1. Introduction

Quantum key distribution (QKD) [1,2] enables two legitimate parties, Alice and Bob, to share secret keys on an un-secure quantum channel controlled by an eavesdropper, Eve. Continuous-variable (CV) QKD encodes information on quadratures of optical fields and extracts information by coherent detections [3–5], thereby possessing the advantages of high secret key rate, low cost and compatible with classical coherent optical communication, which is different from its counterpart discrete variable QKD. Gaussian modulated coherent state (GMCS) protocol [6] is the most mature CVQKD protocol. Theoretically, its unconditional security has been completely demonstrated [7–12], and many of its improvements have also been put forward [13–18]. Experimentally, on the other hand, encouraging developments in terms of high speed, long distance and field test have been achieved in the last one decade [19–26]. In particular, based on the theoretical prove and experimental success, the feasibility of CVQKD parallelly transmitted with classical signals has been demonstrated using multiplexing technique [27–32], which provides a feasible way of CVQKD directly compatible with present coherent optical communication network.

Recently, a new scheme compatible with classical communication, named simultaneous quantum key distribution and classical communication (SQCC) scheme, was suggested, which superposes Gaussian quantum signals on the classical binary phase-shift keying (BPSK) signals [33], enabling CVQKD and classical communication to implement simultaneously on a single wavelength with the same communication infrastructure. More interestingly, a noise-test experiment demonstrated that the SQCC scheme is feasible with classical quadrature phase-shift keying (QPSK) modulation using a locally generated local oscillator (LO), although it is sensitive to the phase noise [34]. In a sense, this scheme provides a simpler alternation method for one way CVQKD coexisting with classical signals in the present optical communication networks and can also extend to the two way case [35,36]. By using classical carrier phase estimation (CPE) algorithm (the Mth-power algorithm), the complexity of the SQCC system can be further simplified as it can eliminate the fast phase drift without the aid of the reference signal [37], which makes the SQCC system more closer and compatible to the practical classical optical communication system.

However, the frequency offset (FO) between the carrier and the true LO, which contributes to the slow phase drift of the carrier phase, is ignored in Ref. [37]. For a relatively large carrier frequency offset (CFO), for example, the level of MHz, the phase deviation caused by the frequency offset will be non-negligible when the symbol rate is 10 Gs/s and the data block length is of the order of 100 or larger, rendering an extra phase noise for the signal data after phase compensation. In practice, The FO between different lasers has always existed inevitably. Therefore, it is necessary to simultaneously perform the estimation of the carrier phase (CP) and the CFO, which can not only eliminate the slow phase drift caused by CFO but also improve the accuracy of the CP compensation. On the other hand, for practical situation with finite-size effects of the CVQKD, the superimposed classical QPSK modulation on the quantum signal may bring new problems to the channel parameter estimation, especially for the satellite-mediated channel with a fluctuated transmittance.

To solve the problems mentioned above, in this paper, we suggest the Kalman filter (KF)-enabled parameter estimation for the SQCC scheme over satellite-mediated link, including the carrier phase and frequency offset estimation, the phase noise estimation after phase compensation and the estimation of channel parameters characterized by the transmittance and excess noise. By employing a improved vector KF algorithm, the CP and the CFO can be estimated with high precision, meaning that the fast and slow drift of optical CP can be tracked accurately. Meanwhile, the phase estimation error is almost approximate to the theoretical mean square error (MSE) limit, which enables us to perform an excellent phase noise estimation after the phase compensation. By utilizing the superimposed classical QPSK modulation and the accurate CP compensation, one can obtain a more accurate estimation of excess noise with disclosing a lower rate of raw keys, which enables us to obtain a higher precision of channel parameter estimation and a better performance of secret key rate in the finite-size case. We perform numerical simulation based on the proposed parameter estimation methods and the elliptic-beam atmospheric channel model, confirming the feasibility of our parameter estimation methods for the SQCC scheme in both the downlink and uplink of the satellite-mediated link. Further, we find that based on the proposed KF algorithm, the SQCC system performance in terms of secret key rate and achievable zenith angle is better than that based on the Mth-power algorithm. Our work suggests the operable parameter estimation methods of the SQCC scheme and demonstrates the requirements for the SQCC scheme to operate on the satellite-mediated coherent optical communication system, which can also extend to fiber or underwater link, paving the way of the SQCC scheme to the practical employment.

This paper is organized as follows: In Sec. 2, we give a brief description of the SQCC scheme over satellite-mediated link and make a detailed noise analysis of it. In Sec. 3, the improved vector KF phase compensation algorithm is introduced, and the phase estimation noise after compensation is derived. Then, the channel parameter estimation method is shown in Sec. 4. We perform numerical simulation in Sec. 5. Finally, conclusions are drawn in Sec. 6.

2. SQCC over satellite-mediated link

In this section, we show the proposed scheme of the SQCC over the satellite-mediated link and the corresponding noise models.

2.1 Description of the SQCC

The schematic illustration of the SQCC over satellite-mediated link is shown in Fig. 1. At Alice’s side, she generates a continuous optical carrier with a central wavelength of $\lambda _A$. This continuous optical carrier is firstly modulated to optical pulses by the first amplitude modulator (AM), and then modulated in the cascaded AM and phase modulator (PM) for achieving the simultaneous QPSK and Gaussian modulation. After that, the optical pulses are attenuated by a variable optical attenuator (VOA) to be the expected coherent state $\left | ({{x}_{A}}+{{e}^{-i{{m}_{A}}\pi }}\alpha _C )+i({{p}_{A}}+{{e}^{-i{{n}_{A}}\pi }}\alpha _C ) \right \rangle$, where $x_A$ and $p_A$ are the Gaussian modulated information with variance $V_A$, $\alpha _C$ is the amplitude of the classical QPSK modulation, and $m_A, n_A \in \{0, 1\}$ are the classical bits. Then, she sends the signals to Bob. At Bob’s side, he amends the polarization of the signals by the polarization controller (PC), and couples the signal pulses with his locally local oscillator (LLO), which has the same wavelength with the signals, for heterodyne detection. The detected results ($x_B$ and $p_B$) are further processed to obtain the expected information by digital signal process (DSP), including carrier phase recovery, parameter estimation, classical and quantum data extraction and the conventional data postprocessing of CVQKD. Note that for the SQCC scheme, Eve has full access and control of the classical and quantum signals. While Bob extracts the classical bits from his received data, the whole system can be seen as a standard GMCS QKD system. Since Eve has the additional power to control the classical bits, the security of the proposed SQCC scheme is at least as strong as the standard GMCS QKD, meaning that the security proofs of the standard GMCS QKD can be applied to this scheme.

