Quantum key distribution (QKD) [1–4] provides an efficient way for two legitimate remote users (Alice and Bob) to implement secure communications over untrusted quantum channels. Various protocols in quantum mechanics are invented to further guarantee the generated secure key against adversaries (Eve) [5]. Until now, QKD can be mainly divided into two families based on encoding and decoding schemes: discrete-variable QKD (DVQKD) and continuous-variable QKD (CVQKD). With the advantages of high key rate and compatibility of classical optical communication systems [6–8], CV-QKD based on coherent detection over an access network has attracted much attention [9,10]. The access network connects multiple endpoints to a common network node, with transmission distances ranging from a few hundred meters to several kilometers. These relatively low-loss transmission scenarios are well compatible with the characteristic of the high channel capacity of CVQKD [9,11,12]. However, high-speed secure access networks require ultra-high downstream rates, typically exceeding 1 Gbps. Therefore, the development of high-rate quantum key distribution is of great importance for its practical application [13,14].

Among many active CVQKD protocols, the Gaussian modulated coherent-state (GMCS) protocol is one of the most popular schemes, which has been experimentally demonstrated in both laboratory and field environments [15–19]. However, high-speed modulation with a high extinction ratio and stability in realistic conditions is challenging, which potentially limits the secret key rate (SKR). The reported GMCS-CVQKD systems fail to satisfy the high-speed access networks [20,21]. To improve the performance and ease of operation of CVQKD, another practical active CVQKD protocol based on a discrete modulation coherent state (DMCS) is proposed [22], which has recently made significant progress in terms of SKR [23,24]. Unfortunately, for the quantum secured gigabit optical access network, the currently accessible secure key rate is still insufficient. Besides, the high-speed modulators and true random number generator (TRNG) are always needed in active CVQKD schemes, inducing significant cost, manufacturing time, and complexity.

To overcome the shortcomings of active CVQKD protocols, the PSP-CVQKD schemes based on a thermal source are proposed [25,26], which eliminates the need of TRNG and active amplitude and phase modulators. Under the assumption that the transmitter on Alice’s side is within a trusted location, it is impossible for Eve to distinguish between PSP-CVQKD and active-state-preparation (ASP) GMCS-CVQKD protocols [26]. In particular, from Eve’s point of view, Alice’s output quantum state is a mixture of all possible coherent states, which can be seen as a thermal state with an average photon number n0=VA/2, where VA represents Alice’s modulation variance. Therefore, the security proof for ASP-GMCS-QKD can be directly used to demonstrate the security of the PSP scheme. In the PSP-CVQKD scheme, the TRNG and active amplitude and phase modulators are replaced by a thermal source, beam splitters, optical attenuators, and homodyne detectors, which offers the prospect of the chip-scale high-rate CVQKD systems with low cost on optical fiber and free space. Recently, the intact PSP-CVQKD was experimentally conducted by transmitting a local oscillator (LO) over fiber and an atmosphere channel [27,28]. However, limited by the low bandwidth of the homodyne detector, which is equivalent to the repetition frequency of traditional ASP-CVQKD schemes, the SKR is not outstanding compared to other CVQKD schemes. More deeply, high-bandwidth detectors require a higher intensity of LO, which may introduce more photon-leakage noise, thereby restraining the increase of SKR. Furthermore, the unique excess noise caused by passive state preparation also extremely leads to the deterioration of SKR.

In this paper, we experimentally realize a high-rate transmitted LO PSP-CVQKD system within an access-network area by an off-the-shelf ASE source. In particular, with the help of high-bandwidth detectors and high-speed data sampling, the PSP-CVQKD can be equivalent to the ASP-CVQKD scheme after high-speed modulation. And to decrease the subsequent deterioration of photon-leakage noise caused by enhanced LO, a polarization beam splitter (PBS) with a high extinction ratio and a polarization monitoring technique are utilized in polarization multiplexing. Meanwhile, the excess noise generated in the passive state preparation, which is one of the most important bottleneck problems for implementing high-rate PSP-CVQKD, is greatly suppressed by increasing the attenuation on the transmitted signal. And by adjusting the output of the thermal source on Alice’s side, we optimize the equivalent modulation variance to maximize the SKR. Based on the above key technological breakthroughs, the asymptotic and non-asymptotic SKRs of the demonstrated PSP-CVQKD scheme within an access-network area reach 1.09 Gbps and 273.25 Mbps, respectively, which fulfills the Gbps key generation rate within an access-network area. Therefore, the investigated transmitted LO PSP-CVQKD scheme has the application potential for a high-speed secure access network and could be a favorable candidate for short-reach secret communication.

