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Abstract
Continuous-variable quantum key distribution with a real local oscillator (LO) has been extensively studied recently due to its security and simplicity. Making it operate automatically and resistant to channel interference is the key to further long-term system operation. To overcome the dynamic changes in the state of polarization (SOP) of quantum signal caused by the random birefringence effect in fiber, Kalman filter is employed to estimate the polarization misalignment, thereby achieving polarization demultiplexing at the data level, and ultimately recovering the quantum signal with the help of a two-step phase compensation. The signal transmission and processing is simulated, which verifies the SOP tracking ability and the immunity to the fast phase drift. A proof-of-principle experiment is also conducted, and results show that it can resist the interference of SOP rotation at $1$ krad/s, and has the ability to distribute $8.4$ kbps secret key rate within $20$ km when only considering SOP tracking imperfection, confirming its feasibility under harsh channel conditions.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Continuous-variable quantum key distribution (CV-QKD) provides a secure way for two remote participants, the sender Alice and the receiver Bob, to establish a secret key through an untrusted channel [1]. Since the proposition of the CV-QKD protocol, its theoretical security [2–8] and practical implementation [9–12] have been greatly developed. Especially in practical use, CV-QKD using coherent states and coherent detection is a promising protocol due to its compatibility with existing telecom equipment and high detection efficiency [13]. In $2015$, to further promote the application of CV-QKD, a real local oscillator (LO) scheme was proposed [14–16]. In this scheme, LO is generated locally instead of the conventional way of transmitting it. The advantages are that the practical security loopholes caused by transmitted LO can be blocked [17,18] and the complexity of the actual implementation can be simplified [15]. Therefore it is promising to be a commercial system solution.
However, there is still an actual problem in the implementation of the local-LO scheme. Considering the random birefringence effect in optical fiber, the state of polarization (SOP) of optical quantum signal will change dynamically over time as it arrives at the receiver. On the one hand, previous experiments have shown that under mechanical vibration the movements of Stokes vector will be quite fast in a coiled patch fiber cable [19]. On the other hand, considering that QKD is expected to be employed in more diverse and complex scenarios, the SOP vibration will be more intensive than in buried fibers [20]. In discrete-variable QKD (DV-QKD) based on polarization encoding, changes in SOP will affect the polarization-encoded qubits, while in phase-encoding DV-QKD, the misalignment of SOP will result in the reduction of interference visibility [21], and both of them will eventually lead to an increase of bit error rate. By contrast, in CV-QKD, the coherent detection, one of the key technologies in CV-QKD, requires quantum signals and LO to remain consistent in SOP, otherwise a considerable portion of the signal power will not be detected, resulting in performance degradation or even obstructing the distribution of secure keys. In previous local-LO experiments, a manual polarization controller (MPC) was employed to control the changes in SOP, which is not conducive to long-term operation of system. Commercial digital polarization controllers (DPC), such as POS-002 (General Photonics), require a strong feedback optical signal to track the SOP [22], and therefore is not suitable for low-power quantum signal.
In this paper, a Kalman filter-based polarization tracking scheme is employed to deal with the receiver-to-transmitter polarization misalignment of the quantum signal in CV-QKD. In particular, in order to calibrate the phase drift due to the non-synchronization of two remote lasers, the simultaneous phase reference signal is prepared and transmitted along with the quantum signal by polarization multiplexing. After two signals are collected, the radius-directed linear Kalman filter (RD-LKF) proposed by Yang et. al. [23], which is immune to frequency offset and carrier phase noise, is used to evaluate the polarization misalignment, thereby achieving polarization demultiplexing at the data level. Then, relying on the phase information of the phase reference, a two-step phase compensation is performed on the quantum signal. Through such a combination, the quantum signal can be accurately recovered at the receiver, overcoming the polarization crosstalk due to the random birefringence effect, ensuring that the system operates automatically for a long time. The simulation results show that the proposed scheme can track the SOP and maintain a low excess noise even under the interference of SOP rotation at $1$ krad/s, and the experiment results show that it can achieve accurate polarization demultiplexing and are promising to have a $8.4$ kbps secret key rate (SKR) output when only considering SOP tracking imperfection.