 

Fig. 1. The schematic diagram of the SQCC over satellite-mediated link. AM, amplitude modulator; PM, phase modulator; LO, local oscillator; P, fiber polarizer; Iso, Isolator; 90$^\circ$ OH, 90$^\circ$ optical hybrid; PC, Polarization controller; HD, homodyne detector. (a)-(e) represent the signal distribution in phase space from preparation to reception to recovery.

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2.2 Noise model

For the convenience of the performance analysis of the proposed scheme, a comprehensive noise analysis and modeling is required. According to the analysis results shown in Refs. [34,38], here we give a detailed description of the noise model for the proposed scheme, including the phase noise, the analog-to-digital converter (ADC) quantization noise and the bit error rate (BER)-introduced noise. There are also many other kinds of noise haven discussed in the previous literatures of CVQKD and therefore not expounded here.

Phase excess noise Phase noise is one of the main excess noises for all the LLO-based scheme as two independent laser sources are used for the legitimate parts. If the residual phase noise after Bob’s phase compensation is $\sigma _{ph}^2$, then the phase excess noise can be given by [34]

ADC quantization noise As the output detection results will be quantized by an ADC, there will be an additional error generated in the quantized value of quadratures of the signal. The ADC quantization noise, which is depended on the maximal amplitude of the modulated signal [4,38], has the following form as

BER-introduced noise The BER in classical communication will contribute a extra excess noise given by [34]

Other untrusted excess noise Except the analyzed noise mentioned above, there are some other kinds of noise fully considered in the conventional CVQKD, such as relative-intensity noise, modulation noise, background noise, excess noise caused by the Eve’s attack and so on. In order to facilitate the analysis, we will assume that the sum of these noises $\xi _0$ to be 0.01 (in shot noise units) in the rest content.

It is worthy to note that there are two noise models considered in the previous SQCC scheme, i.e. the untrusted and trusted noise model [34,37]. In the untrusted noise model, the phase excess noise is assumed to be controlled by Eve. Actually, from a more practical point of view, the phase excess noise is contributed from the phase compensation error, which can be monitored by the algorithm in real time. Consequently, in the trusted noise model, the phase excess noise is regarded as a trusted noise. In these situations, we will consider both of the two noise models in the following performance analysis.

3. Phase compensation and phase noise estimation

After Bob gets the measurements of two quadratures, the first step of his DSP is CP recovery. The phases of the carrier and the LO will be drifted fast with two random walk processes (modeled by a Wiener process), which is caused by the linewidth of the two lasers at Alice and Bob’s side. Meanwhile, the nonzero CFO between the two lasers will render a slowly extra phase drift. Therefore, both the fast and slow drift of CP should be compensated. In Ref. [37], it has adopted the Mth-power algorithm to perform the CPE. To reduce the effect of additive noise on phase estimation error, this algorithm uses an equal-tap-weight transversal filter to average the estimated phase over a sequence of symbols with the block length of $N_b$, which should be optimized so that the phase compensation error is minimum. However, the above Mth-power algorithm ignores the beat-frequency offset. To guarantee the offset-induced phase deviation in a block is smaller than $5^{\circ }$ [41], $N_b$ can not be too large. For example, if the symbol rate is 10 GS/s, $N_b = 100$, the CFO should be smaller than 1.4 MHz. In the analysis of Ref. [37], the optimal block length $N_b$ is about the order of 100, indicating that the CFO between the sender and receiver’s lasers should lower than the order of MHz.

To suppress the impact of frequency offset, one of approaches is to split the carrier recovery into two individual operations of CFO compensation and CP compensation. However, the performance of the CFO compensation has a significant impact on the CP compensation. Therefore, a better way to do carrier recovery is to simultaneously estimate the CP and CFO. Fortunately, the KF algorithm, which estimates the carrier phase based on the principle of MSE, is an excellent method to simultaneously track the CP and CFO with low complexity and high accuracy [42]. In Ref. [43], the vector KF is used to estimate the reference phase and the residual optical frequency difference for the LLO-CVQKD. To make it compatible with the proposed SQCC scheme, we give the improved vector KF algorithm as follows.

For the received complex signal $r = x_B + ip_B$, it is divided into fixed-length blocks with block length of $N_b$. CFO can be considered as a constant over tens block of signals since the laser linewidth is much smaller than the symbol rate. Consequently, according to the models described in Refs. [42,43], the state equation for the kth block can be given by

As practical measurements of the phase drifts are contaminated by noises from channel and detector, the measurement equation of phase drift can be given by

The above state equations and the measurement equation can be rewritten into matrix forms, given by

For practical heterodyne detection, the measurement noise is arising from three aspects: the noise superposed on the QBSK signals before heterodyne detection, the measurement noise of the homodyne detector and the vacuum noise entering the empty port of the input beamsplitter corresponding to the two arms. Consequently, the measurement noise $\sigma _n^2$ can be modeled as $\sigma _n^2 = \frac {1 + \xi _{el} + \eta T_q(V_A + \xi _{ADC} + \xi _0)/2}{4\eta T_c(\sqrt {2}\alpha _C)^2/N_0} = \frac {1 + \xi _{el} + \sigma _{n_0}^2}{4\eta T_c(\sqrt {2}\alpha _C)^2/N_0}$ [44], where $\sigma _{n_0}^2 = \eta T_q(V_A + \xi _{ADC} + \xi _0)/2$.

