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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <title>Secret key rate bounds for quantum key distribution with faulty active phase randomization | EPJ Quantum Technology | Full Text</title> <meta name="citation_abstract" content="Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios."/> <meta name="journal_id" content="40507"/> <meta name="dc.title" content="Secret key rate bounds for quantum key distribution with faulty active phase randomization"/> <meta name="dc.source" content="EPJ Quantum Technology 2023 10:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="SpringerOpen"/> <meta name="dc.date" content="2023-12-15"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2023 The Author(s)"/> <meta name="dc.rights" content="2023 The Author(s)"/> <meta name="dc.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="dc.description" content="Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios."/> <meta name="prism.issn" content="2196-0763"/> <meta name="prism.publicationName" content="EPJ Quantum Technology"/> <meta name="prism.publicationDate" content="2023-12-15"/> <meta name="prism.volume" content="10"/> <meta name="prism.number" content="1"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="26"/> <meta name="prism.copyright" content="2023 The Author(s)"/> <meta name="prism.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="prism.url" content="https://epjquantumtechnology.springeropen.com/articles/10.1140/epjqt/s40507-023-00210-0"/> <meta name="prism.doi" content="doi:10.1140/epjqt/s40507-023-00210-0"/> <meta name="citation_pdf_url" content="https://epjquantumtechnology.springeropen.com/counter/pdf/10.1140/epjqt/s40507-023-00210-0"/> <meta name="citation_fulltext_html_url" content="https://epjquantumtechnology.springeropen.com/articles/10.1140/epjqt/s40507-023-00210-0"/> <meta name="citation_journal_title" content="EPJ Quantum Technology"/> <meta name="citation_journal_abbrev" content="EPJ Quantum Technol."/> <meta name="citation_publisher" content="SpringerOpen"/> <meta name="citation_issn" content="2196-0763"/> <meta name="citation_title" content="Secret key rate bounds for quantum key distribution with faulty active phase randomization"/> <meta name="citation_volume" content="10"/> <meta name="citation_issue" content="1"/> <meta name="citation_publication_date" content="2023/12"/> <meta name="citation_online_date" content="2023/12/15"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="26"/> <meta name="citation_article_type" content="Research"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1140/epjqt/s40507-023-00210-0"/> <meta name="DOI" content="10.1140/epjqt/s40507-023-00210-0"/> <meta name="size" content="970478"/> <meta name="citation_doi" content="10.1140/epjqt/s40507-023-00210-0"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1140/epjqt/s40507-023-00210-0&api_key="/> <meta name="description" content="Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios."/> <meta name="dc.creator" content="Sixto, Xoel"/> <meta name="dc.creator" content="Currás-Lorenzo, Guillermo"/> <meta name="dc.creator" content="Tamaki, Kiyoshi"/> <meta name="dc.creator" content="Curty, Marcos"/> <meta name="dc.subject" content="Quantum Physics"/> <meta name="dc.subject" content="Quantum Information Technology, Spintronics"/> <meta name="dc.subject" content="Nanotechnology and Microengineering"/> <meta name="citation_reference" content="citation_journal_title=Rev Mod Phys; citation_title=Secure quantum key distribution with realistic devices; citation_author=F Xu, X Ma, Q Zhang, HK Lo, JW Pan; citation_volume=92; citation_publication_date=2020; citation_doi=10.1103/RevModPhys.92.025002; citation_id=CR1"/> <meta name="citation_reference" content="citation_journal_title=Adv Opt Photonics; citation_title=Advances in quantum cryptography; citation_author=S Pirandola, UL Andersen, L Banchi, M Berta, D Bunandar, R Colbeck; 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It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig1_HTML.jpg"/> <script type="application/ld+json">{"mainEntity":{"headline":"Secret key rate bounds for quantum key distribution with faulty active phase randomization","description":"Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios.","datePublished":"2023-12-15T00:00:00Z","dateModified":"2023-12-15T00:00:00Z","pageStart":"1","pageEnd":"26","license":"http://creativecommons.org/licenses/by/4.0/","sameAs":"https://doi.org/10.1140/epjqt/s40507-023-00210-0","keywords":["Quantum key distribution","Decoy state","Phase randomization","Source imperfections","Quantum Physics","Quantum Information Technology","Spintronics","Nanotechnology and 