The paper is organized as follows. In Section 2, the principles of scheme, including CV-QKD with simultaneous phase reference pulse, RD-LKF, and two-step phase compensation are introduced. In Section 3, by simulating the whole signal processing, the tolerance of SOP rotation rate and the immunity to frequency offset and carrier phase noise are analysed. In Section 4, a proof-of-principle experiment is conducted to verify the feasibility of this scheme. The paper is concluded in Section 5.
2. Scheme principle
2.1 CV-QKD with simultaneous phase reference signal
The key to implementing the local LO scheme is to eliminate the phase drift caused by the non-synchronization of lasers at both ends. Since the repetition rate of the CV-QKD system is lower compared to coherent optical communication, the phase drift is very large after a symbol interval, causing the classical phase compensation scheme to be unavailable. For this, in the beginning, an interval phase reference signal scheme is proposed [14–16]. To further eliminate the phase uncertainty between the quantum signal and the reference signal, these two signals are designed to be generated and detected simultaneously, and transmitted through the channel by various classical multiplexing schemes to ensure that they can be completely separated at the receiver [24–27]. Here the polarization-multiplexing scheme is adopted, which is described in Fig. 1(a). Alice first generates a pulse signal using a laser with a center wavelength of $\omega _{sig}$, and splits it into two pulses to produce a quantum signal and a phase reference signal, respectively. The quantum signal carries the key information by modulation, while the phase reference signal is designed to carry the original phase information without modulation. Then, these two signals are combined by orthogonal polarization multiplexing and transmitted through the optical fiber. At Bob’s side, due to the random birefringence effect in fiber, the SOP of received signal is rotated dynamically, resulting in the aliasing of the quantum signal and the phase reference signal in both paths after passing through a polarization beam splitter (PBS). A local LO with a center wavelength $\omega _{LO}$ is splitted into two parts, and then used for heterodyne (X and P) detection with two signals respectively (which is also called as intradyne detection in the field of coherent optical communication). The entire signal transmission process can be expressed as
Fig. 1. (a) CV-QKD with simultaneous phase reference. BS: beam splitter; Mod: modulation; PBC: polarization beam collector; PBS: polarization beam splitter; BD: balanced detector. (b) Schematic diagram of radius-directed linear Kalman filter. $\boldsymbol{Z}(k)$: input; $\boldsymbol{U}(k)$: measurement prediction; $\boldsymbol{U}_{c}(k)$: constraint; $\Delta \boldsymbol{U}(k)$: residual; $\boldsymbol{S}(k)$: state vector; $\boldsymbol{S}^{-}(k)$: state vector prediction; $\boldsymbol{P}^{-}(k)$: priori estimate error covariance; $\boldsymbol{P}(k)$: posteriori estimate error covariance.
2.2 Radius-directed linear Kalman filter
According to Eq. (2), one can find that it is the Jones matrix that causes the quantum signal and the phase reference signal to be aliased. In order to recover the transmitted signal, the polarization demultiplexing process should be
Fig. 2. (a) Constraints on the ideal circle. Enforcing the measurements onto the ideal circle. (b) Two-step phase compensation. The red dots represent the ideal symbols of quantum signal, the blue dot represents the ideal symbols of phase reference signal, and the purple dot represents the received symbols. $\theta _{ref}$ is for the first rotation, while $\theta _{s}$ is for the second rotation.
