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arXiv:2608.01723

Ring Optimized M-APSK Modulation for Discrete Modulated CV-QKD

Seonguk Kim, Jun Heo

quant-phSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
arXivHTMLPDF

Abstract

This paper proposes a multi ring M-APSK constellation optimization method for discrete-modulated continuous variable quantum key distribution. Unlike conventional APSK structures with fixed ring spacing and predefined ring probabilities, the proposed method optimizes the ring radius ratio and ring probability to improve the finite-size secret key rate. A method based on the Gram matrix is used to calculate the nonzero spectrum of the average state $τ$, and fidelity is used to compare the optimized discrete average state with the Gaussian average state. The results show that the proposed structure extends the maximum transmission distance of 16-APSK by approximately 15% compared with the conventional binomial APSK structure. The optimization gain is larger for small size APSK constellations, where the average state has a larger structural gap from Gaussian modulation.

Comments
16 pages, 6 figures, 2 tables, An earlier conference version covering the 16-APSK results has been accepted for presentation at APWCS 2026

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This paper proposes a multi ring M-APSK constellation optimization method for discrete-modulated continuous variable quantum key distribution. Unlike conventional APSK structures with fixed ring spacing and predefined ring probabilities, the proposed method optimizes the ring radius ratio and ring probability to improve the finite-size secret key rate. A method based on the Gram matrix is used to calculate the nonzero spectrum of the average state τττ, and fidelity is used to compare the optimized discrete average state with the Gaussian average state. The results show that the proposed structure extends the maximum transmission distance of 16-APSK by approximately 15% compared with the conventional binomial APSK structure. The optimization gain is larger for small size APSK constellations, where the average state has a larger structural gap from Gaussian modulation.
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.
In this work we study the performance of M-symbol (M-)Amplitude and Phase Shift Keying (APSK) constellations for Continuous-Variable Quantum Key Distribution systems. Despite the gap still observed between the performance of higher order regular M-APSK constellations with binomial distribution for probabilistic shaping and the optimal performance of Gaussian Modulation, regular 128-APSK’s performance is better than irregular 256-APSK’s and almost the same as for regular 256-APSK. Moreover, higher-order regular M-APSK is secure for longer distances and higher values of mean number of photons per symbol, and is more resistant to excess noise.
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.
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.
Here we report the performance of a discrete modulated Continuous Variable Quantum Key Distribution protocol based on eight and sixteen state optimal Amplitude Phase Keying constellations with pure loss quantum channel transmission.Our constellations use the same amplitudes and state probability as in an optimal four-state unidimensional constellation model and apply amplitude/phase modulation to the quadratures of coherent states.We compare the obtained results with the performance of conventional m-PSK and m-APK constellations.The results presented show that protocols that use the proposed constellations outperform the ones based on conventional constellations in both secret key rate and link distance.In special, our protocol is able to accomplish viable SKR's in low transmittance regime, τ < 0.2.
We experimentally demonstrate a high-rate discretely modulated continuous-variable quantum key distribution over 80-km standard single-mode fiber with a 2.5 Gbaud, 16-symbol, two-ring constellation. With the help of well-designed digital signal processing algorithms, the excess noise of the system can be effectively suppressed. The achieved secret key rates are 49.02 Mbits/s, 11.86 Mbits/s, and 2.11 Mbits/s over 25-km, 50-km, and 80-km optical fiber, respectively, and achieve 67.4%, 70.0%, and 66.5% of the secret key rate performance of a Gaussian-modulated protocol. Our work shows that it is feasible to build a high-performance, long-distance continuous-variable quantum key distribution system with only a small constellation size.