqrt{T_A}\right)}^{b^{†}b}\mid \alpha \rangle =\mid \sqrt{T_A}\alpha \rangle ,$$

(9)

where $\mid \alpha \rangle$ denotes the coherent state. We find that the ZPC operation happens to be an NA process with attenuation coefficient $g=\sqrt{T_A}$. In addition, the ZPC operation acting on a coherent state is a probability process with success probability

$$P=\frac{2}{1+T_A+RV},$$

(10)

where $R=1-T_A$. Figure 4 shows the relationship between ABS transmittance $T_A$ and attenuation coefficient g with the corresponding success probability P. We find that both attenuation coefficient and success probability saw an upward trend with the increasing transmittance.

We note that the covariance matrix of the ZPC-involved attenuation state can be derived by its characteristic function (CF), as shown in Appendix D. The form of the covariance matrix can be expressed as

$${\gamma }_{A{B}_1}=\left[\begin{array}{cc}X\Pi & Y{\sigma }_z\\ Y{\sigma }_z& X\Pi \end{array}\right],$$

(11)

where the factors X and Y are listed as follows

$$X=\frac{2V-RV+R}{1+T_A+RV},$$

(12)

$$Y=\frac{2\sqrt{T_A(V^2-1)}}{1+T_A+RV}.$$

(13)

Then, the attenuation state transfers the uplink channel for quadrature measurement. The final covariance matrix of arriving attenuation state is given by

$${\gamma }_{A{B}_2}=\left[\begin{array}{cc}X\Pi & \langle \sqrt{T}\rangle Y{\sigma }_z\\ \langle \sqrt{T}\rangle Y{\sigma }_z& \langle T\rangle (X+{\chi }_{\mathrm{line}})\Pi \end{array}\right].$$

(14)

4. Performance Analysis

In this section, we perform the security analysis of the NA-based uplink CV-QKD protocol for the asymptotic case and the finite-size regime. Since the ZPC operation belongs to a kind of Gaussian operation (see Equation (9)), we can derive the secret key rate via the conventional method of the Gaussian protocol. The simulation parameters are shown in Table 1. We consider a collimated Gaussian beam with an initial radius $W_0$ transmitted onto a telescope with radius a, and illustrate the results of clear night-time. Note that CV protocol has the potential to reach an all-day communication [36,37], and the performance of the day-time situation can be derived by updating the atmospheric parameters, i.e., refraction index structure parameter $C_n^2$, and the mean number of scatterer particles $n_0$.

4.1. Asymptotic Analysis

As shown in Figure 5a, it demonstrates the secret key rate $K_A$ as a function of zenith angle $\zeta$ for various satellite altitudes. The attenuation coefficient g is optimal for each zenith angle. In addition, the single-photon catalysis (SPC) protocol, which can be considered as another noiseless attenuation operation [34], is also plotted for comparison. We find that the NA-based scheme improves performance. The allowable zenith angle is improved by $6^{\circ }$ and $8^{\circ }$ at $H=100$ km and $H=120$ km, respectively. Note that the satellite-mediated CV-QKD system has a strong dependence on the instantaneous position between the moving satellite and the ground station. The effective time of a communication cycle (satellite orbits the earth) is limited to the period when the satellite passes above the ground station. Therefore, a larger allowable zenith angle ensures a longer effective communication time so, as a consequence, more total secret keys. Figure 5b reveals that the NA operation increases the effective time. The calculation is based on Newton’s second law with the earth mass $M=5.965\times {10}^{24}$ kg and the earth radius $r_0=6371$ km.

Another observation in Figure 5 is that, at a small zenith angle, the bound of the secret key rate of the NA-based CV-QKD protocol coincident with that of the coherent-state protocol. As the zenith angle increases, so does the strength of turbulence, the improvement caused by the NA operation starts to appear. This phenomenon can be attributed to the optimization of attenuation coefficient. Figure 6 illustrates the optimal attenuation coefficient g varying with the zenith angle $\zeta$ for various satellite altitudes. The optimal attenuation coefficient equals 1 for the case of small zenith angle ($\zeta <3^{\circ }$ and $\zeta <{26}^{\circ }$ for 100 km and 120 km, respectively), and then shows a downward trend with the increasing zenith angle. In other words, the protocol without NA operation may perform better than the NA-based one under weak turbulence effects, while the NA-based protocol can reach higher secret key rate when the turbulence effects become stronger. Moreover, the NA operation may be required upon higher satellite altitude at $H=120$ km.

In the following, we show the effect of the NA operation on the security of the CV-QKD system. In Figure 7, we plot the reconciliation-stricken mutual information ${\beta }_r{I}_{AB}$ and Holevo bound ${\chi }_E$ as a function of attenuation coefficient g. Without loss of generality, we select $\zeta =0^{\circ }$ and $\zeta ={32}^{\circ }$ to represent small zenith angle case and large zenith angle case, respectively. In addition, the satellite altitude is set to 100 km. We find that the use of the NA operation acts a negative effect on mutual information, but at the same time reduces the Holevo bound. For $\zeta =0^{\circ }$, the secret key rate, represented by the gap between the green line and purple line, keeps increasing as the growth attenuation coefficient. For a zenith angle of ${32}^{\circ }$, the secret key rate reaches the maximum at $g=0.834$, and then decreases with the growth of attenuation coefficient. In a word, the NA operation can improve the secret key rate under a large zenith angle by relatively reducing the amount of information available to Eve, which is accomplished by selecting an optimal attenuation coefficient.

