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Chinese Physics B

Continuous-variable quantum secure direct communication based on N-APSK with Boltzmann–Maxwell distribution

Zhengwen Cao, Yujie Zhang, Geng Chai, Zhang-Tao 章韬 Liang 梁, Xinlei Chen, Lei 磊 Wang 王, Yu-Jie 禹杰 Wang 王

Published Jan 3, 2025• 4 citations
Publisher

Abstract

Abstract Continuous-variable quantum secure direct communication (CVQSDC) with Gaussian modulation (GM) demands a considerable quantity of random numbers during the preparation process and encodes them separately on the quadrature components of the quantum states. Hence, high-speed random number generators are required to satisfy this demand, which is difficult to implement in practical applications. CVQSDC with discrete modulation (DM), correspondingly, employs a finite number of quantum states to achieve encoding, which can circumvent the shortcomings of the GM scheme. Based on the advantages of DM, the issue of attaining the most optimal secrecy capacity and communication distance remains to be resolved. Here, we propose a CVQSDC protocol based on N-symbol amplitude phase shift keying (N-APSK), which exploits the Boltzmann–Maxwell distribution assisted probability shaping technique. In comparison with the uniform distribution, according to 32-APSK CVQSDC, the proposed scheme extends the communication distance by about 38%, while obtaining a higher secrecy capacity at the same communication distance. Furthermore, increasing the value of N will concurrently increase the quantity of rings in the constellation, thereby facilitating enhancements of communication distance. This work incorporates the modulation approaches prevalently employed in classical communication into the realm of quantum communication, attaining gratifying advancements in communication distance and secrecy capacity, and concurrently facilitating the integrated development of quantum communication and classical communication.