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Sixto","affiliation":[{"name":"University of Vigo","address":{"name":"Vigo Quantum Communication Center, University of Vigo, Vigo, Spain","@type":"PostalAddress"},"@type":"Organization"},{"name":"University of Vigo","address":{"name":"Escuela de Ingeniería de Telecomunicación, Department of Signal Theory and Communications, University of Vigo, Vigo, Spain","@type":"PostalAddress"},"@type":"Organization"},{"name":"University of Vigo","address":{"name":"atlanTTic Research Center, University of Vigo, Vigo, Spain","@type":"PostalAddress"},"@type":"Organization"}],"email":"xsixto@vqcc.uvigo.es","@type":"Person"},{"name":"Guillermo Currás-Lorenzo","affiliation":[{"name":"University of Vigo","address":{"name":"Vigo Quantum Communication Center, University of Vigo, Vigo, Spain","@type":"PostalAddress"},"@type":"Organization"},{"name":"University of Vigo","address":{"name":"Escuela de Ingeniería de Telecomunicación, Department of Signal Theory and Communications, University of Vigo, Vigo, 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2023</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">Secret key rate bounds for quantum key distribution with faulty active phase randomization</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Xoel-Sixto-Aff1-Aff2-Aff3" data-author-popup="auth-Xoel-Sixto-Aff1-Aff2-Aff3" data-author-search="Sixto, Xoel" data-corresp-id="c1">Xoel Sixto<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a>,<a href="#Aff3">3</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Guillermo-Curr_s_Lorenzo-Aff1-Aff2-Aff3-Aff4" data-author-popup="auth-Guillermo-Curr_s_Lorenzo-Aff1-Aff2-Aff3-Aff4" data-author-search="Currás-Lorenzo, Guillermo">Guillermo Currás-Lorenzo</a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a>,<a href="#Aff3">3</a>,<a href="#Aff4">4</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Kiyoshi-Tamaki-Aff4" data-author-popup="auth-Kiyoshi-Tamaki-Aff4" data-author-search="Tamaki, Kiyoshi">Kiyoshi Tamaki</a><sup class="u-js-hide"><a href="#Aff4">4</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 4 authors for this article" title="Show all 4 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Marcos-Curty-Aff1-Aff2-Aff3" data-author-popup="auth-Marcos-Curty-Aff1-Aff2-Aff3" data-author-search="Curty, Marcos">Marcos Curty</a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a>,<a href="#Aff3">3</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">EPJ Quantum Technology</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 10</b>, Article number: <span data-test="article-number">53</span> (<span data-test="article-publication-year">2023</span>) <a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">1117 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="citation-count"> <p class="c-article-metrics-bar__count">3 <span class="c-article-metrics-bar__label">Citations</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">1 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/10.1140/epjqt/s40507-023-00210-0/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </div> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient solution to handle multi-photon signals emitted by laser sources, and provides the same secret key rate scaling as ideal single-photon sources. It requires, however, that the phase of each emitted pulse is uniformly random. This might be difficult to guarantee in practice, due to inevitable device imperfections and/or the use of an external phase modulator for phase randomization in an active setup, which limits the possible selected phases to a finite set. Here, we investigate the security of decoy-state QKD when the phase is actively randomized by faulty devices, and show that this technique is quite robust to deviations from the ideal uniformly random scenario. For this, we combine a novel parameter estimation technique based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the parameters that determine the achievable secret key rate. In doing so, we demonstrate that our analysis can significantly outperform previous results that address more restricted scenarios.</p></div></div></section> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Quantum key distribution (QKD) is a method for securely establishing symmetric cryptographic keys between two distant parties (so-called Alice and Bob) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title=" Xu F, Ma X, Zhang Q, Lo HK, Pan JW. Secure quantum key distribution with realistic devices. Rev Mod Phys. 2020;92:025002. https://doi.org/10.1103/RevModPhys.92.025002 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR1" id="ref-link-section-d131444011e475">1</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title=" Lo HK, Curty M, Tamaki K. Secure quantum key distribution. Nat Photonics. 