Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Emerging Imaging and Sensing Technologies for Security and Defence

Modulation variance optimization in discrete modulated CV-QKD systems

Margarida Almeida, Armando N. Pinto, Nuno A. Silva

Published Oct 17, 2023• 3 citations
Publisher

Abstract

The distribution of symmetric keys over optical networks is possible through the implementation of continuous variables quantum key distribution (CV-QKD) systems. Despite the rate of key extraction in CV-QKD being maximized for Gaussian modulation (GM), its practical implementation is demanding due to the finite extinction ratio of the electro-optic modulators. The use of discrete modulation (DM) allows for a simpler practical implementation. Provided that higher-order constellations can be used, DM can approximate the performances of GM. Higher-order DM has been shown to be feasible experimentally in CV-QKD. Notwithstanding, such implementations are usually obtained in offline without applying all the post-processing steps, particularly the information reconciliation, turning the validity of the results dependent on a null frame error rate after information reconciliation. Besides that, a fixed value is typically assumed for the reconciliation efficiency value, being the modulation variance optimized under such assumptions. This disregarding the dependency of the reconciliation efficiency parameter on the method for information reconciliation and on the modulation variance. In this work we study the impact that such assumptions have on the optimization of the modulation variance in a CV-QKD system using probabilistic shaped higher-order DM. Moreover, we analyse how optimizing the modulation variance can affect the extraction key rates of symmetric keys at the output of the CV-QKD system. Besides the assumptions of null frame error rate and of the reconciliation efficiency, we study how different code rates may also affect the optimization of the modulation variance. This considering M-symbol quadrature amplitude modulation and multidimensional reconciliation.

DOI
10.1117/12.2683577

Related papers

Reconciliation Efficiency Impact on Discrete Modulated CV-QKD Systems Key Rates

Margarida Almeida, Daniel Pereira, M. Facão +2 more

Continuous variable quantum key distribution (CV-QKD) allows the distribution of symmetric keys in a secure manner. CV-QKD systems can extract keys at its maximum rate when using Gaussian modulation (GM). Nonetheless, GM demands high-capacity random number sources and tends to be very hard to approach in practice. To circumvent these disadvantages, higher-order probabilistic -shaped discrete modulation (DM) can be used. State-of-the-art works compute the key rates of DM-CV-QKD systems considering a fixed value for the reconciliation efficiency and do not take into account the frame error rate (FER) of the system, thus over or under estimating the key rates. In this work, we study the security bounds of CV-QKD systems considering probabilistic shaped DM formats with 256-symbols in the finite-size regime. This accounting for the true value of the reconciliation efficiency, and the FER of the information reconciliation step. Both conventional and hexagonal 256-quadrature amplitude modulation (QAM) constellations yield higher key rates than 256-amplitude and phase shift keying (APSK) constellations, with 256-QAM constellations being indistinguishable in performance with GM for high transmission distances. Minimum signal-to-noise ratio (SNR) values were fixed from a FER analysis through a CV-QKD simulation allowing for key extraction considering different FER levels. Lower FER values are associated with higher SNRs in the system and thus lower achievable transmission distances. Nonetheless, the FER maximizing the extraction key rate is not null. Our results show that the extraction key rate is maximized by SNR adjustment which should have in account both the reconciliation efficiency and the FER.

Journal of Lightwave Technology•May 25, 2023•14 citations

Optimization of continuous variables quantum key distribution using discrete modulation

Daniel Pereira, Nuno A. Silva, Margarida Almeida +1 more

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.

Emerging Imaging and Sensing Technologies for Security and Defence•Nov 4, 2022•0 citations

Secret Key Rate Analysis of Distribution Matching Algorithms for Discrete-Modulated CV-QKD

Micael Dias, Caroline Alves, Gabrielly Roman +1 more

Continuous variable quantum key distribution protocols (CV-QKD) with discrete modulation have been intensively investigated to bridge the gap between ideal Gaussian modulation and modern coherent optical communication systems. To mitigate the penalty of discrete modulation, probabilistic constellation shaping (PCS) is applied to the modulation format and is typically performed by distribution matching (DM) algorithms. In this paper, we address the application of DM algorithms to perform PCS in CV-QKD protocols. We investigate the impact of approximating optimized Maxwell-Boltzman distributions with DM algorithms based on Huffman (HDM) and constant composition (CCDM) codes on the protocol's secret key rate (SKR) and tolerance to excess noise. Our results show that specifically symbol-by-symbol HDM degrades the SKR by at least 30\%, whereas CCDM matches the optimal SKR with code length of $10^3$ or more symbols. Furthermore, we also provide a statistical analysis of symbol dependence for both approaches, showing that CCDM must operate with blocks of at least $10^5$ symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.