2.3 Two-step phase compensation
After polarization demultiplexing, the quantum signal and the phase reference signal are separated into two components in $U_{k}$. However, due to the frequency offset and carrier phase noise, the phase of the quantum signal at both sides are not aligned. In [33], a two-step phase compensation method is proposed aimed at fast and slow phase drift respectively. Here it is employed for phase recovery and depicted in Fig. 2(b). First, since the quantum signal and the phase reference signal are simultaneously prepared and detected, they carry the same phase $e^{j( \Delta \omega t + \theta _n(t))}$, so the phase angle of the reference signal is used to rotate the quantum signal:
in which ${U}_{H}^{1}(k)$ denotes the results after rotation. Then, due to the phase difference caused by the path difference at both sides, there is still a fixed phase deviation existing. According to [33], some of the corresponding data of Alice and Bob is disclosed to estimate the phase deviation $\theta _{s}$, which is then used to do the second phase rotation on the undisclosed portion of the data: where ${U}_{H}^{2}(k)$ refers to the recovered quantum signal. It should be noted that the above parameters all use the complex representation for convenience, where the real part represents the X-quadrature component and the imaginary part represents the P-quadrature component. Finally, utilizing the recovered quantum signal and the corresponding Alice modulation data, the secret key can be ultimately generated by reconciliation and LDPC decoding.3. Performance analysis
3.1 Signal transmission and processing
According to the principles described in Sec. 2, the simulation of the signal transmission and processing is carried out. The whole process is displayed in Fig. 3, where the left part of each figure represents the distribution of the signal on the constellation in H-polarization, and the right part represents the distribution in V-polarization. Figure 3(a) shows the modulated quantum signal and the unmodulated phase reference signal. Here, the SNR of quantum signal is set to $-3$ dB, which meets the low SNR requirement in CV-QKD, while the SNR of phase reference is set to $30$ dB to ensure the accuracy of phase compensation. The symbol rate is set to $50$ MHz, which is consistent with the current CV-QKD experiments. Figure 3(b) depicts the phase drift due to the unsynchronization of lasers and the phase difference caused by different paths. Here, the frequency offset is set to $1$ MHz, the total linewidth is $10$ kHz, and the phase difference is $\frac {2}{3} \pi$. Figure 3(c) displays the polarization crosstalk due to the time-varying Jones matrix. Here the Jones matrix is assumed as $\boldsymbol{J}=[\cos (kwT_{s}) \sin (kwT_{s}); -\sin (kwT_{s}) \cos (kwT_{s})]$, where $w$ takes $1$ krad/s, characterizing the polarization rotation angular frequency. It can be seen that after the above transmission process, the phase reference signal and the quantum signal are completely aliased together and cannot be distinguished.
Fig. 3. Signal transmission and processing. The SNR of quantum signal is set as $-3$ dB, and the SNR of reference signal is $30$ dB. The shot noise and the electronic noise in two detectors are assumed to be equal here and has the proportion $10 \lg (v_{el} / N_{0}) = -20$ dB. The total loss $\eta T$ is set as $1$ for convenience. Other parameters are set as: $\Delta \omega =1$ MHz, $\Delta v = 10$ kHz, $\theta _{ch}=\frac {2}{3} \pi$, $w=1$ krad/s.
Next, RD-LKF is employed to achieve channel equalization. Figure 3(d) illustrates that after this process, the quantum signal and the phase reference signal have been polarization demultiplexed. Here, according to [23], the tuning parameters $Q$ and $R$ are experimentally tested to have the best performance when set as $10^{-6}$ and $0.1$ respectively. In Fig. 3(e), relying on Eq. (15), the first phase compensation is performed. It can be seen that as the phase reference signals return to the initial phase value, the quantum signals carrying different phase modulation information are also separated. Finally, we eliminate the bias in both quadratures and then perform the second rotation (see Fig. 3(f)). During the simulation, we found that $\theta _{s}$ were not exactly the same as $\theta _{ch}$, which would be discussed in Sec. 3.3. After these procedure, we estimate the excess noise existing in the recovered quantum signal. The estimation method we adopt here is based on the conditional variance in each quadrature, i. e. $\varepsilon _A =\varepsilon _B / \eta T = 2 \cdot (V_{B|A}-1-v_{el}) / \eta T$ [25], where $\varepsilon _A$ and $\varepsilon _B$ represent the excess noise at Alice’s side and Bob’s side respectively, and all these parameters are normalized by the shot noise $N_{0}$.