Based on the models described by Eqs. (8) and (9), we can establish an iteration algorithm to acquire an optimal estimation of the state vector $\mathbf {s}(k)$, which is denoted by $\hat {\mathbf {s}}(k)$. For the initialization stage, we set $\mathbf {P}(0) = \mathbf {0}$, $\hat {\mathbf {s}}(0) = \mathbf {0}$. For the prediction stage, the prior estimation of the current state vector $\hat {\mathbf {s}}^{-}(k)$ and its error covariance matrix $\mathbf {P}^{-}(k) = \mathbb {E}[(\mathbf {s}(k) - \hat {\mathbf {s}}^{-}(k))(\mathbf {s}(k) - \hat {\mathbf {s}}^{-}(k))^T]$ are given as

 

Fig. 2. The schematic diagram of the improved vector KF algorithm and the data recovery of the classical bit and quantum information.

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Since the transmittance of the satellite-mediated link is fluctuated in a frequency of the order of kHz, the measurement noise $\sigma _n^2$ will be fluctuated with the transmittance. It is reasonable to assume that the transmittance is a constant value in a coherent time of $10^{-4}$s. Therefore, to obtain a accurate estimation of the measurement noise, we should make a relative precise estimation of transmittance in each coherent time before the CPE. This estimation method is shown in Section 4. Then, the measurement noise $\mathbf {R}(j)$ can be updated for the $j$th coherent time.

Based on the above improved vector KF algorithm, we acquire an excellent tracking capability of CFO even for the FO of 50 MHz, as shown in Fig. 3(a), where we assume $\Delta f_A = \Delta f_B = 1$ MHz, the symbol rate is 10 Gs/s, the channel coherent time is $10^{-4}$ s, the parameter $\sigma _{\varrho }^2 = 5\times 10^{-9}$, the BER is $10^{-5}$ (which is the expected BER to obtain the displacement $\alpha _C$), the modulation variance $V_A = 4$ (shot noise units), the transmittance is 0.3 ($T_c$ and $T_q$), the block length $N_b = 5$, and the detector’s electronic noise and efficiency are 0.1 (shot noise units) and 0.6 respectively. Distinctly, the impact from the CFO can be almost completely eliminated and the estimation accuracy of carrier phase can be determined by the algebraic Riccati equation applied in the Ref. [43]. Therefore, after Bob’s phase compensation, we can estimate the residual phase noise as [43]

 

Fig. 3. The simulation results of the FO and the MSE of the phase estimation for the proposed KF algorithm. (a) The estimated FO. (b) The real-time phase estimation MSE.

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According to Eqs. (1) and (18), the phase excess noise increases with the spectral linewidth, the block length $N_b$, the power of the classical QPSK signal and the modulation variance $V_A$. To control the magnitude of phase noise and take advantage of the parallelism of the KF algorithm, we should choose the laser with smaller spectral linewidth and a overall consideration of block length $N_b$. From Eq. (3), the BER-introduced noise decreases with the increase of the power of the classical QPSK signal. To clearly see this point, we plot the phase excess noise, the sum of the phase excess noise and the BER-introduced noise and the BER as a function of the power of the classical QPSK signal in Fig. 4, where the phase excess noise based on the result in Ref. [37] is also shown (dash-dotted line) for comparison. We can see that the phase excess noise based on our improved vector KF algorithm is lower than that of the Mth-power algorithm in Ref. [37], which means a more accurate phase compensation can be achieved by simultaneously performing the CP and CFO estimation. Meanwhile, although a low classical QPSK signal power can help us to suppress the phase excess noise, the BER-introduced noise will increase remarkably while the classical QPSK signal power is lower than a certain value. Therefore, while the spectral linewidth of laser and the block length $N_b$ is selected properly, the classical QPSK modulation displacement $\alpha _C$ should be optimized for each zenith angle at a fixed height or at least be optimized at a fixed height for a overall consideration of all zenith angles.

 

Fig. 4. The sum of the phase excess noise and the BER-introduced noise, the phase excess noise and the BER as a function of the power of the classical QPSK signal. The linewidth of lasers is 1 kHz, the symbol rate is 10 Gs/s, the block length is $N_b = 5$, the modulation variance $V_A = 4$, $\eta = 0.6$, $\xi _{el} = 0.1$, the channel transmittance is 0.25. The left coordinate is corresponding to the noise variance, and the right coordinate is used to depict the BER of classical communication. We also use the method described in Ref. [37] to calculate the corresponding phase excess noise (optimizing the block length with Eq. (12) in Ref. [37]), which is displayed by the dash-dotted line.

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4. Channel parameter estimation

After finishing the carrier phase recovery, the next step of DSP is the classical bit recovery and the channel parameter estimation, in which the obtained channel parameters are indispensable for the following quantum information extraction and performance evaluation.

The channel is characterized by two parameters, the transmittance and the excess noise. For the SQCC scheme, the classical bits and the QKD signals may be controlled by Eve so that the transmittances of them may be different. For example, in the case of intercept-and-resent attack, Eve has some freedom to control the loss/noise of the QKD signals and the classical bits separately when she regenerates the overall quantum states. Therefore, one has to estimate both the transmittances of the classical bits $T_c$ and the QKD signals $T_q$ separately. This can be reached as follows.