The experimental schematic of the transmitted LO PSP-CVQKD scheme is depicted in Fig. 1. First, instead of actively preparing a coherent state, Alice splits the output of a thermal source into two spatial modes. One is locally measured by using heterodyne detection (here Att2 denotes the local loss in Alice’s signal channel) to obtain the values of quadratures xA and pA, which are the quantum random numbers from the intrinsic field fluctuations of a thermal source (see Appendix A). The other mode is attenuated with Att1 and then sent to Bob with the LO through PBS and an optical fiber channel based on the polarization multiplexing technology. Second, after the signal and LO are decoupled by the PBS, Bob performs heterodyne detection on the incoming quantum states transmitted from Alice to generate received quadratures xB and pB. A 99:1 BS is deployed before the 90° OH to monitor the polarization leakage. Third, after frame synchronization and phase compensation, Alice and Bob perform parameter estimation. Finally, a string of secure secret keys is generated by reconciliation and privacy amplification.

The secret key rate can be calculated as [29] (see Appendix A) SKR=f(1−FER)n(1−α)N[βIAB−χBE−Δ(n)],(1)where f is the repetition rate, FER is the frame error rate of the reconciliation, n and N represent the block length for final key distillation and the entire sampling length, respectively, α is the overhead for frame synchronization and phase compensation, β is the reconciliation efficiency, IAB denotes the Shannon mutual information between Alice and Bob, Δ(n) is an offset term due to privacy amplification in the finite-size regime, and χBE is the Holevo bound between Eve and Bob. In the PSP-CVQKD scheme, the repetition rate is dependent on the minimum value of the bandwidth of the homodyne detectors and sampling frequency, which is an advantage of PSP-CVQKD compared to ASP-GMCS-CVQKD. Therefore, to achieve maximum SKR, coherent detection with a high bandwidth and high-speed data sampling are utilized in the experiment. Besides, to achieve maximum SKR, the overall excess noise must be effectively suppressed, which determines IAB and χBE.

For any high-rate CVQKD system, the low excess noise is always required. Compared with ASP-CVQKD, PSP-CVQKD can prepare the quantum states without active modulation by using amplitude and phase modulators, and TRNG. However, the passive state preparation will lead to a unique excess noise, which greatly deteriorates the overall excess noise and decreases the SKR. In the transmitted LO PSP-CVQKD scheme, the complete excess noise model could be constructed as follows [28]: ϵ=ϵpsp+ϵleak+ϵphase+ϵchannel+O(ϵmode),(2)where ϵpsp,ϵleak,ϵphase,ϵchannel,O(ϵmode) represent the excess noises caused by passive state preparation, photon leakage, phase drift, channel instability, and out-of-band light’s mismatching, respectively. Among them, O(ϵmode) is so small that it can be ignored. ϵchannel, which depends on the environment, is uncontrollable. Therefore, in order to increase the SKR, the ϵpsp,ϵleak,ϵphase must be effectively decreased. The ϵpsp originates from the quantum-state-preparation step in PSP-CVQKD (see Appendix A), which can be calculated by ϵpsp=2VA(veleA+1)VAηxAηs+2ηA(veleA+1)ηA,(3)where VA=ηAn0 is the equivalent modulation variance of the mode sent to Bob, which can be adjusted to the desired value and pre-measured before each experiment, ηA represents the attenuation factor of the attenuator, n0 is the average photon number of the thermal source, ηs is the transmittance of the SMF spool, and ηxA and veleA are the quantum efficiency and variance of the electrical noise of Alice’s homodyne detectors, respectively. Therefore, Alice can reduce the ϵpsp by adjusting the transmittance ηA of the attenuator to a certain extent. Here ϵpsp is classified as untrusted excess noise as other components, but it can be evaluated before the parameter estimation process.

Notably, to optimize the equivalent modulation variance, the average photons of a thermal source need to be increased together with attenuation. In our experiment, unlike previous PSP-CVQKD schemes, we set a quite high attenuation on the transmitted signal to extremely suppress the excess noise during the PSP step. However, this leads to a significant power difference between the signal and LO, which may increase the photon leakage noise ϵleak. And to achieve coherent detection with high efficiency, low noise, and high bandwidth, the high-intensity LO is required, which explains why the LO cannot be attenuated. Fortunately, the thermal polarization multiplexing transmission LO technology with high isolation can make up for this shortcoming. By utilizing the multifunction polarization controller (MPC-201), the practical isolation is much higher than the finite polarization isolation of the polarization beam splitter (PBS). The leaked photons per second are distributed at each sampling point, thereby further decreasing the photon leakage noise. To remove the practical security loopholes induced by transmitted LO, a simple counter-measure is employed here to monitor the fluctuation of LO. Moreover, we can develop a local LO PSP-CVQKD scheme [30] to completely eliminate the practical security vulnerabilities and photon leakage noise induced by transmission of LO. And to further suppress the excess noise due to phase drift through the lossy and noisy channel, a fine-grained phase compensation algorithm [28] is employed to improve it (see Appendix A).