3.2 Resistance against polarization changes
The SOP tracking ability of RD-LKF algorithm is analyzed here. We first randomly choose a constant Jones matrix and set the initial state vector $[a, b, c, d]$ to $[0.5, 0.5, 0.5, 0.5]$. In Fig. 4(a), one can clearly see how the parameters change with the number of input symbols. After just $750$ symbols, $[a, b, c, d]$ converges to a constant value, reflecting the fast convergence of Kalman filtering. These $750$ symbols are the minimum samples to perform the tracking. If the parameter $Q$ increases, the minimum sampling points will be further reduced. In Fig. 4(b), we set the polarization rotation angle frequency $w=1$ krad/s, and show the variation of $[a, b, c, d]$. It can be seen that as the SOP rotates, $[a, b, c, d]$ also fluctuate at the same frequency, indicating that the SOP is dynamically tracked. Considering that the excess noise is a key parameter in determining the performance of CV-QKD, the capability of the resistance against SOP changes is measured according to $\varepsilon _A$. In Fig. 4(c), we estimate the excess noise at different polarization rotation frequency, and considering the low SNR characteristics of quantum signal in CV-QKD, different low SNR is chosen to simulate. The excess noise is evaluated using $10^{6}$ symbols, with the first $1000$ symbols discarded since the SOP has not been tracked yet. It can be seen that when the rotation frequency is less than $1$ krad/s, the power of the excess noise is substantially constant. However, when the rotation frequency surpasses $1$ krad/s, the excess noise will increase dramatically, and eventually exceed the threshold of the key generation that is calculated according to [34]. Sometimes the excess noise will be lower than $0$, which is caused by the statistical errors with finite samples. It is worth noting that here $Q = 10^{-6}$ depends on both the tracking speed and the estimation accuracy. If $Q = 10^{-5}$, the noise due to estimation inaccuracy will exceed the threshold. Besides, it can be seen that the fundamental excess noise will be larger when the optical SNR is higher. For example, when the SNR is $0$ dB, $5$ dB, $10$ dB, the average excess noise is $0.018$, $0.058$, $0.103$. The main reason for this phenomenon is that due to the residual phase deviation, the more intense signal power is, the larger excess noise will be [27]. As for the phase reference signal, the large SNR not only ensures the accuracy of polarization demultiplexing, but also improves the accuracy of subsequent phase compensation. For example, when the SNR is $20$ dB, $25$ dB, and $30$ dB, the residual excess noise after polarization demultiplexing and phase compensation is $0.055$, $0.025$, and $0.018$, respectively.
Fig. 4. (a) Fast convergence performance with constant SOP deviation. (b) Tracking ability with SOP rotation at 1krad/s. (c) Excess noise changes with polarization rotation angular frequency. The threshold of key generation within $20$ km is calculated according to [34] with typical parameters: $V_{A}=0.5$, $\eta =0.6$, $\beta =95\%$, $v_{el}=0.01$, $\alpha =0.2$ dB/km.
3.3 Immunity to phase drift
Since the convergence constraint of RD-LKF is the entire circle formed by the rotation of the symbol, it is inherently insensitive to carrier phase noise and frequency offset, which are verified by simulation here. In Fig. 5(a), we estimate the excess noise at different laser linewidths and different frequency offsets. It can be seen that the effect of laser linewidth and frequency offset is almost negligible, confirming its immunity to the phase drift. However, RD-LKF only constrains the signal amplitude on each polarization channel, while the phase is not constrained. Therefore, the recovered signal and the original signal will have a certain phase deviation, which is related to the initial state vector $[a, b, c, d]$. For example, if the initial state vector is set to $[1, 0, 0, 0]$, the constellation distribution obtained after the first phase compensation is just like Fig. 5(b). When the initial state vector is $[0.5, 0.5, 0.5, 0.5]$, the signal distribution is shown in Fig. 5(c), and one can find the difference in the phase deviation. This can be explained that since the signal phase is not constrained, different initial state vector will eventually converge to different result during the Kalman filtering process. Fortunately, owing to the second phase compensation, such a phase deviation are considered in $\theta _{s}$ and can be compensated. In practice, we find that disclosing $1/10$ of data for the second phase compensation is enough, and this part will be calculated in the system overhead.
Fig. 5. (a) Excess noise under different frequency offset and laser linewidth. (b) Constellation distribution with initial [a,b,c,d]=[1,0,0,0]. (c) Constellation distribution with initial [a,b,c,d]=[0.5,0.5,0.5,0.5].