Before the phase compensation, the quadratures of the received signal in Heisenberg picture are given by

Based on the estimated $\sqrt {T_c}$, we can further estimate the transmittance of the QKD signals $T_q$. Extract the classical displacement from $x_B'$, we have

Note, as the estimation of $\sqrt {T_c}$ is unbiased and the variance of the estimator $\sqrt {\hat {T}_c}$ is of the order of $O(\frac {1}{mV_C})$, the noise coming from the imperfect classical displacement extraction is negligible. Denote $B_i'$ the realizations of $x_B''$. The covariance of $x_A$ and $x_B''$ is $\texttt {Cov}(x_A,x_B'') = \sqrt {\frac {\eta T_q}{2}}V_A =: C_{AB}$. Then, the transmittance of the QKD signals $T_q$ can be estimated as

The next step is to estimate the excess noise. Although the superposed classical QPSK modulation can increase the total noise of the SQCC system, we can take advantage of this double-modulated scheme for the estimation of excess noise, which has been demonstrated in Ref. [45]. Furthermore, different from the method presented in Ref. [45], the estimation of excess noise can be finished without any classical communication to disclose the second modulation. Thus, to obtain the estimation of the excess noise $\hat {V}_n$, we use two maximum likelihood estimators $\hat {V}_{n_1}$ and $\hat {V}_{n_2}$. For the former one, it has the following form

For the latter one, it can be expressed as

We can see that $B_i' - \sqrt {\frac {\eta T_q}{2}}A_i$ and $B_i - \sqrt {\frac {\eta T_c}{2}}Q_i$ are normally distributed with variances $V_{N_1}$ and $V_{N_2}$, respectively. Consequently, $Y_1 := \Sigma _{i = 1}^{m}(\frac {B_i' - \sqrt {\frac {\eta T_q}{2}}A_i}{\sqrt {V_{N_1}}})^2$ and $Y_2 := \Sigma _{i = 1}^{m}(\frac {B_i - \sqrt {\frac {\eta T_c}{2}}Q_i}{\sqrt {V_{N_2}}})^2$ will be $\chi ^2$ distributed: $Y_1(Y_2) \sim \chi ^2(m)$. Thus, the mean of estimators $\hat {V}_{n_1}$ and $\hat {V}_{n_2}$ can be expressed as

Then, the estimator $\hat {V}_n$ can be obtained as

Note that given the transmittances of the classical and quantum signals, Eve adds noise to the classical bits will be equivalent to add it on the quantum signals as the classical and quantum modulations can be extracted out accurately. Consequently, it is feasible to take advantage of the dual modulation for the estimation of excess noise.

Now, let us resolve the left problem in the previous section. In the KF algorithm, to obtain a more accurate estimation of measurement noise, we need to estimate the transmittance of the classical bits $T_c$ and the noise $\sigma _{n_0}^2$ for each coherent time before the phase compensation. This can be achieved by two steps. Firstly, one can assume that the transmittances of the classical bits and the QKD signals are equal, so that one is able to obtain a coarse estimation of the measurement noise. Since the carrier phase is unknown for the originally received complex signals before phase compensation, we need to choose a quantity that can obtain the information of the transmittance but is insensible to the CP. For this purpose, we define $R_B = x_B^2 + p_B^2$, and the mean of it can be given by

In the above derivation, we have used the relationships $x_C^2 + p_C^2 = 2\alpha _C^2/N_0$, $\mathrm {Var}(x_C) = \mathrm {Var}(p_C) = \alpha _C^2/N_0 = V_C$. Based on the quantity of $R_B$, we can define the maximum likelihood estimator $\hat {R}_B = \frac {1}{m}\sum _{i = 1}^{m}R_i$, where $R_i$ denotes the realizations of $R_B$ ($i \in {1, 2, \ldots, m}$). Thus, the mean of $\hat {R}_B$ is given as

The channel transmittance $T$ (assuming that $T_c = T_q = T$) can be estimated by

Then, the measurement noise can be approximated as $\sigma _{n}^2 = \frac {1 + \xi _{el} + \eta \hat {T}_1V_A/2}{4\eta \hat {T}_1(\sqrt {2}\alpha _C)^2/N_0}$. With this approximated measurement noise, one can use the proposed KF algorithm to achieve a coarse carrier phase recovery with a relatively large block length (to achieve a fast carrier phase recovery), which can still reach a very low BER but result in a larger phase noise. After that, a high-precise estimation of $T_c$ can be obtained according to Eq. (25). Secondly, the noise $\sigma _{n_0}^2$ is able to estimated from the originally received complex signals, given by

Based on the estimated noise $\hat {\sigma }_{n_0}^2$, one can perform our improved KF algorithm again with an appropriate block length to obtain a more accurate estimation of carrier phase.

5. Performance analysis

In this section, we show the performance of the proposed scheme in terms of secret key rate based on the above noise model and the parameter estimation methods. The global simulation parameters involved in the following analysis are shown in the Table 1. We have consider the satellite-mediated link, where the probability distribution of the channel transmittance is obtained by the elliptic-beam model [46–49] shown in the Appendix A. Meanwhile, an extra extinction factor contributed from the absorption and back-scattering should be taken into account to obtain the final distribution of the channel transmittance. The extinction factor has the form $T_{ext} = \exp [-\vartheta \sec (\zeta )]$, where $\zeta$ is the zenith angle, and $\vartheta = 0.7$ [46,49]. We also take three atmospheric conditions into account, including the clear day, the clear night and the fog night. The detailed calculations of the secret key rate are shown in Appendix B. As the results corresponding to the atmospheric condition of clear day are close to that of the clear night, we don’t show them in the following simulations.

Table 1. Simulation parameters.

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We first show the numerical results for the untrusted noise model. In this noise model, although the residual phase noise can be estimated essentially from the proposed phase compensation algorithm, it is assumed to be controlled by Eve for a conservative evaluation. We show the performance of the proposed SQCC scheme in various altitude for both downlink and uplink in Fig. 5, where the laser linewidths of the transmitter and receiver are set to 1 kHz and 10 kHz, respectively. For each altitude, the disclosed raw key rate f and the classical signal power (shot noise unit) are optimized to get the maximum value of the sum of secret key rates throughout all the zenith angles. From the results shown in the figures, we can see that with our proposed parameter estimation methods, the SQCC scheme is feasible over satellite-mediated link for a typical data length of $10^9$ with various atmospheric conditions but requires, however, very narrow laser linewidth. Although the atmospheric thickness is identical for both the downlink and the uplink, the performance of the uplink is much worse as the turbulence of atmosphere effects the signals at the beginning of its transmission. For the laser linewidth of 1 kHz, the tolerable zenith angle of downlink, where the secret key rate is positive in the atmospheric condition of clear night, can be up to about 45 and 28 degree while the satellite altitude is 160 km and 300 km (black solid lines in the first row of the figures), respectively. The tolerable zenith angle reduces to about 44 and 23 degree (green solid lines in the first row of the figures) in the atmospheric condition of fog night. For the uplink, the tolerable zenith angle reaches about 33 degree in the clear night for the altitude of 100 km (black solid line in the lower left of the figures). Non positive secret key rate can be obtained when the atmospheric condition turns into fog night or the altitude increases to 150 km. If the laser linewidth becomes to 10 kHz, the performance of the both cases of downlink and uplink has a relatively large degradation. In this case, the tolerable zenith angle of downlink at the altitude of 160 km reduce to about 31 and 28 degree for the clear night and fog night (red and blue solid lines in the top left of the figures), respectively. At the altitude of 300 km, we can not obtain a positive secret key rate. The tolerable zenith angle of uplink at the altitude of 100 km degrades to only about 12 degree for the clear night (red solid line in the lower left of the figures). No secret key can be acquired in the other cases for the uplink when the laser linewidth is 10 kHz. The results of asymptotical secret key rates are also displayed in Fig. 5 to show the upper bound of performance for the untrusted noise model.

