High-speed Gaussian modulated continuous-variable quantum key distribution with a local local oscillator based on pilot-tone-assisted phase compensation

A high-speed Gaussian modulated continuous-variable quantum key distribution (CVQKD) with a local local oscillator (LLO) is experimentally demonstrated based on pilot-tone-assisted phase compensation. In the proposed scheme, the frequency-multiplexing and polarization-multiplexing techniques are used for the separate transmission and heterodyne detection between quantum signal and pilot tone, guaranteeing no crosstalk from strong pilot tone to weak quantum signal and different detection requirements of low-noise for quantum signal and high-saturation limitation for pilot tone. Moreover, compared with the conventional CVQKD based on homodyne detection, the proposed LLO-CVQKD scheme can measure X and P quadrature simultaneously using heterodyne detection without need of extra random basis selection. Besides, the phase noise, which contains the fast-drift phase noise due to the relative phase of two independent lasers and the slow-drift phase noise introduced by quantum channel disturbance, has been compensated experimentally in real time, so that a low level of excess noise with a 25km optical fiber channel is obtained for the achievable secure key rate of 7.04 Mbps in the asymptotic regime and 1.85 Mbps under the finite-size block of 10^7.


Introduction
Continuous-variable quantum key distribution (CVQKD) allows the sender (Alice) and the receiver (Bob) to distribute the key information securely through an insecure quantum channel controlled by the eavesdropper (Eve) [1,2]. Moreover, CVQKD has attracted significant interest in the practical implementation of quantum communications, because it has the advantages of high quantum efficiency based on coherent detection and good compatibility with commercial off-the-shelf components [3][4][5]. The Gaussian modulated coherent state (GMCS) CVQKD protocol can be easily implemented in practice experiment, and meanwhile it also has been proven secure against collective attacks [6,7] and coherent attacks [8,9]. Therefore, it is foreseeable that GMCS CVQKD will become one of the main contenders of future QKD development [10][11][12].
In recent years, two important CVQKD schemes have been proposed according to different arrangement of local oscillator (LO) [13][14][15][16][17], named by the transmitting LO (TLO) and local LO (LLO) CVQKD, respectively. The TLO-CVQKD preserve good coherence between the quantum and LO signal that generated from the same laser, and the homodyne detection can be performed efficiently at Bob's site [18][19][20][21][22][23]. However, the TLO-CVQKD scheme suffers from the TLO intensity bottleneck and the potential security loopholes. Moreover, the TLO-CVQKD scheme based on homodyne detection requires extra random basis selection to realize the measurement of both quadrature X and P. By contrast, the LLO-CVQKD fundamentally removes the intensity bottleneck and the security loopholes of the TLO scheme by arranging the LO at Bob's site [24,25], and it can flexibly operate at heterodyne detection instead of homodyne detection for the simultaneous measurement of both quadrature X and P in the absence of random basis selection [26,27]. However, to realize the coherent detection efficiently in LLO-CVQKD scheme, a reliable phase compensation is required for reducing the dominate phase noise resulted from the phase drift of two independent lasers and fiber channel disturbance in the practical implementation, so that an tolerable excess noise can be guaranteed for the secure key distribution.
In order to achieve reliable phase compensation in LLO-CVQKD scheme, the pilot-pulseassisted methods, such as the pilot-sequential [28], pilot-delayline [29] and pilot-displacement [30] methods, have been proposed and experimentally demonstrated respectively, in which a bright pilot pulse is generated and propagated along with weak quantum signal by time multiplexing. However, the same balanced receiver has been used to measure the quantum signal and pilot pulse in the proposed methods [28][29][30], while the low saturation limit required for efficient detection of weak quantum signal restricts the detection of bright pilot pulse. Subsequently, the pilot-pulse-assisted methods are improved by separate transmission and detection of the quantum signal and pilot pulse by combining time and polarization multiplexing techniques [26,27]. Nevertheless, the high-speed LLO-CVQKD system based on time multiplexing requires extra physical synchronization pulses and complicated electrooptical devices for reducing the time crosstalk from the strong pilot pulse to the weak quantum signal in fiber channel, which limits its practice. So, a pilot-tone-assisted method based on frequency and polarization multiplexing has been proposed to solve timemultiplexing problem [31][32][33][34]. However, most reported schemes are experimentally demonstrated based on discrete modulated CVQKD protocol, which are not conducive to ensure maximum mutual information between Alice and Bob close to fiber channel capacity and sensitively monitor the eavesdropper Eve for the long distance CVQKD compared with GMCS CVQKD [5]. Therefore, the phase compensation methods that are capable of reducing the phase noise in LLO-CVQKD based on GMCS protocol, and at the same time avoiding the crosstalk between quantum and pilot signals are of great interest.