4. Experiment results
We conduct an experiment to verify the feasibility of RD-LKF in CV-QKD system with a local LO. The experimental setup is depicted in Fig. 6. At Alice’s side, a commercial frequency-stabilized CW laser with $150$ kHz linewidth (Wavelength Reference, Clarity-NLL-1542-HP) is employed as the signal laser. One Lithium Niobate electro-optic amplitude modulator (EOSpace) is used to generate a pulse train with $2$ns pulse width and at a repetition rate of $50$ MHz. Then, a $50:50$ beamsplitter (BS) is used to split into the signal path and the reference path. In the signal path, a phase modulator is adopted to achieve the QPSK modulation format, followed by a variable optical attenuator (VOA) to adjust to the required modulation variance $V_{A}$. In the reference path, the reference pulse is delayed by a delay line so that it can aligned to the signal pulse in the time domain. A second VOA is used to adjust its intensity. Next, these two pulses are converged by a polarizing beam collector (PBC) and then transmitted through a $20$ km SMF-28 fiber spool with a tested $3.82$ dB attenuation to the receiver. Here the PBC is the same device as the PBS, only the direction of light transmission is opposite. At Bob’s side, the mixed signal are separated through a PBS. Meanwhile, a LO is generated from another frequency-stabilized CW laser that has a frequency offset about $20$ MHz compared with Alice’s laser, and then divided into two parts by a BS with a splitting ratio of $99:1$. The pigtails of these devices are polarization-maintaining. On the one hand, the large one is interfered with the pulses from the signal path in a commercial $90$ degree optical hybrid (Optoplex), followed by two $350$ MHz balanced detector (Thorlabs, PDB130C) to achieve heterodyne detection. On the other hand, the small portion of the LO along with the pulses from the reference path is sent to another $90$ degree optical hybrid, followed by another two $350$ MHz balanced detectors (Thorlabs, PDB130C) to detect both X-quadrature and P-quadrature components. All the modulation signals are generated by an Arbitrary Waveform Generator (Tektronix, AWG5204), and the output signals are collected by a four-channel oscilloscope with $5$ GS/s sampling rate (LeCroy, Wavemaster 806Zi-A).
Fig. 6. Set-up of our experiment. CW laser: continuous-wave laser; AM: amplitude modulator; PM: phase modulator; BS: beamsplitter; DL: delay line; VOA: variable optical attenuator; PBC: polarizing beam collector; AWG: arbitrary waveform generator; SMF: single mode fiber; PSA: polarization state analyzer; MPC: manual polarization controller; PBS: polarizing beamsplitter; BD: balanced detector.
We first inject the optical signal transmitted through the fiber spool into a polarization state analyzer (General Photonics, PSY-201) to investigate the change in the SOP. Taking into account the scaling range of the polarization analyzer, the input optical power is controlled at $-20$ dBm. We tested the SOP for $6$ hours and found that the Stokes vector on the Poincare sphere changes very slowly (see Fig. 7(a)). In order to verify the feasibility of the above polarization tracking scheme, the interference of SOP rotation is digitally loaded onto the original signal [23,35]. The specific experimental steps are as follows: a. Firstly, we cut off the quantum signal path through VOA1, and adjust the power of the phase reference signal through VOA2. By observing through the oscilloscope, we adjust the MPC to ensure the phase reference signal is injected into the reference path as much as possible, avoiding entering the signal path and interfering with the LO. b. Keep the MPC unchanged, adjust VOA1 to release a weak quantum signal. These signals are collected by the oscilloscope with each frame containing $10^{5}$ pulse symbols. c. We extract the data of the peak point of pulses as the quantum signal data and the phase reference data. At the same time, considering that the fluctuation of the LO intensity will lead to the variation of the shot noise variance, here we propose a scheme to monitor the shot noise in real time, that is, the shot noise power is evaluated by sampled data between two signal pulses. Since the electronic noise is basically constant, it is calibrated in advance. d. We compensate the quantum signal according to the two-step phase compensation and evaluate the SNR of quantum signal and phase reference signal. According to the excess noise estimation method introduced in Sec. 3.1, the excess noise existing in the quantum signal after phase recovery is evaluated as $\varepsilon _{B1}$. e. Then, we add the SOP rotation on the original signal data, which is achieved by digitally multiplying the rotation Jones matrix $\boldsymbol{J}=[\cos (kwT_{s}) \sin (kwT_{s}); -\sin (kwT_{s}) \cos (kwT_{s})]$ with $w=1$ krad/s. f. Finally, polarization demultiplexing and phase recovery are performed, and then residual excess noise is evaluated. By subtracting $\varepsilon _{B1}$ from this noise, the excess noise $\varepsilon _{B2}$ caused by the imperfection of SOP tracking can be obtained.