DOI
10.1088/1674-1056/ada549

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Abstract Continuous-variable quantum secure direct communication (CVQSDC) with Gaussian modulation (GM) demands a considerable quantity of random numbers during the preparation process and encodes them separately on the quadrature components of the quantum states. Hence, high-speed random number generators are required to satisfy this demand, which is difficult to implement in practical applications. CVQSDC with discrete modulation (DM), correspondingly, employs a finite number of quantum states to achieve encoding, which can circumvent the shortcomings of the GM scheme. Based on the advantages of DM, the issue of attaining the most optimal secrecy capacity and communication distance remains to be resolved. Here, we propose a CVQSDC protocol based on N-symbol amplitude phase shift keying (N-APSK), which exploits the Boltzmann–Maxwell distribution assisted probability shaping technique. In comparison with the uniform distribution, according to 32-APSK CVQSDC, the proposed scheme extends the communication distance by about 38%, while obtaining a higher secrecy capacity at the same communication distance. Furthermore, increasing the value of N will concurrently increase the quantity of rings in the constellation, thereby facilitating enhancements of communication distance. This work incorporates the modulation approaches prevalently employed in classical communication into the realm of quantum communication, attaining gratifying advancements in communication distance and secrecy capacity, and concurrently facilitating the integrated development of quantum communication and classical communication.
A continuous-variable quantum key distribution (CVQKD) can be realized over the seawater channel, but the transmission of quantum signals in seawater media exhibits significant attenuation effects. Therefore, we propose an N-symbol amplitude and phase shift keying (N-APSK) modulation scheme to enhance the transmission performance of the CVQKD over the seawater channel. Specifically, an optimal N-APSK modulation scheme is designed based on the principle of maximizing the minimum Euclidean distance (MED). The simulation results show that the CVQKD protocol based on N-APSK modulation achieves a longer transmission distance over the seawater channel compared to the Gaussian modulation protocol. Additionally, increasing the value of N simultaneously expands the number of rings in the constellation diagram, further enhancing the communication distance. This study transfers modulation methods from the field of classical communications to the field of quantum communications, achieving a substantial improvement in communication distance and thereby promoting the integration of quantum communications and classical communications.
Continuous-variable quantum secret sharing (CVQSS) based on discrete modulation is a viable alternative to Gaussian-modulated CVQSS due to its implementation simplicity. However, its theoretical performance remains far below the Gaussian-modulated counterparts. In this work we propose a CVQSS scheme based on constellation shaping, which enables near-Gaussian modulation effects by combining probabilistic and geometric constellation shaping with quadrature amplitude modulation (QAM). For the probabilistic shaping, binomial distribution and Boltzmann-Maxwell (BM) distribution are selected to adjust the occurrence probabilities of quantum states, and for the geometric shaping, triangular quadrature amplitude modulation (TQAM) is employed to optimize the geometric positions of the constellation points. Numerical simulations indicate that compared with other discrete-modulated CVQSS schemes, the proposed BM-distributed 16-TQAM scheme achieves the highest asymptotic secret key rate and the lowest detection error probability. Moreover, the optimal modulation variance of this scheme is higher than that of other schemes, demonstrating that the players can employ a larger modulation variance when preparing coherent states, thereby enhancing the signal-to-noise ratio of the whole system and providing a potential advantage in long-distance transmission.
Continuous Variables Quantum Key Distribution (CV-QKD) tackles the problem of the generation and distribution of cryptographic keys without assuming any computational limitations while employing standard telecom equipment. Gaussian Modulation (GM) theoretically maximizes the information a CV-QKD system is capable of transmitting while exhibiting a higher resistance to excess channel noise. However, GM-CV-QKD protocols put an extreme burden on the transmitter’s random number source and tend to be more susceptible to imperfect state preparation. Due to these difficulties, most experimental implementations of CV-QKD have used Discrete Modulation (DM). The closer the DM constellation approaches a GM one, the closer the theoretical performance of the associated system will be to the optimum value. To achieve this, high-cardinality constellations, coupled with probabilistic shaping, can be explored. However, choosing a too complex constellation will cause the modulation stage imperfections to again become apparent. Thus, the choice of the constellation format is not direct and is of high importance. In this work we present a methodology to determine the optimum constellation for a given DM-CV-QKD system, taking into account the limitations of the modulation stage, choosing from a variety of M-QAM and M-APSK constellations coupled with probabilistic shaping. Our obtained methodology will allow for the optimum modulation format for each specific system to be selected.
Discrete modulation continuous variable quantum key distribution (DM-CV-QKD) is highly considered in real implementations to avoid the complexity of Gaussian modulation (GM), which is optimum in terms of the key rate. DM-CV-QKD systems usually consider M-symbol phase shift keying (M-PSK) constellations. However, this type of constellation cannot reach transmission distances and key rates as high as GM, limiting the practical implementation of CV-QKD systems. Here, by considering M-symbol amplitude and phase shift keying (M-APSK) constellations, we can approximate GM. Indeed, considering finite-size effects, 4 ring 64-APSK can reach 52.0 km, only 7.2 km less than GM and 282% the maximum achievable transmission distance for 8-PSK.
Quantum key distribution (QKD) offers a promising solution by providing information-theoretic security based on the principles of quantum mechanics. Beyond theoretical experiments, QKD network construction and verification are actively progressing in various locations worldwide. Continuousvariable QKD (CVQKD) has emerged as a strong candidate for integration into optical networks, particularly due to its compatibility with coherent detection schemes used in existing optical communication systems. Gaussian-modulated (GM) CVQKD enables simpler security analysis through covariance matrices, but its practical implementation typically requires highperformance optical components, leading to increased cost and complexity. In contrast, discrete-modulated (DM) CVQKD using modulation formats, such as quadrature amplitude modulation (QAM), can reduce implementation costs by utilizing the existing optical components. However, its security analysis requires more advanced techniques due to the non-Gaussian nature of the quantum states involved. This study evaluates the performance of GM- and DM-CVQKD under a common optical fiber channel model. Our results show that the secure key rates (SKRs) of DM-CVQKD with probabilistic constellation shaping (PCS) 64QAM and PCS-256QAM drop below zero at transmission distances of 277 km and 399 km, respectively. The results confirm the feasibility of DM-CVQKD for long-distance secure communication over optical fiber networks when appropriate modulation formats and PCS are employed.
We propose a discrete-modulation continuous-variable quantum key distribution protocol based on coherent-state quadrature amplitude modulation combined with probabilistic amplitude shaping. Designed to overcome the practical limitations of Gaussian modulation schemes, the proposed protocol employs finite-energy shaped constellations and enables efficient implementation using standard telecom components while retaining compatibility with homodyne detection and reverse reconciliation. Assuming a linear quantum channel model, characterized by a linear input–output relation between quadratures, and considering collective attacks in the asymptotic regime, we evaluate the secret key rate, achievable transmission distance, optimal launch power, and excess noise tolerance for different constellation sizes, comparing the results with those of the benchmark GG02 protocol. Our results show that probabilistic shaping significantly enhances the performance of discrete-modulation schemes, allowing high-order constellations to closely approach the performance of GG02 in terms of secret key rate, transmission distance, optimal launch power, and excess noise tolerance while preserving practical implementability. By leveraging mature coherent optical communication technologies, the proposed approach provides a realistic pathway toward experimentally feasible high-rate continuous-variable quantum key distribution systems.