In this paper, we experimentally demonstrate a high-speed Gaussian modulated LLO-CVQKD scheme by using pilot-tone-assisted phase compensation. The demonstrated LLO-CVQKD scheme not only properly guarantees the different detection requirements for quantum signal and pilot tone, but also eliminates the crosstalk between them with the frequency-multiplexing and polarization-multiplexing techniques. Moreover, the scheme realizes simultaneous measurement of both quadrature (X and P) by heterodyne detection, avoiding the use of extra random basis selection. Besides, the dominant phase noise of the LLO-CVQKD can be well compensated in the experiment by the proposed reliable pilot-toneassisted phase compensation method, achieving a secure key rate of 7.04 Mbit/s in the asymptotic regime and a secure key rate of 1.85 Mbps under the finite-size block of N=10 7 with an low level of excess noise at the transmission distance of 25 km.

Scheme setup
and φ B represent the central frequency and phase of Bob laser, respectively. A lo1 and A lo2 are the amplitudes of the LLO signals E lo1 and El o2 generated from the Bob laser, respectively. R i and η i denote the equivalent resistance and quantum efficiency of two BHDs (i=1, 2), respectively. φ sig and φ ref represent the slow-drift phase of quantum signal and pilot tone introduced by quantum channel disturbance, respectively. It is observed from Eq. (3a) that the state received by Bob is degraded by the fast-drift laser phase and slow-drift channel phase with respect to the GMCS prepared by Alice. Therefore, a pilot-tone-assisted phase compensation method is proposed for eliminating the phase drift in the LLO-CVQKD scheme, which is based on the pilot assisted channel equalization recently applied in optical fiber LLO-CVQKD [25,30]. Firstly, the heterodyning products of the quantum signal and the pilot tone are orthogonally down-converted to the desired baseband components according to Eq. (3a) and (3b) respectively, which can be written as: where Δφ slow =φ ref -φ sig means the slow-drift relative channel phase between quantum signal and pilot tone. Next, the fast-drift laser phase Δφ fast is compensated based on Eqs. (4) and both quadrature values of the quantum signal can be simultaneously obtained as: where N 0 corresponds the shot noise variance of the LLO signals E lo1 . It is seen from Eq. (5a) and (5b) that both quadratures of the quantum signal are normalized to shot noise units.
Finally, the slow-drift channel phase Δφ slow can be estimated by a bit of training sequence disclosed in quantum signal, and the received quadratures X B and P B at Bob's site can be more precisely compensated as It is worth noting that the proposed LLO-CVQKD scheme avoids the crosstalk from strong pilot tone to weak quantum signal by properly separating the weak quantum signal and the strong pilot signal in different frequency sideband and orthogonal polarization state. Meanwhile, the scheme has the advantage of guaranteeing the sufficient optical power to achieve the low-noise detection for quantum and high-saturation limitation detection for pilot by providing a "locally" local laser, which gets rid of the intensity bottleneck and security hole in the TLO-CVQKD [30]. Moreover, one can see from Eq. (6a) and (6b) that both quadratures (X and P) are simultaneously obtained by heterodyne detection, avoiding extra random basis selection in the conventional CVQKD based on homodyne detection. Besides, the main advantage of the proposed scheme with respect to the time-multiplexing LLO-CVQKD scheme is that the pilot tone for phase compensation is shared by using high-resolution frequency-multiplexing technique, facilitating the promotion of the spectral efficiency and the repetition rate for a high-speed CVQKD.