quant-pharXiv:2607.06783•Jul 9, 2026•0 citations

A feasible two-step optimization method for continuous-variable quantum key distribution

Li Ma, Yang Li, Jie Yang +8 more

Continuous-variable quantum key distribution (CV-QKD) offers the advantages of high secret key rates in metropolitan areas. Optimization of modulation variance is an efficient method to improve the secret key rate of CVQKD system. However, in practical CV-QKD system, inevitable slight parameter fluctuation could occur after the modification of modulation variance, and controlling the modulation variance with arbitrary accuracy is also difficult. In this paper, we propose a two-step optimization for practical CV-QKD. The first step is to determine the optimal working state by combining the modulation variance optimization with error correction matrix optimization. The second step is to optimize the rate-adaptive reconciliation parameters to compensate the loss of secret key rate caused by inaccuracy modulation variance. Our results show that the secret key rate can be improved by 17.8% in comparison to one-step optimization method. Our method can be conveniently applied to CV-QKD protocol with homodyne and heterodyne detection, which will pave the way to the deployment of high stable and high performance for CV-QKD.

SPIE Conference on Quantum and Nonlinear Optics•Nov 27, 2022•0 citations

Impact of Limited Classical Channel Bandwidth on the Secret Key Rate of a CV-QKD System

Nuno A. Silva, Margarida Almeida, Nelson Jesus Muga +1 more

Continuous-variable quantum key distribution (CV-QKD) can extract secret keys in an efficient and practical manner using off-the-shelf telecommunication equipment. This sets CV-QKD as a key technology for implementation in optical fiber networks already deployed in the field. Notwithstanding, the extraction of secret keys in CV-QKD requires the exchange of information, not only on a quantum channel, but also on an authenticated and public classical channel. The classical channel is required to exchange part of the coherent states to aid with the parameter estimation step to conclude on the secrecy of the extracted key, and to exchange side information essential to the reconciliation step. Here, we study the bandwidth demand of the information exchanged in the classical channel and how it can impact the extraction key rate of a CV-QKD system. The post-processing step consuming the highest bandwidth of the classical channel is the reconciliation step, due to the exchange of the side information. We consider different dimensions for the multidimensional reconciliation demanding different bandwidths on the classical channel and compute the allowed extraction key rate depending on the performance of the method, in terms of frame error rate.

International Conference on Transparent Optical Networks•Jul 14, 2024•0 citations

Secret key rate of multi-ring M-APSK continuous variable quantum key distribution

Margarida Almeida, Daniel Pereira, Nelson Jesus Muga +3 more

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.

