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Journal of Lightwave Technology

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

Margarida Almeida, Daniel Pereira, M. Facão, Armando N. Pinto, Nuno A. Silva

Published May 25, 2023• 14 citations
Publisher

Abstract

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.

DOI
10.1109/jlt.2023.3280076

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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.
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.
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.
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.
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.
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 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.