Experimental demonstration
We experimentally performed the proposed LLO-CVQKD scheme based on the setup shown in Fig. 1. At Alice's site, a continuous optical carrier comes from Alice laser (NKT Photonic Basik) with narrow linewidth and low relative intensity noise (RIN), which is split into two branches of the HI by a BS with intensity ratio of 99:1. The weak optical carrier in the upper branch is sent into an amplitude modulator (AM1, EOSPACE) and modulated to be an optical pulse signal with repetition frequency of 100 MHz. The generated optical pulse is injected into another amplitude modulator (AM2, EOSPACE) for the required Rayleigh distribution modulation and then sent into two cascaded phase modulators (PM1 and PM2, EOSPACE) for the required uniform distribution modulation. So, the Rayleigh distribution and uniform distribution modulation are cascaded to provide a centered Gaussian distribution modulation. A VOA is used to adjust the GMCS with an optimized variance of V A . The pulse modulation electrical signal and the Gaussian modulation electrical signal are generated from two arbitrary waveform generators (Keysight M8195A and Ceyear 1652A). The strong optical carrier in the lower branch is modulated in a Mach-Zehnder modulator (MZM, EOSPACE) according to a sinusoidal microwave signal generated from microwave source (MS, Rohde & Schwarz SMB) for forming CS-DSB modulated pilot tone. At Bob's site, the quantum signal and the pilot tone are separately detected using two commercial BHDs (THORLABS PDB480C), and the corresponding LLO signals are supplied by Bob laser (NKT Photonic Basik). Moreover, the polarization controller (PC, General Photonics PSY PSY-201) is used to eliminate the deterioration of the polarization due to fiber channel disturbance in favor of the clear separation between the quantum signal and the pilot tone at Bob's site. Besides, two BHDs' electrical outputs are monitored and collected with a real-time oscilloscope (Keysight DSAZ254A) and an electrical spectrum analyzer (Rohde & Schwarz FSW43) for the subsequent offline digital signal processing (DSP). A pilot-tone-assisted phase compensation method in the offline DSP is designed to extract the phase sharing of the pilot tone for compensating the raw data of the quantum signal, from which the final key is distilled. All the optical components are connected with angle polished finishes to reduce the residual reflection as much as possible.   Gaussian modulated quantum signal and the expected optical-frequency deviation in the experiment. Thirdly, the band-pass filtered quantum signal is orthogonally down-converted with frequency of 0.69 GHz (Δf AB ) to the baseband for extracting the in-phase component X sig and the quadrature component P sig . Similarly, the in-phase component X ref and quadrature component P ref of the pilot tone also can be obtained by the orthogonal down-conversion with frequency 1.31 GHz (f m -Δf AB ) and baseband filtering. Finally, after the clock recovery, both quadratures of the quantum signal can be simultaneously obtained and normalized to shot noise units according to Eq. (5a) and (5b) after the compensation for the fast-drift laser phase Δφ fast from the phase sharing of the pilot tone. Subsequently, both quadratures of the quantum signal can be more precisely compensated based on Eq. (6a) and (6b) by estimating slow-drift channel phase Δφ slow from a bit of training sequence disclosed in the raw data of the quantum signal. As is shown in Fig. 4, both quadratures (X and P) between Alice and Bob are compared before and after phase compensation, which verifies the proposed pilot-toneassisted phase compensation method.

Excess noise and secure key rate
The excess noise estimation is a crucial step in CVQKD, which has direct influence on the secure key rate of the designed system. In order to estimate the excess noise and evaluate the secure key rate, the LLO-CVQKD system is considered to be a normal linear model y T x z η = + for Alice and Bob's correlated variables (x i , y i ) i=1,2,…,n, where η is the quantum efficiency and T is the transmittance efficiency, and z follows a centered normal distribution with a variance of σ 2 =ηTε+1+υ el . Note that the excess noise ε and the electronic noise υ el are normalized to shot noise units. For the proposed LLO-CVQKD scheme, one of the most critical contributions to the excess noise is the phase noise including laser phase noise and channel phase noise. The laser phase noise ε fast =2πV A (Δν A +Δν B )/f rep is resulted from the fast-drift laser phase between Alice laser and Bob laser, where Δν A and Δν B are the linewidths of Alice laser and Bob laser respectively [30]. In our scheme, the laser phase noise can be eliminated by sharing fast-drift laser phase Δφ fast of the pilot tone. The channel phase noise ε slow is introduced from different fiber channel disturbance between quantum signal and pilot tone, which can be compensated by sharing slow-drift channel phase Δφ slow from a bit of training sequence disclosed in the raw data of the quantum signal. In addition to the phase noise, the other noise sources, such as modulation noise due to finite dynamics and ADC quantization noise, should be reduced as much as possible for a tolerable excess noise ε in the experiment.