The measured minimum overall noise varying with the attenuation ηA is shown in Fig. 3. At first, ϵpsp constitutes a high proportion and reduces with the attenuation, resulting in a decrease in overall excess noise. As the attenuation continues to increase, the residual excess noise including ϵleak is the main part of excess noise; the photon leakage increases due to the power difference between the signal and LO, thereby deteriorating the overall excess noise. Despite the absence of strict monotony for the experimental results, a concave function with a minimum value can be used to fit the variation. The fitted curve is based on a polynomial function and fortunately indicates there exits an appropriate attenuation that minimizes the overall excess noise. Notably, the signal cannot be infinitely attenuated to maintain the correlation between Alice’s and Bob’s data.

The experimental excess noise measured with 25 raw data blocks of length 5×105 for a distance of 5.005 km is shown in Fig. 4. Due to the instability of polarization leakage and phase drift, a worst-case estimator of excess noise compatible with the experimental data is also provided for each data point. The channel input has an average experimental excess noise of 0.047 and a worst-case excess noise estimator value of 0.098. Besides, for the high-speed PSP-CVQKD system, we confirm the upper bound of the excess noise at 5.005 km allowing for generating a secret key. It should be mentioned here that precise frame synchronization is necessary for Alice and Bob to evaluate the excess noise and generate a final key, especially for the high-rate CVQKD scheme. This explains why the same 5.005-km SMF spool is added in Alice’s thermal light transmission channel, despite the slight increase in excess noise. When Alice and Bob use independent data sampling modules and calibrate the transmission delay, the SMF spool can be eliminated.

To maximize the SKR, the homodyne detectors of 23 GHz are utilized in the high-rate PSP-CVQKD scheme, which increases the repetition rate (f), as mentioned in Eq. (1). However, limited by the sampling clock frequency of the oscilloscope, for the homodyne detectors of 23-GHz sampling rate, we can perform up-sampling (40 GHz) or down-sampling (20 GHz). In the high-rate PSP-CVQKD experiment, when the sampling clock frequency is lower than the signal frequency, the displayed waveform may not be the actual frequency and amplitude. Interpolation calculation is utilized in the oscilloscope, which may lead to severe data distortion. Therefore, we perform up-sampling of the raw data with a 40-GHz rate. Notably, for the practical high-rate CVQKD system, a sampling module with the same sampling frequency as the detector may be required. Although we perform up-sampling of the raw data with a 40-GHz rate, the equivalent repetition rate for the CW-mode communication in our experiment is still 23 GHz. Here highly efficient post-processing is needed to distill a secure secret key. Fortunately, for the low-loss quantum channel, there are effective reconciliation algorithms that are sufficient to achieve reconciliation efficiencies above 96% over a wide range of SNR (see Appendix A). Although the real-time post-processing is not implemented in the experiment, it can be achieved through multiple GPUs theoretically. The experimental secret key generation results based on Eq. (1) are shown in Fig. 5. In the high-rate PSP-CVQKD system, the SKR over transmission distances of 5.005 km can reach 1.09 Gbps. When the finite-size effects are taken into account, the SKR is 273.25 Mbps (see Appendix A). The simulated secret key rates based on a larger data block is also depicted. The larger data block can reduce the finite-size effects, thus improving the secret key rate and transmission distance. Moreover, we can also utilize the EDFA multi-stage amplification to increase the signal power to the optimized value, thereby increasing the transmission distance.

To conclude, we experimentally investigate a high-rate transmitted LO continuous-variable quantum key distribution system through optical fiber. In particular, we implement the passive state preparation with an equivalent perfect thermal source, which eliminates the need of TRNG and active amplitude and phase modulators. And by synthetically suppressing the excess noise induced in the whole communication including the passive-state-preparation stage, we realize the coherent detection with high efficiency, low noise, and high bandwidth. With the high-speed data sampling and effective post-processing technique, the secure key rate over transmission distances of 5.005 km can reach 1.09 Gbps, which allows the Gbps high-speed one-time-pad quantum cryptography communication within an access-network area. The demonstrated transmitted LO PSP-CVQKD scheme could be a favorable candidate to implement high-speed, chip-based, and even sunlight-based CVQKD with less cost and complexity.