Fig. 7. (a) 6 hours polarization state test. (b) Excess noise $\varepsilon _{B1}$ with different SNR of received signal. (c) Excess noise $\varepsilon _{B2}$ and corresponding achievable key rate. the average $\bar {\varepsilon _{B1}}=0.0397$ is treated as Bob’s trusted noise, $\bar {\varepsilon _{B2}}=0.0095$ is considered as channel excess noise. Length of each frame is $10^{5}$, in which $10 \%$ of data is used for second phase compensation. The modulation variance $V_{A}$ is assumed as $0.5$. the reconciliation efficiency $\beta$ is assumed as $95 \%$. Other parameters are calibrated as: the quantum efficiency $\eta =0.58$, the electronic noise $v_{el}=0.18$, the attenuation of fiber spool is $3.82$ dB, the repetition rate is $50$ MHz.
In Fig. 7(b) we first exhibit the relationship between the SNR of quantum signal and phase reference signal and the excess noise $\varepsilon _{B1}$. Since the VOA is manually adjusted, it is hard to accurately control the modulation variance $V_{A}<1$. Therefore, the SNR of quantum signal is just controlled in the range of $[-4,3]$ dB, and the phase reference signal is in the range of $[26,28]$ dB. The jitter of SNR is mostly caused by the fluctuation of the laser intensity. In this case, the estimated excess noise $\varepsilon _{B1}$ is in the range of $[0.01, 0.09]$. Such noise is derived from the leakage of the reference signal, limited phase compensation accuracy due to the finite reference signal power, and limited sampling quantization accuracy. Besides, one can find that when the SNR of quantum signal decreases, $\varepsilon _{B1}$ is also reduced, confirming that the noise is mainly caused by the phase deviation [27]. Considering that in the actual four-state CV-QKD experiment, especially in the case of long distance, the SNR is extremely low, it is promising that $\varepsilon _{B1}$ can be further suppressed. Then we load the SOP rotation on the data, and perform the data recovery and the evaluation of $\varepsilon _{B2}$ (see Fig. 7(c)). It can be seen that this part of noise is relatively small compared with $\varepsilon _{B1}$, verifying the effectiveness of our polarization tracking scheme. If we consider $\varepsilon _{B2}$ as untrusted excess noise and $\varepsilon _{B1}$ derived from other noise sources as trusted noise, we can evaluate the effect of SOP tracking imperfection on the final SKR. Such means of treating different noise sources separately in actual conditions is inspired by [36], which we think highly helpful to analyze the effects of certain noise source. In this case, based on the average $\bar {\varepsilon _{B1}}=0.0397$ and $\bar {\varepsilon _{B2}}=0.0095$, the achievable SKR is evaluated as $8.4$ kbps within $20$ km under $1$ krad/s SOP rotation interference. In the next step, higher polarization isolation devices will be employed to avoid noise caused by the leakage of phase reference signal, and electronically controlled VOAs will help us to adjust the modulation variance precisely, thus achieving a real four-state CV-QKD. On the other hand, in order to verify that this scheme can resist the randomly added SOP variations, we will try to conduct the experiment under the actual buried fiber and use analog-digital converters with large storage capacity for long-term data acquisition, which can evaluate the practical feasibility under real conditions.
5. Conclusion
In conclusion, to resist the disturbance of SOP rotation and ensure the long-term operation of the system, a polarization tracking scheme based on RD-LKF is employed in local-LO CV-QKD. The SOP tracking ability and the immunity to the fast phase drift are verified through simulation. Besides, we have conducted a proof-of-principle experiment using standard optical devices. Experiment results show that it can withstand the SOP changes at $1$ krad/s and achieve a key rate of $8.4$ kbps within $20$ km optical fiber when only considering SOP tracking imperfection. Since QKD is expected to operate in more and more complex scenarios, this scheme can work for harsh channel conditions and long-term automatic operation.
Funding
National Natural Science Foundation of China (61332019, 61631014, 61671287); National key research and development program (2016YFA0302600).
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