With the repetition frequency f p =100 MHz, the quantum efficiency η=0.56, the ratio of electronic noise and shot noise υ el /N 0 =0.042, the modulation variance V A =3.246, the secure transmission distance L=25 km and the reverse reconciliation efficiency β=95% [35], the experimentally measured excess noise on the block of size 5×10 6 and the worst excess noise in the considering of the finite-size effect are estimated over 30 minutes and shown in Fig. 5. Moreover, one can see from Fig. 5 that the mean of measured excess noise is around 0.022 and the mean of the worst excess noise under the finite-size effect is round 0.048, which are used to calculate the final key rate under the infinite-size and the finite-size effect, respectively. Note that the rest excess noise is probably originated from the polarization drift due to inaccurate polarization control, ADC quantization noise, the modulation noise because of finite dynamic and the residual phase noise resulted by imperfect phase compensation scheme [26]. Besides, the tolerable maximal value of excess noise at the secure transmission distance of 25 km is calculated and defined in the yellow solid line in Fig. 5. The experimental and simulation results of secure key rate corresponding to different secure transmission distance are plotted in Fig. 6. The red solid line denotes the secure key rate as a function of secure transmission distance under finite-size block of N=10 7 and the black dash line defines the calculated secure key rate as a function of secure transmission distance in infinite-size scenarios. As is shown in Fig. 6, the experimental secure key rate with infinite size and finite-size block of N=10 7 are calculated to be 7.04 Mbit/s and 1.85 Mbit/s, respectively. In addition, the previous experiment results are also shown in Fig. 6 for comparison, highlighting the proposed higher rate GMCS LLO-CVQKD.  Secure key rate versus secure transmission distance curves. The red solid line represents secure key rate at different secure transmisstion distance under finite-size block of N=10 7 and the black dash line represents secure key rate at different secure transmission distance in infinite-size scenarios, while the blue circle and the yellow square represent the experimental secure key rate with block size N=10 7 and infinite-size, respectively. The reverse reconciliation efficiency β is 95%, the quantum efficiency η is 0.56, the ratio of electronic noise and shot noise υ el /N 0 is 0.042 and the modulation variance V A is 3.246 in the experiment.
Note that the secure key rate is calculated under the finite-size effect, which can be expresses as [26,27] where V A denotes modulation variance. PE BE S ∈ represents the Holevo bound defining the maximum information available to Eve on Bob's key with finite-size effect, which can be calculated as 2 5 where G(x)=(x+1)log 2 (x+1)-xlog 2 x. Symplectic eigenvalues λ i can be derived from the covariance, which can be given by where 1 1

Conclusion
We have experimentally demonstrated a high-speed pilot-tone-assisted GMCS LLO-CVQKD system. In order to compensate the dominate phase noise of the proposed LLO-CVQKD scheme, a pilot-tone-assisted method is proposed based on the frequency-multiplexing and polarization-multiplexing techniques. Superior to the former pilot-pulse-assisted method, the pilot tone is separately transmitted and detected with quantum signal in different frequency band and orthogonal polarization state, guaranteeing no crosstalk from the strong pilot tone to the weak quantum signal and different detection requirements of low-noise for quantum signal and high-saturation limitation for pilot tone. Moreover, compared with the conventional CVQKD based on homodyne detection, the proposed scheme based on heterodyne detection not only avoids the need of the random base selection, but also realizes the simultaneous measurement of both quadrature (X and P). Besides, the phase noise caused by the fast-drift laser phase of two independent lasers and the slow-drift channel phase from fiber channel disturbance can be compensated in real time by sharing phase of the pilot tone and the disclosed quantum training sequence, so that a low level of excess noise at the secure transmission distance of 25 km can be obtained for the GMCS LLO-CVQKD with an asymptotic secure key rate of 7.04 Mbit/s and an secure key rate of 1.85 Mbps under the finite-size block of N=10 7 . It is worth noting that the final secure key rate might be further improved by robustly increasing the repetition rate of the proposed LLO-CVQKD, benefiting from the pilot-tone-assisted method based on the high-resolution frequency-multiplexing technique.