2014;8(8):595–604. https://doi.org/10.1038/nphoton.2014.149 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR3" id="ref-link-section-d131444011e478">3</a>]. Its security is based on principles of quantum mechanics, such as the no-cloning theorem [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title=" Wootters WK, Zurek WH. A single quantum cannot be cloned. Nature. 1982;299:802–3. https://doi.org/10.1038/299802a0 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR4" id="ref-link-section-d131444011e481">4</a>], which guarantee that any attempt by an eavesdropper (Eve) to learn information about the distributed key inevitably introduces detectable errors. Importantly, when combined with the one-time-pad encryption scheme [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title=" Vernam GS. Cipher printing telegraph systems for secret wire and radio telegraphic communications. Trans Am Inst Electr Eng. 1926;XLV:295–301. https://doi.org/10.1109/T-AIEE.1926.5061224 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR5" id="ref-link-section-d131444011e484">5</a>], QKD provides information-theoretically secure communications.</p><p>The field of QKD has made much progress in recent years, both theoretically and experimentally, leading to the first deployments of metropolitan and intercity QKD networks [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title=" Sasaki M, Fujiwara M, Ishizuka H, Klaus W, Wakui K, Takeoka M et al.. Field test of quantum key distribution in the Tokyo QKD network. Opt Express. 2011;19(11):10387. https://doi.org/10.1364/oe.19.010387 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR6" id="ref-link-section-d131444011e490">6</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title=" Chen YA, Zhang Q, Chen TY, Cai WQ, Liao SK, Zhang J et al.. An integrated space-to-ground quantum communication network over 4,600 kilometres. Nature. 2021;589:214–9. https://doi.org/10.1038/s41586-020-03093-8 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR9" id="ref-link-section-d131444011e493">9</a>]. Despite these remarkable achievements, there are still certain challenges that need to be overcome for the widespread adoption of this technology. One of these challenges is to close the existing security gap between theory and practice. This is so because QKD security proofs, typically consider assumptions that the actual experimental implementations do not satisfy. Such discrepancies could create security loopholes or so-called side channels, which might be exploited by Eve to compromise the security of the generated key without being detected.</p><p>Indeed, practical QKD transmitters usually emit phase-randomized weak coherent pulses (PR-WCPs) generated by laser sources. These pulses might contain more than one photon prepared in the same quantum state. In this scenario, Eve is no longer limited by the no-cloning theorem, because multi-photon signals provide her with perfect copies of the signal photon. As a result, it can be shown that the secret key rate of the BB84 protocol [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title=" Bennett CH, Brassard G. Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE international conference on computers, systems, and signal processing. 1984. p. 175–9. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR10" id="ref-link-section-d131444011e499">10</a>] with PR-WCPs scales quadratically with the system’s transmittance due to the photon-number-splitting (PNS) attack [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Huttner B, Imoto N, Gisin N, Mor T. Quantum cryptography with coherent states. Phys Rev A. 1995;51:1863–9. https://doi.org/10.1103/PhysRevA.51.1863 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR11" id="ref-link-section-d131444011e502">11</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Brassard G, Lütkenhaus N, Mor T, Sanders BC. Limitations on practical quantum cryptography. Phys Rev Lett. 2000;85:1330. https://doi.org/10.1103/PhysRevLett.85.1330 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR12" id="ref-link-section-d131444011e505">12</a>]. This attack provides Eve with full information about the part of the key generated with the multi-photon pulses, without introducing any error.