Optics Express•Oct 26, 2021•37 citations
The distribution of symmetric keys over optical networks is possible through the implementation of continuous variables quantum key distribution (CV-QKD) systems. Despite the rate of key extraction in CV-QKD being maximized for Gaussian modulation (GM), its practical implementation is demanding due to the finite extinction ratio of the electro-optic modulators. The use of discrete modulation (DM) allows for a simpler practical implementation. Provided that higher-order constellations can be used, DM can approximate the performances of GM. Higher-order DM has been shown to be feasible experimentally in CV-QKD. Notwithstanding, such implementations are usually obtained in offline without applying all the post-processing steps, particularly the information reconciliation, turning the validity of the results dependent on a null frame error rate after information reconciliation. Besides that, a fixed value is typically assumed for the reconciliation efficiency value, being the modulation variance optimized under such assumptions. This disregarding the dependency of the reconciliation efficiency parameter on the method for information reconciliation and on the modulation variance. In this work we study the impact that such assumptions have on the optimization of the modulation variance in a CV-QKD system using probabilistic shaped higher-order DM. Moreover, we analyse how optimizing the modulation variance can affect the extraction key rates of symmetric keys at the output of the CV-QKD system. Besides the assumptions of null frame error rate and of the reconciliation efficiency, we study how different code rates may also affect the optimization of the modulation variance. This considering M-symbol quadrature amplitude modulation and multidimensional reconciliation.
Continuous variable quantum key distribution (CV-QKD) allows the distribution of symmetric keys in a secure manner. CV-QKD systems can extract keys at its maximum rate when using Gaussian modulation (GM). Nonetheless, GM demands high-capacity random number sources and tends to be very hard to approach in practice. To circumvent these disadvantages, higher-order probabilistic -shaped discrete modulation (DM) can be used. State-of-the-art works compute the key rates of DM-CV-QKD systems considering a fixed value for the reconciliation efficiency and do not take into account the frame error rate (FER) of the system, thus over or under estimating the key rates. In this work, we study the security bounds of CV-QKD systems considering probabilistic shaped DM formats with 256-symbols in the finite-size regime. This accounting for the true value of the reconciliation efficiency, and the FER of the information reconciliation step. Both conventional and hexagonal 256-quadrature amplitude modulation (QAM) constellations yield higher key rates than 256-amplitude and phase shift keying (APSK) constellations, with 256-QAM constellations being indistinguishable in performance with GM for high transmission distances. Minimum signal-to-noise ratio (SNR) values were fixed from a FER analysis through a CV-QKD simulation allowing for key extraction considering different FER levels. Lower FER values are associated with higher SNRs in the system and thus lower achievable transmission distances. Nonetheless, the FER maximizing the extraction key rate is not null. Our results show that the extraction key rate is maximized by SNR adjustment which should have in account both the reconciliation efficiency and the FER.
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.
Continuous variable quantum key distribution protocols (CV-QKD) with discrete modulation have been intensively investigated to bridge the gap between ideal Gaussian modulation and modern coherent optical communication systems. To mitigate the penalty of discrete modulation, probabilistic constellation shaping (PCS) is applied to the modulation format and is typically performed by distribution matching (DM) algorithms. In this paper, we address the application of DM algorithms to perform PCS in CV-QKD protocols. We investigate the impact of approximating optimized Maxwell-Boltzman distributions with DM algorithms based on Huffman (HDM) and constant composition (CCDM) codes on the protocol's secret key rate (SKR) and tolerance to excess noise. Our results show that specifically symbol-by-symbol HDM degrades the SKR by at least 30\%, whereas CCDM matches the optimal SKR with code length of 10310^3103 or more symbols. Furthermore, we also provide a statistical analysis of symbol dependence for both approaches, showing that CCDM must operate with blocks of at least 10510^5105 symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.
Continuous-variable quantum key distribution (CV-QKD) offers the advantages of high secret key rates in metropolitan areas. Optimization of modulation variance is an efficient method to improve the secret key rate of CVQKD system. However, in practical CV-QKD system, inevitable slight parameter fluctuation could occur after the modification of modulation variance, and controlling the modulation variance with arbitrary accuracy is also difficult. In this paper, we propose a two-step optimization for practical CV-QKD. The first step is to determine the optimal working state by combining the modulation variance optimization with error correction matrix optimization. The second step is to optimize the rate-adaptive reconciliation parameters to compensate the loss of secret key rate caused by inaccuracy modulation variance. Our results show that the secret key rate can be improved by 17.8% in comparison to one-step optimization method. Our method can be conveniently applied to CV-QKD protocol with homodyne and heterodyne detection, which will pave the way to the deployment of high stable and high performance for CV-QKD.
Continuous-variable quantum key distribution (CV-QKD) can extract secret keys in an efficient and practical manner using off-the-shelf telecommunication equipment. This sets CV-QKD as a key technology for implementation in optical fiber networks already deployed in the field. Notwithstanding, the extraction of secret keys in CV-QKD requires the exchange of information, not only on a quantum channel, but also on an authenticated and public classical channel. The classical channel is required to exchange part of the coherent states to aid with the parameter estimation step to conclude on the secrecy of the extracted key, and to exchange side information essential to the reconciliation step. Here, we study the bandwidth demand of the information exchanged in the classical channel and how it can impact the extraction key rate of a CV-QKD system. The post-processing step consuming the highest bandwidth of the classical channel is the reconciliation step, due to the exchange of the side information. We consider different dimensions for the multidimensional reconciliation demanding different bandwidths on the classical channel and compute the allowed extraction key rate depending on the performance of the method, in terms of frame error rate.
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.