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arXiv:2603.02870

Discrete-modulation continuous-variable quantum key distribution with probabilistic amplitude shaping over a linear quantum channel

Emanuele Parente, Michele N. Notarnicola, Stefano Olivares, Enrico Forestieri, Luca Potì, Marco Secondini

quant-phcs.ITSubmitted Mar 3, 2026• Announced Mar 4, 2026• 0 citations
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Abstract

The practical implementation difficulties arising from the Gaussian modulation of the GG02 protocol lead us to investigate the possibilities offered by the combination of probabilistic amplitude shaping technique and quadrature amplitude modulation formats in the context of continuous variable quantum key distribution systems. Our interest comes from the fact that quadrature amplitude modulation and probabilistic shaping can be implemented with current technologies and are widely used in classical telecom equipment. In this treatment, we assume to work in the scenario of a linear quantum channel and we analyze maximum achievable secure key rates, maximum reachable distances and the resilience to noise of our discrete-modulation based protocol with respect to GG02, which is taken as a benchmark. In particular, we deal with the infinite key size regime, consider a homodyne detection scheme, and analyze what happens for different cardinalities of the input alphabet at different distances, in the case of collective attacks and in the reverse reconciliation picture. We find that our protocol, beyond being easily reproducible in the laboratory, provides a way to closely approach the theoretical performance offered by GG02 and, at the same time, preserves the ability to assure an unconditional security level.

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Paper submitted to IEEE Transactions on Communications

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The practical implementation difficulties arising from the Gaussian modulation of the GG02 protocol lead us to investigate the possibilities offered by the combination of probabilistic amplitude shaping technique and quadrature amplitude modulation formats in the context of continuous variable quantum key distribution systems. Our interest comes from the fact that quadrature amplitude modulation and probabilistic shaping can be implemented with current technologies and are widely used in classical telecom equipment. In this treatment, we assume to work in the scenario of a linear quantum channel and we analyze maximum achievable secure key rates, maximum reachable distances and the resilience to noise of our discrete-modulation based protocol with respect to GG02, which is taken as a benchmark. In particular, we deal with the infinite key size regime, consider a homodyne detection scheme, and analyze what happens for different cardinalities of the input alphabet at different distances, in the case of collective attacks and in the reverse reconciliation picture. We find that our protocol, beyond being easily reproducible in the laboratory, provides a way to closely approach the theoretical performance offered by GG02 and, at the same time, preserves the ability to assure an unconditional security level.
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.
To achieve the maximum information transfer and face a possible eavesdropper, the samples transmitted in continuous-variable quantum key distribution (CV-QKD) protocols are to be drawn from a continuous Gaussian distribution. As a matter of fact, in practical implementations the transmitter has a finite (power) dynamics and the Gaussian sampling can be only approximated. This requires the quantum protocols to operate at small powers. In this paper, we show that a suitable probabilistic amplitude shaping of a finite set of symbols allows to approximate at will the optimal channel capacity also for increasing average powers. We investigate the feasibility of this approach in the framework of CV-QKD, propose a protocol employing discrete quadrature amplitude modulation assisted with probabilistic amplitude shaping, and we perform the key generation rate analysis assuming a wiretap channel and lossless homodyne detection.
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.
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.
Most continuous-variable quantum key distribution schemes are based on the Gaussian modulation of coherent states followed by continuous quadrature detection using homodyne detectors. In all previous schemes, the Gaussian modulation has been carried out in conjugate quadratures thus requiring two independent modulators for their implementations. Here, we propose and experimentally test a largely simplified scheme in which the Gaussian modulation is performed in a single quadrature. The scheme is shown to be asymptotically secure against collective attacks, and considers a specific attack using asymmetric preparation and excess noise. We find that this protocol is considerably more sensitive to noise than other CVQKD schemes, as a consequence of the simplified implementation. A single-quadrature modulation approach renders the need for a costly amplitude modulator unnecessary, and thus facilitates commercialization of continuous-variable quantum key distribution, provided that the low noise requirement can be achieved.
We have developed a continuous-variable quantum key distribution (CV-QKD) system that employs discrete quadrature-amplitude modulation and homodyne detection of coherent states of light. We experimentally demonstrated automated secure key generation with a rate of 50 kbps when a quantum channel is a 10 km optical fibre. The CV-QKD system utilises a four-state and post-selection protocol and generates a secure key against the entangling cloner attack. We used a pulsed light source of 1550 nm wavelength with a repetition rate of 10 MHz. A commercially available balanced receiver is used to realise shot-noise-limited pulsed homodyne detection. We used a non-binary LDPC code for error correction (reverse reconciliation) and the Toeplitz matrix multiplication for privacy amplification. A graphical processing unit card is used to accelerate the software-based post-processing.