Continuous-variable quantum key distribution (CVQKD) offers a promising approach to secure communication by sharing a secret key between sender (Alice) and receiver (Bob) with information-theoretical security. Its potential for high key rates and compatibility with commercial components makes it particularly attractive for broadband metropolitan and access networks. However, current CVQKD systems are limited to secret key rates (SKRs) of a few Mbps, falling short of the requirements for applications like high-speed secure access networks which demand the much higher rates needed for one-time-pad encryption. Therefore, developing ultra-high SKR CVQKD is crucial for practical applications. Two main CVQKD schemes exist: one based on Gaussian modulation coherent states (GMCS) and another based on discrete modulation coherent states (DMCS). While GMCS has shown significant progress, high-rate implementation requires high-speed, high-linearity digital-to-analog converters (DACs) and analog-to-digital converters (ADCs), potentially limiting the SKR. DMCS, such as the four-state protocol, offers advantages at low signal-to-noise ratios (SNRs) and with low linearity, potentially leading to improved SKRs. High-speed DMCS CVQKD, often using the local local oscillator (LLO) scheme, avoids security loopholes and transmission limitations of external local oscillators. However, challenges remain including precise phase noise compensation (PNC), robust excess noise elimination (from photon leakage, modulation, detection, and quantization), and the need for more robust security analysis beyond the linear channel assuming (LCA) method. Furthermore, high-efficiency, high-speed post-processing is essential for practical key extraction. This paper addresses these issues to demonstrate a sub-Gbps key rate four-state DMCS LLO-CVQKD system.

The literature reveals significant progress in continuous-variable quantum key distribution (CVQKD), particularly in Gaussian-modulated coherent state (GMCS) systems. High key rates have been reported, but these often rely on high-speed, high-linearity components that may limit scalability and practical deployment. Discretely modulated coherent state (DMCS) approaches, especially four-state protocols, offer potential advantages in terms of robustness to noise and simpler implementation. The use of a local local oscillator (LLO) has been explored to address security and transmission limitations. However, previous studies have demonstrated only relatively low secret key rates (SKRs), typically in the Mbps range, and often rely on the less robust linear channel assuming (LCA) security analysis. There's a clear need for improved phase noise compensation techniques and more comprehensive security analysis methods, such as semidefinite programming (SDP), to account for more general attacks. The lack of high-efficiency post-processing for practical key extraction also presents a major obstacle. This paper contributes to addressing these gaps.

This paper presents a novel experimental setup for a four-state discretely modulated LLO-CVQKD system. The system uses frequency- and polarization-multiplexing to transmit both a weak quantum signal and an intense pilot tone over a single fiber. At Alice's site, a continuous optical carrier is split into two paths. The upper path is modulated using quadrature phase-shift keying (QPSK) with a 5 GBaud symbol rate by a high-speed arbitrary waveform generator (AWG). The signal is then attenuated to create the four-state quantum signal. The lower path is directly attenuated to create the pilot tone. The quantum signal and pilot tone, with different frequencies and orthogonal polarizations, are transmitted and separated at Bob's site using a polarization beam splitter (PBS) and a polarization synthesis analyzer (PSA). Two balanced homodyne detectors (BHDs) detect the quantum signal and pilot tone respectively, using independent local oscillators (LOs). The design minimizes noise sources: the independent generation and detection of signal and pilot reduce modulation and quantization noise; frequency separation avoids photon leakage; and the orthogonal polarizations minimize crosstalk between the signal and pilot. A novel fast-slow phase noise compensation (PNC) scheme is implemented. The pilot tone enables fast drift phase recovery, while a least mean square (LMS) algorithm compensates for slow drifts. This achieves ultra-low excess noise. Post-processing is crucial for extracting the final key. A high-efficiency post-processing setup uses multidimensional reconciliation with a multi-edge-type low-density parity check (MET-LDPC) code for error correction, and Toeplitz matrices for privacy amplification. Three different parity check matrices are designed for 5 km, 10 km, and 25 km transmission distances, optimized for low SNR using a density evolution algorithm. Rate-adaptive reconciliation efficiencies exceeding 95% are achieved. The entire post-processing is implemented using a GPU (NVIDIA TITAN Xp) for high throughput. The secret key rate (SKR) is evaluated using both the linear channel assuming (LCA) and semidefinite programming (SDP) methods, the latter being more robust against general attacks.