 

Fig. 5. The secret key rate of the proposed SQCC scheme as a function of the zenith angle under the untrusted noise model. The laser linewidths of 1 kHz and 10 kHz are considered for both the downlink and uplink case. The solid lines represent the secret key rates of finite-size case. The black lines denote the case of 1 kHz linewidth in the clear night. The red lines are the case of 10 kHz in the clear night. The green lines represent the case of 1 kHz in the fog night. The blue lines express the case of 10 kHz in the fog night. The dash-dotted lines are the secret key rates of the asymptotical case. CN, clear night; FN, fog night.

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We now consider the trusted noise model where the phase noise after phase compensation is not controlled by Eve. We set the laser linewidths to 10 kHz and 1 MHz, respectively. The numerical simulation results for various altitudes and atmospheric conditions are shown in Fig. 6. Similarly, the asymptotical secret key rates of the trusted noise model are also shown for comparison. We can see that the obtained performance and the tolerable laser linewidth are better and larger than that of the untrusted model above. For the laser linewidth of 10 kHz, the tolerable zenith angles of downlink at the altitude of 160 km can reach 52 degree for both the clear night and fog night (red and blue solid lines in the top left of the figures). Even the altitude is up to 500 km, the tolerable zenith angle at which the key can be distributed still reaches about 28 degree (red and blue solid lines in the top right of the figures). For uplink, the reachable zenith angle is close to 40 degree at the altitude of 100 km for the clear night (red solid lines in the lower left of the figures). At the altitude of 160 km, the tolerable zenith angle can be up to near 22 degree for the clear night (red solid line in the lower right of the figures). No key can be obtained for other atmospheric conditions. While the laser linewidth increases to 1 MHz, different from the situation of the untrusted noise model, we don’t find a significant performance degradation compared with that of 10 kHz laser linewidth, meaning that the standard distributed feedback lasers (linewidths of order of MHz) can be used in this case. From a practical point of view, since the phase noise can be tracked in real time with our improved vector KF algorithm, it is reasonable to consider the trusted noise model and thus feasible to deploy the SQCC scheme over satellite-mediated link with the commercial communication devices.

 

Fig. 6. The secret key rate of the proposed SQCC scheme as a function of the zenith angle under the trusted noise model. The laser linewidths of 10 kHz and 1 MHz are shown for both the downlink and uplink case. The solid lines represent the secret key rates of finite-size case. The red lines denote the case of 10 kHz linewidth in the clear night. The blue lines are the case of 10 kHz in the fog night. The cyan lines represent the case of 1 MHz in the clear night. The magenta lines express the case of 1 MHz in the fog night. The dash-dotted lines are the secret key rates of the asymptotical case. CN, clear night; FN, fog night.

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We remark that the optimal disclosed raw key rate f is lower than that of the case that only uses one estimator to estimate the excess noise, meaning that one can get a higher estimation precise of excess noise with sacrificing a low rate of raw keys. In other words, if take advantage of the extra information carried by the classical QPSK signals, one can not only perform a better excess noise estimation, but also acquire a higher secret key rate. Benefit from this, when the data length is $10^{9}$, the secret key rate considering finite-size effect is close to that of the asymptotical case, as shown in Fig. 5 and 6. Moreover, the block length $N_b$ is set to 5 for all the simulations above. One can obtain a higher precise of phase compensation by setting a smaller value of $N_b$, which results in a better system performance but at the expense of increasing the time complexity of the phase estimation. Besides, when the zenith angle approaches to the maximal tolerable zenith angle, we find that the BER of classical communication remains of the order of $10^{-5}$, which is acceptable for the practical communication.

With our proposed channel parameter estimation method, we also extend the SQCC scheme suggested in Ref. [37] to the satellite-mediated link and show its performance in Fig. 7, where the solid lines represent the results of our proposed scheme. We can see that for both the untrusted and the trusted noise model, the performance of secret key rate and the achievable zenith angle based on our improved vector KF algorithm is better than that based on the Mth-power algorithm, which results from the fact that the phase excess noise of the former is lower than that of the latter (see Fig. 4). Similar results can be obtained for other satellite altitudes.

 

Fig. 7. The secret key rate of the SQCC scheme using the KF algorithm or the Mth-power algorithm as a function of the zenith angle under finite-size effect. (a) Untrusted noise model with laser linewidth of 1 kHz. (b) Trusted noise model with laser linewidth of 1 MHz. The solid lines represent the results using our proposed KF algorithm. The dotted lines denote the results using the Mth-power algorithm in Ref. [37]. The block size $N_b$ is 5 for the case using the KF algorithm, while it is optimized for the case using the Mth-power algorithm. Other simulation parameters are the same with in Table 1. The atmospheric conditions here are all assumed to be clear night.