4.2. Finite-Size Regime

In the asymptotic case mentioned above, the secure analysis is performed with the premise that the length of the data is infinite, which is impossible in practice. Therefore, considering the finite-size effect in security analysis is necessary for practical implementations [38,39,40]. In the finite-size regime, we need to estimate the unknown parameters, including the channel transmittance and channel noise. In practice, these parameters are estimated using finite-sized data samples so that the secret key rate will be reduced. Although the finite-size effect in fiber link has been well studied, the analysis in free space is still a subject to be focused on since the channel fluctuation has a significant negative impact on the parameter estimation. To compensate for imprecise estimation, a suppression method, which divides the overall data into several clusters for fluctuation suppression, has been proposed [41]. However, it also increases the difficulty of the software and hardware at the receiver.

In this section, we consider the case that overall data are included for security analysis. With the overall data, the covariance matrix ${\gamma }_{A{B}_2}$ can be equivalently parameterized as originating from a stable link with fixed transmittance $\eta ={\langle \sqrt{T}\rangle }^2$, which is given by

$${\gamma }_{A{B}_2}=\left[\begin{array}{cc}X\Pi & \sqrt{\eta }Y{\sigma }_z\\ \sqrt{\eta }Y{\sigma }_z& \eta (X-1)+{\xi }^2\end{array}\right],$$

(15)

where ${\xi }^2=(\langle T\rangle -{\langle \sqrt{T}\rangle }^2)(X-1)+1+\langle T\rangle ϵ$. Following the rewritten covariance matrix above, the calculation of the secret key rate in finite-size regime can be expressed as

$$K_F=\varsigma \times \left\{r[K_A({\eta }_{min},{\xi }_{max}^2)-\Delta \left(n\right)]\right\},$$

(16)

where r is the proportion of data used for secret key rate generation, $\Delta \left(n\right)$ is related to the security of the privacy amplification given by

$$\Delta \left(n\right)=7\sqrt{\frac{{log}_2(2/\overline{ϵ})}{r·N}}+\frac{2}{r·N}{log}_2(1/{ϵ}_{PA}),$$

(17)

where N represents the total exchanged signals, $\overline{ϵ}$ is the smoothing parameter, and ${ϵ}_{PA}$ is the failure probability of the privacy amplification procedure. Based on the derived results in [38], the value of ${\eta }_{min}$ and ${\xi }_{max}^2$ can be calculated by the maximum-likelihood estimators, which has the form

$${\left(\sqrt{\eta }\right)}_{min}\approx \sqrt{\eta }-Z_{{ϵ}_{PE}/2}\sqrt{\frac{1+\eta \xi }{N(1-r)(V-1)}},$$

(18)

$${\xi }_{max}^2\approx 1+\eta \xi +Z_{{ϵ}_{PE}/2}\frac{(1+\eta \xi )\sqrt{2}}{\sqrt{N(1-r)}},$$

(19)

where $Z_{{ϵ}_{PE}/2}=\sqrt{2}{\mathrm{erf}}^{-1}(1-{ϵ}_{PE})$ with the failing probability of parameter estimation ${ϵ}_{PE}$ and inverse error function ${\mathrm{erf}}^{-1}$. Here, erf represents the error function defined as

$$\mathrm{erf}=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}\mathrm{d}t.$$

(20)

Figure 8 shows the secret key rates $K_F$ as a function of the zenith angle $\zeta$ for various satellite altitudes, where the attenuation coefficient is optimal for each zenith angle. The parameters $\overline{ϵ}$, ${ϵ}_{PE}$, and ${ϵ}_{PA}$ are ${10}^{-10}$, and half of the signals are used for parameter estimation (r = 0.5). We find that the fewer signals exchanged, the more pronounced of finite-size effect. When the number of the exchanged signals is ${10}^6$, the performance shows a big gap compared with other scenarios, especially in the case of $H=120$ km (no secret key rate in that case). As the block length is increased, more signals can be used for parameter estimation and key extraction, so as a consequence, the secret key rate becomes closer to the asymptotic scenario.

5. Conclusions

We have proposed an NA-based scheme for improving the uplink CV-QKD protocol. Specifically, the NA process is realized by ZPC operation, which happens to be a zero-energy-input attenuation that does not cause the consumption of external energy. We find that this scheme shows better performance when the zenith angle is above ${26}^{\circ }$ and $3^{\circ }$ for 100 km and 120 km, respectively. Numerical simulations show that the ZPC operation improves the performance by relatively reducing the eavesdropper’s information. In addition, the ZPC-involved scheme performs better than the scheme with SPC operation, which is considered as another NA process. The research confirms the role of the NA operation on the uplink CV-QKD protocol and is conducive to making the CV-QKD protocol more practical. The limit of the proposed approach is that the effective time within a single communication cycle is still below 30 s. The next step of our work is to reach a longer effective time.