[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83142-en":3,"doc-seo-83142-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83142,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Secret Key Rate Analysis of Distribution Matching Algorithms for Discrete-Modulated CV-QKD","Continuous-variable quantum key distribution (CV-QKD) with discrete modulation is studied to close the performance gap with ideal Gaussian modulation for coherent optical communication. Probabilistic constellation shaping (PCS) is implemented via distribution matching (DM) algorithms, focusing on how approximating Maxwell–Boltzmann distributions using Huffman-based DM (HDM) and constant composition DM (CCDM) affects the secret key rate (SKR) and excess-noise tolerance. Symbol-by-symbol HDM reduces SKR by at least 30%, while CCDM achieves the optimal SKR with code length ≥ 103. Statistical symbol-dependence analysis shows CCDM needs ≥ 105-symbol blocks for correlations to become negligible, and an algorithm is proposed for near-optimal independent symbols.","1  \nThis work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be  \naccessible.  \nSecret Key Rate Analysis of Distribution Matching Algorithms for Discrete-Modulated CV-QKD  \nMicael Dias, Caroline Alves, Gabrielly Roman, and Søren Forchhammer, Member, IEEE  \narXiv :2607 .06783v1 [ quant-ph] 7 Jul 2026  \nAbstract—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 103 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 105 symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.  \nIndex Terms—Continuous-variable quantum key distribution, distribution matching, probabilistic constellation shaping.  \nI. INTRODUCTION  \nQUantum key distribution (QKD) protocols allow the  \ngeneration of secret keys with information-theoretic security against a malicious eavesdropper (Eve) [1], [2] . Over the years, QKD protocols have undergone continuous progress in both theory and experiment, leading to field trials with existing telecommunication infrastructure and significant steps toward commercial deployment [3]–[6], with continuousvariable QKD (CV-QKD) protocols playing a pivotal role in this development. In these protocols, the secret key is encoded in the field quadratures of the states of light, such that it integrates naturally with conventional optical modulators and coherent detection, making compatibility seamless [7],[8] . Beyond that, the application of advanced digital signal processing  \nManuscript received month XX, 2026; revised August XX, 20XX. This work was supported in part by the European Union (HORIZON-MSCA- 2023 Postdoctoral Fellowship, 101153602 - COCoVaQ) and by the project Analysis and Development of Distribution Matching Algorithms for CVQKD, supported by QuIIN – Quantum Industrial Innovation, the EMBRAPIICIMATEC Competence Center in Quantum Technologies, with financial resources from the PPI IoT/Industry 4.0 of the MCTI, grant number 053/2023, signed with EMBRAPII.  \nMicael Dias and Søren Forchhammer are with the Department of Electrical and Photonics Engineering, Technical University of Denmark (DTU), 2800 Lyngby, Denmark. ([e-mail: mandi@dtu.dk](e-mail: mandi@dtu.dk); [sofo@dtu.dk](sofo@dtu.dk)).  \nCaroline Alves and Gabrielly Roman are with the QuIIN – Quantum Industrial Innovation, EMBRAPII CIMATEC Competence Center in Quantum Technologies, SENAI CIMATEC, Av. Orlando Gomes 1845, Salvador, 41650-010, BA, Brasil. ([email: caroline.morais@fieb.org.br](email: caroline.morais@fieb.org.br); [gabrielly.roman@fbter.org.br](gabrielly.roman@fbter.org.br)).  \nalgorithms enables the mitigation of channel impairments, increasing the signal-to-noise ratio [9]–[11] .  \nAlthough Gaussian modulated protocols benefit from more robust security proofs [12]–[14], discrete modulation is less resource demanding, requiring lower random number generation rates, and improving compatibility with finite resolution digi","cbCaiahcBZa5zVFl","https://ap.wps.com/l/cbCaiahcBZa5zVFl","pdf",876876,2,1,12,"English","en",105,"# Introduction\n## Quantum key distribution and CV-QKD background\n## Discrete modulation and PCS\n## Distribution matching (Huffman, constant composition)\n# Proposed study and evaluation\n## HDM vs CCDM impact on secret key rate\n## Excess-noise tolerance\n## Symbol dependence and block-length requirements\n# Independent-symbol generation algorithm","[{\"question\":\"What problem does the document address in discrete-modulated CV-QKD?\",\"answer\":\"It addresses the secret-key-rate penalty caused by discrete modulation and how to mitigate that penalty using probabilistic constellation shaping implemented through distribution matching algorithms.\"},{\"question\":\"How do the two distribution matching approaches compare (HDM vs CCDM) in terms of SKR?\",\"answer\":\"Symbol-by-symbol Huffman-based DM (HDM) degrades the secret key rate by at least 30%, while constant composition DM (CCDM) matches the optimal SKR when the code length is 103 symbols or more.\"},{\"question\":\"What block length requirement does CCDM have to make symbol correlations negligible?\",\"answer\":\"The document’s statistical analysis indicates CCDM must operate with blocks of at least 10^5 symbols for correlations to become negligible.\"}]",1784185572,30,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"secret-key-rate-analysis-of-distribution-matching-algorithms-for-discrete-modulated-cv-qkd","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/secret-key-rate-analysis-of-distribution-matching-algorithms-for-discrete-modulated-cv-qkd/83142/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address in discrete-modulated CV-QKD?","Question",{"text":75,"@type":76},"It addresses the secret-key-rate penalty caused by discrete modulation and how to mitigate that penalty using probabilistic constellation shaping implemented through distribution matching algorithms.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the two distribution matching approaches compare (HDM vs CCDM) in terms of SKR?",{"text":80,"@type":76},"Symbol-by-symbol Huffman-based DM (HDM) degrades the secret key rate by at least 30%, while constant composition DM (CCDM) matches the optimal SKR when the code length is 103 symbols or more.",{"name":82,"@type":73,"acceptedAnswer":83},"What block length requirement does CCDM have to make symbol correlations negligible?",{"text":84,"@type":76},"The document’s statistical analysis indicates CCDM must operate with blocks of at least 10^5 symbols for correlations to become 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