The experimental results demonstrate significant improvements in CVQKD performance. The system achieves sub-Gbps secret key rates (SKRs) within a metropolitan area setting, representing a substantial increase over previous studies. Specific SKRs obtained are: * **5 km:** 233.87 Mbps (SDP) / 190.54 Mbps (LCA) * **10 km:** 137.76 Mbps (SDP) / 133.6 Mbps (LCA) * **25 km:** 21.53 Mbps (SDP) / 52.48 Mbps (LCA) These results highlight the effectiveness of the key technological advancements: * **Frequency and polarization multiplexing:** Effectively reduced various noise components associated with signal generation, transmission, and detection. * **Precise fast-slow PNC scheme:** Enabled ultra-low excess noise levels, crucial for high SKRs. * **High-efficiency post-processing:** Achieved reconciliation efficiencies above 95%, maximizing the final key extraction. The experimental excess noise levels are exceptionally low (around 0.0075 SNU), significantly below the thresholds for zero SKR (0.0176 SNU for SDP, and 0.0563 SNU for LCA at 5 km). The study also includes a detailed analysis of the various noise components contributing to the overall excess noise, demonstrating the impact of each element and the efficiency of the noise reduction techniques employed. This detailed breakdown allows for refinement and optimization in future developments. A comparison with previous literature demonstrates the superior SKR obtained in this work, primarily attributed to the higher repetition rate (5 GBaud), ultra-low excess noise, and high reconciliation efficiency. The use of the SDP security analysis method, offering more robust security guarantees compared to the LCA method, further strengthens the findings.

The results demonstrate the feasibility of sub-Gbps CVQKD within a metropolitan area, overcoming several key limitations of previous systems. The ultra-low excess noise achieved is critical for high-rate CVQKD, pushing the boundaries of practical implementation. The use of the SDP security analysis method provides more robust security guarantees compared to the LCA method, making the system more resistant to potential attacks. The high reconciliation efficiency is vital for efficient key extraction, contributing significantly to the achieved SKR. This work showcases the advantages of DMCS and LLO approaches, highlighting their potential for high-rate, long-distance secure communication. The detailed analysis of different noise sources provides valuable insights for future system improvements and optimization. While the focus of this study was on asymptotic SKR, future work should consider the impact of finite-key size effects on practical security. The development of real-time post-processing capabilities is also crucial for widespread deployment. Investigating higher-order modulation schemes or Gaussian modulation could potentially further enhance the SKR and transmission distance.

This work successfully demonstrates a sub-Gbps key rate four-state continuous-variable quantum key distribution (CVQKD) system within a metropolitan area. Key improvements include the use of frequency and polarization multiplexing for reduced noise, a precise fast-slow phase noise compensation scheme for ultra-low excess noise, and a high-efficiency post-processing system achieving >95% reconciliation efficiency. The system achieved SKRs exceeding previous work, highlighting the potential of this approach for high-rate and secure broadband networks. Future work will focus on real-time post-processing and exploring higher-order modulation schemes for further performance enhancements.

The current implementation uses off-line post-processing. Real-time post-processing is needed for practical applications. The study focuses on asymptotic key rates; finite-key effects should be investigated for a more complete security analysis. While the SDP method offers stronger security guarantees, it is computationally more demanding. Finally, although the experimental setup achieved ultra-low excess noise, further refinements to reduce noise and improve system stability may improve the SKR further.