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6. Conclusion

We have proposed the KF-enabled parameter estimation for the SQCC scheme over satellite-mediated link. Using the improved KF CPE algorithm, the carrier phase and FO referred to the true LO can be simultaneously estimated and recovered with high precise, which enables the phase estimation error of the proposed KF algorithm can be lower than that of the Mth-power algorithm and be almost approximate to the theoretical MSE limit. Since Eve has full access and control of the classical bits, the transmittances of the classical signals and the quantum signals need to be estimated separately. Taking advantage of the accurate phase estimation and the dual modulation of the SQCC scheme, the excess noise can be estimated with not only a higher precise but also a lower rate of sacrificed raw keys. Based on our parameter estimation methods, numerical simulations demonstrate the feasibility and confirm the conditions required for the SQCC scheme in both the downlink and uplink. Further, benefiting from the lower phase estimation error, the performance of our proposed scheme in terms of the secret key rate and the achievable zenith angle is better than that of the SQCC scheme based on the Mth-power algorithm. Our work suggests the operable parameter estimation methods of the SQCC scheme that is applied into the satellite-mediated coherent optical communication system. These parameter estimation methods can also be used in other channels, such as fiber and under water channels.

Appendix

A. Elliptic-beam model

In this section, we give a brief review of the elliptic-beam model used for describing the probability distribution of the transmissivity of the satellite-mediated link [46–49]. In this model, the received beam is assumed to have the profile of ellipse and can be modeled by a random vector $\upsilon = (x_0, y_0, W_1, W_2, \varphi )$, where $(x_0, y_0)$ is the beam-centroid coordinates of the arriving beam in the received aperture, $W_1$ and $W_2$ are the two half axis of the elliptical beam, $\varphi$ is the angle of the half axis $W_1$ relative to the x-axis. With the knowledge of the mean values and covariance matrix of the vector $\upsilon$, one is able to estimate the probability density function of the transmissivity via the Monte Carlo method. In the Monte Carlo simulation, the transmittance T is expressed as [46–49]

(36)$$T = {{T}_{0}}\exp \left\{ -{{\left[ \frac{r/a}{R\left( \frac{2}{{{W}_{eff}}\left(\varphi - \phi\right)} \right)} \right]}^{\lambda \left( \frac{2}{{{W}_{eff}}\left(\varphi - \phi\right)} \right)}} \right\},$$

where

(37)$$\lambda \left( \tau \right)= 2a^{2}{{\tau }^{2}}\frac{\exp \left({-}a^{2}{{\tau }^{2}} \right){{I}_{1}}\left( a^{2}{{\tau }^{2}} \right)}{1-\exp \left({-}a^{2}{{\tau }^{2}} \right){{I}_{0}}\left( a^{2}{{\tau }^{2}} \right)}\left[ \ln \left( 2\frac{1-\exp \left( -\frac{1}{2}a^{2}{{\tau }^{2}} \right)}{1-\exp \left({-}a^{2}{{\tau }^{2}} \right){{I}_{0}}\left( a^{2}{{\tau }^{2}} \right)} \right) \right],$$

(38)$$R\left(\tau\right)=\left[\mathrm{ln}\left( 2\frac{1-\exp \left( -\frac{1}{2}a^{2}{\tau }^{2} \right)}{1-\exp \left({-}a^{2}\tau^{2} \right){I}_{0}\left( a^{2}\tau^{2} \right)} \right) \right]^{-\frac{1}{\lambda \left( \tau \right)}},$$

(39)$$\begin{aligned}{{T}_{0}}=&1-{{I}_{0}}\left( a^{2}\frac{W_{1}^{2}-W_{2}^{2}}{W_{1}^{2}W_{2}^{2}} \right)\exp \left[{-}a^{2}\frac{W_{1}^{2}+W_{2}^{2}}{W_{1}^{2}W_{2}^{2}} \right]- 2\left\{ 1-\exp \left[ -\frac{a^{2}}{2}{{\left( \frac{1}{{{W}_{1}}}-\frac{1}{{{W}_{2}}} \right)}^{2}} \right] \right\}\\ &\times \exp \left\{ -\left[ \frac{\frac{\left( {{W}_{1}}+{{W}_{2}} \right)^{2}}{\left| W_{1}^{2}-W_{2}^{2} \right|}}{R\left( \frac{1}{{W}_{1}}-\frac{1}{{W}_{2}} \right)}\right]^{\lambda \left( \frac{1}{{W}_{1}}-\frac{1}{{W}_{2}} \right)} \right\}, \end{aligned}$$

(40)$$W_{eff}\left(\tau\right) = 2a \left[ \omega\left( \frac{4a^{2}}{{{W}_{1}}{{W}_{2}}}e^{ \frac{a^{2}}{W_{1}^{2}}\left( 1+2{{\cos }^{2}}\tau \right) + \frac{a^{2}}{W_{2}^{2}}\left( 1+2{{\sin }^{2}}\tau \right)} \right) \right]^{{-}1/2},$$

$r = x_0^2 + y_0^2$, $\phi$ is the angle of $\textbf {r} = (x_0, y_0)$ in the polar coordinates $(r, \phi )$, a is the receiving lens radius, and $\omega (\cdot )$ denotes the Lambert W function.

To apply the Monte Carlo method, one needs to obtain the mean values and covariance matrix of the vector $\upsilon$. Since the parameter $\varphi$ is assumed to be a uniform distribution in $[0, \pi /2]$ and has no correlation with other parameters [47,48], the mean values of the vector $\upsilon$ reduces to the form [46,49]

(41)$$\bar{\upsilon} = (\langle x_0\rangle, \langle y_0\rangle, \langle W_1^2\rangle, \langle W_2^2\rangle),$$

where the elements $\langle x_0\rangle = \langle y_0\rangle = 0$. The covariance matrix of $\upsilon$ can be given as

(42)$$\Sigma = \left( \begin{array}{cccc} \langle\Delta x_0^2\rangle & 0 & 0 & 0\\ 0 & \langle\Delta y_0^2\rangle & 0 & 0\\ 0 & 0 & \langle\Delta W_1^2\rangle & \langle\Delta W_1\Delta W_2\rangle\\ 0 & 0 & \langle\Delta W_1\Delta W_2\rangle & \langle\Delta W_2^2\rangle \end{array} \right).$$

For downlink case,

(43)$$\langle\Delta x_0^2\rangle_{\textrm{down}} = \langle\Delta y_0^2\rangle_{\textrm{down}} = \varpi H\sec(\zeta_1),$$

(44)$$\langle W_i^2\rangle_{\textrm{down}} = \frac{4H^2\sec^2(\zeta_1)}{k^2W_0^2}[1 + \frac{\pi}{24}n_0W_0^2\sec(\zeta_1)\frac{H_0^3}{H^2} + 1.1C_n^2k^2W_0^{5/3}\sec(\zeta_1)\frac{H_0^{8/3}}{H^{5/3}} ],$$

(45)$$\langle\Delta W_i\Delta W_j\rangle_{\textrm{down}} = (2\delta_{ij} - 0.8)\frac{4.14H_0^{8/3}C_n^2H^{7/3}\sec^5(\zeta_1)}{k^2W_0^{7/3}}[1 + \frac{\pi}{24}n_0W_0^2\sec(\zeta_1)\frac{H_0^3}{H^2}],$$

where $k$ is the optical wave number, $\varpi$ the boresight error, $\zeta _1$ the zenith angle, $H$ the altitude of the satellite, $H_0$ the thickness of the atmosphere, $W_0$ the initial Gaussian beam radius, $n_0$ the scatterer particles per unit volume in the equivalent atmosphere channel and $C_n^2$ the atmospheric index-of-refraction structure constant in the equivalent channel. In this paper, we consider three atmospheric conditions, and the values of $C_n^2$ and $n_0$ under these three conditions are shown in Table 2. For uplink case,

(46)$$\langle\Delta x_0^2\rangle_{\textrm{up}} = \langle\Delta y_0^2\rangle_{\textrm{up}} = 1.16H_0C_n^2W_0^{{-}1/3}H^2\sec^3(\zeta_2),$$

(47)$$\langle W_i^2\rangle_{\textrm{up}} = \frac{4H^2\sec^2(\zeta_2)}{k^2W_0^2}[1 + \frac{\pi}{8}n_0W_0H\sec(\zeta_2) + 1.8C_n^2k^2W_0^{5/3}H_0\sec(\zeta_2) ],$$

(48)$$\langle\Delta W_i\Delta W_j\rangle_{\textrm{up}} = (2\delta_{ij} - 0.8)\frac{11H_0C_n^2H^{4}\sec^5(\zeta_2)}{k^2W_0^{7/3}}[1 + \frac{\pi}{8}n_0W_0^2H_0\sec(\zeta_2)],$$

where $\zeta _2$ is the zenith angle.

Table 2. The parameters of the equivalent atmosphere channel [46,49].

View Table | View all tables in this article

B. Calculations of the secret key rate

For the GMCS CVQKD protocol, the equivalent entanglement-based scheme of it is more convenient for security analysis. The covariance matrix of the ensemble-average state at the output of the fluctuating channel is given by [50–52]

(49)$$\Gamma_{AB} = \left( \begin{array}{cc} V\mathbf{I} & \sqrt{T_e}\sqrt{V^2 - 1}\sigma_Z\\ \sqrt{T_e}\sqrt{V^2 - 1}\sigma_Z & T_e(V + \chi_{line})\mathbf{I} \end{array} \right)$$

with the parameters

(50)$$V = V_A + 1,$$

(51)$$T_e = \langle\sqrt{T_q}\rangle^2,$$

(52)$$\chi_{line} = \frac{1 - T_e}{T_e} + \varepsilon_e - \langle\xi_{\sigma_{ph}}\rangle^{trust},$$

(53)$$\varepsilon_e = \frac{X_1(V - 1) + 2\langle V_{\varepsilon}\rangle/\eta}{T_e},$$

(54)$$X_1 = \langle T_q\rangle - \langle\sqrt{T_q}\rangle^2,$$

where $\mathbf {I} = \bigg (\begin {array}{cc} 1 & 0 \\ 0 & 1 \end {array}\bigg )$, $\sigma _Z = \bigg (\begin {array}{cc} 1 & 0 \\ 0 & -1 \end {array}\bigg )$, $T_e$ and $\chi _{line}$ are the effective transmittance and channel added excess noise, respectively. For the trusted noise model, $\langle \xi _{\sigma _{ph}}\rangle ^{trust} = \langle \xi _{\sigma _{ph}}\rangle = \int P(T_q)\xi _{\sigma _{ph}}(T_q)$, where $P(T_q)$ is the probability density function of $T_q$. Otherwise, $\langle \xi _{\sigma _{ph}}\rangle ^{trust} = 0$. We assume that $\langle V_{\varepsilon }\rangle = \eta \langle T_q\rangle \xi _t/2$ for simplicity. In practical implementation, $\langle V_{\varepsilon }\rangle$ should be regarded as $\int P(T_q)V_{\varepsilon }(T_q)$.

To get the secret key rate in the finite-size case, we first calculate the asymptotical secret key rate of the GMCS CVQKD. For the reverse reconciliation case, the asymptotical secret key rate read as [34]

(55)$$K_{asy} = \beta I_{AB} - \chi_{BE},$$

where $I_{AB}$ is the Shannon mutual information between Alice and Bob, $\beta$ is the reconciliation efficiency, $\chi _{BE}$ is the Holevo bound on Eve’s information about Bob’s secret key. The mutual information between Alice and Bob can be given by [34]

(56)$$I_{AB} = \log\left(\frac{V + \chi_{tot}}{1 + \chi_{tot}}\right),$$

where

(57)$$\chi_{tot} = \chi_{line} + \frac{\chi_{det}}{T_e},$$

(58)$$\chi_{det} = \frac{1 + (1 - \eta) + 2\xi_{el}}{\eta} + T_e\langle\xi_{\sigma_{ph}}\rangle^{trust}.$$

The Holevo bound between Eve and Bob is given as [34]

(59)$$\chi_{BE} = \sum_{j = 1}^{2}G\left(\frac{\lambda_j - 1}{2}\right) - \sum_{j = 3}^{5}G\left(\frac{\lambda_j - 1}{2}\right),$$

where $G(x) = (x + 1)\log _2(x + 1) - x\log _2x$. The symplectic eigenvalues $\lambda _{1,2}$ take the form

(60)$$\lambda_{1,2}^2 = \frac{1}{2}[\mathbb{A} \pm \sqrt{\mathbb{A}^2 - 4\mathbb{B}}]$$

with

(61)$$\mathbb{A} = V^2 + T_e^2(V + \chi_{line})^2 - 2T_e(V^2 - 1),$$

(62)$$\mathbb{B} = \left(VT_e(V + \chi_{line}) - T_e^2(V^2 - 1)\right)^2.$$

The symplectic eigenvalues $\lambda _{3,4}$ have the form

(63)$$\lambda_{3,4}^2 = \frac{1}{2}[\mathbb{C} \pm \sqrt{\mathbb{C}^2 - 4\mathbb{D}}],$$

where

(64)$$\mathbb{C} =\frac{1}{(T_e(V + \chi_{tot}))^2}[\mathbb{A}\chi_{det}^2 + \mathbb{B} + 2T_e(V^2 - 1) + 1 + 2\chi_{det}(V\sqrt{\mathbb{B}} + T_e(V + \chi_{line}))],$$

(65)$$\mathbb{D} =\left(\frac{V + \sqrt{\mathbb{B}}\chi_{det}}{T_e(V + \chi_{tot})}\right)^2.$$

The symplectic eigenvalue $\lambda _5 = 1$.

Under the finite-size case, the secret key rate is given by [45,51]

(66)$$K_{fini} = \frac{m_2}{m}[K_{asy}(T_e^{low}, X_1^{up}, \langle V_{\varepsilon}\rangle^{up}) - \Delta(m_2)],$$

where $X_1^{up} = X_1 + \gamma \sqrt {\textrm {Var}(X_1)}$, $\langle V_{\varepsilon }\rangle ^{up} = \langle V_{\varepsilon }\rangle + \gamma \sqrt {\langle \sigma _{\hat {V}_n}^2\rangle }$ [45], $\Delta (m_2)$ is given as [9]

(67)$$\Delta(m_2) = 7\sqrt{\frac{\mathrm{log_2}(2/\epsilon_{sm})}{m_2}} + \frac{2}{m_2}\mathrm{log_2}(1/\epsilon_{PA}),$$

where $\epsilon _{sm}$ is a smoothing parameter, $\epsilon _{PA}$ is the failure probability of the privacy amplification procedure. Here, $\gamma$ is a factor such that for a normally distributed variable $Z$ with mean $\mu$ and standard derivation $\sigma$, $P_{prob}(\mu - \gamma \sigma < Z < \mu + \gamma \sigma ) = 1 - \delta$, where $P_{prob}(\cdot )$ represents probability, $\delta$ denotes the significance level of the symmetric confidence interval around $\mu$. Consequently, $\gamma = \Psi ^{-1}(1 - \delta /2)$, where $\Psi$ is the cumulative distribution function of the standard normal distribution. For a usual magnitude of error $10^{-10}$, $\delta /2 = 10^{-10}$ and then $\gamma \approx 6.5$, which is used in this paper. To get $T_e^{low}$, we introduce another Gaussian variable $X_2:= \langle T_q\rangle + \langle \sqrt {T_q}\rangle ^2$ [51]. Thus, $T_e^{low} = (X_2^{low} - X_1^{up})/2$, where $X_2^{low} = X_2 - \gamma \sqrt {\textrm {Var}(X_2)}$. The variances of $X_1$ and $X_2$ are given by

(68)$$\textrm{Var}(X_1) = \textrm{Var}(X_2) = \sigma_{\langle T_q\rangle}^2 + 2\sigma_{\langle \sqrt{T_q}\rangle}^4[1 + 2\frac{\mu_{\sqrt{T_q}}^2}{\sigma_{\langle \sqrt{T_q}\rangle}^2}],$$

where $\sigma _{\langle T_q\rangle }^2 = \int (P(T_q))^2\sigma _{\hat {T}_q}^2$, $\mu _{\sqrt {T}_q} = \int P(T_q)\mathbb {E}\left (\sqrt {\hat {T}_q}\right )$, $\sigma _{\langle \sqrt {T_q}\rangle }^2 = \int (P(T_q))^2\textrm {Var}\left (\sqrt {\hat {T}_q}\right )$. For estimator $\sqrt {\hat {T}_q}$, we have

(69)$$\sqrt{\hat{T}_q} = \sqrt{\frac{2(\hat{C}_{AB})^2}{\eta V_A^2}}$$

Therefore, the mean value and variance of estimator $\sqrt {\hat {T}_q}$ are given as

(70)$$\mathbb{E}(\sqrt{\hat{T}_q}) = \sqrt{T_q},$$

(71)$$\textrm{Var}\left(\sqrt{\hat{T}_q}\right) = \frac{2\mathrm{Var}(\hat{C}_{AB})}{\eta V_A^2}.$$

We assume that $\epsilon _{sm} = \epsilon _{PA} = 10^{-10}$ in this paper.

We note that due to the present of turbulence, the laser beam wavefront arriving at the receiver will be distorted. This will lead to spot motion or image dancing at the focal place of the receiver. This effect is called angle-of-arrival fluctuation, which may lead to communication interruption. However, it can be compensated by the use of adaptive optics or fast beam steering mirror [53]. Therefore, in this paper, we have assumed that the angle-of-arrival fluctuation has been compensated and its effect is not considered.

Funding

National Natural Science Foundation of China (61801522, 61871407, 61872390); Postgraduate Scientific Research Innovation Project of Hunan Province, China (CX20200209); Postgraduate Independent Exploration and Innovation Project of Central South University (2020zzts136).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61871407, 61872390, 61801522), the Postgraduate Scientific Research Innovation Project of Hunan Province, China (Grant No. CX20200209) and the Postgraduate Independent Exploration and Innovation Project of Central South University (Grant No. 2020zzts136).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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