</p><p>To overcome this limitation, the most efficient solution today is undoubtedly the decoy-state method [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title=" Hwang WY. Quantum key distribution with high loss: toward global secure communication. Phys Rev Lett. 2003;91(5):057901. https://doi.org/10.1103/physrevlett.91.057901 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR13" id="ref-link-section-d131444011e511">13</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title=" Lo HK, Ma X, Decoy CK. State quantum key distribution. Phys Rev Lett. 2005;94(23):230504. https://doi.org/10.1103/physrevlett.94.230504 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR15" id="ref-link-section-d131444011e514">15</a>], in which Alice varies at random the intensity of the PR-WCPs that she sends to Bob. This allows them to better estimate the behavior of the quantum channel. Indeed, using the observed measurement statistics associated to different intensity settings, Alice and Bob can tightly estimate the yield and phase error rate of the single-photon pulses, from which the secret key is actually distilled. As a result, the decoy-state method delivers a secret key rate that scales linearly with the channel transmittance [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title=" Hwang WY. Quantum key distribution with high loss: toward global secure communication. Phys Rev Lett. 2003;91(5):057901. https://doi.org/10.1103/physrevlett.91.057901 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR13" id="ref-link-section-d131444011e517">13</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title=" Lim CCW, Curty M, Walenta N, Xu F, Concise ZH. Security bounds for practical decoy-state quantum key distribution. Phys Rev A. 2014;89:022307. https://doi.org/10.1103/physreva.89.022307 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR16" id="ref-link-section-d131444011e520">16</a>], matching the scaling achievable with ideal single-photon sources. This technique has been extensively demonstrated in multiple recent experiments [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title=" Zhao Y, Qi B, Ma X, Lo HK, Qian L. Experimental quantum key distribution with decoy states. Phys Rev Lett. 2006;96:70502. https://doi.org/10.1103/PhysRevLett.96.070502 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR17" id="ref-link-section-d131444011e523">17</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title=" Boaron A, Boso G, Rusca D, Vulliez C, Autebert C, Caloz M et al.. Secure quantum key distribution over 421 km of optical fiber. Phys Rev Lett. 2018;121:190502. https://doi.org/10.1103/PhysRevLett.121.190502 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR23" id="ref-link-section-d131444011e527">23</a>], including satellite links [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title=" Liao SK, Cai WQ, Liu WY, Zhang L, Li Y, Ren JG et al.. Satellite-to-ground quantum key distribution. Nature. 2017;549(7670):43–7. https://doi.org/10.1038/nature23655 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR24" id="ref-link-section-d131444011e530">24</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title=" Liao SK, Cai WQ, Handsteiner J, Liu B, Yin J, Zhang L et al.. Satellite-relayed intercontinental quantum network. Phys Rev Lett. 2018;120:030501. https://doi.org/10.1103/PhysRevLett.120.030501 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR25" id="ref-link-section-d131444011e533">25</a>] and the use of photonic integrated circuits [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title=" Sibson P, Erven C, Godfrey M, Miki S, Yamashita T, Fujiwara M et al.. Chip-based quantum key distribution. Nat Commun. 2017;8:13984. https://doi.org/10.1038/ncomms13984 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR26" id="ref-link-section-d131444011e536">26</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title=" Marco ID, Woodward RI, Roberts GL, Paraïso TK, Roger T, Sanzaro M et al.. Real-time operation of a multi-rate, multi-protocol quantum key distribution transmitter. Optica. 2021;8(6):911–5. https://doi.org/10.1364/OPTICA.423517 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR29" id="ref-link-section-d131444011e539">29</a>]. Also, decoy-state QKD setups are currently offered commercially by several companies [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title=" ID Quantique SA. https://www.idquantique.com/ . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR30" id="ref-link-section-d131444011e542">30</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title=" Quantum Telecommunications Italy S.R.L. https://www.qticompany.com . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR34" id="ref-link-section-d131444011e546">34</a>], which highlights its importance.</p><p>Importantly, standard decoy-state security proofs assume perfect phase randomization, <i>i.e.</i>, that the phase, <i>θ</i>, of each generated WCP is uniformly random in <span class="mathjax-tex">$[0,2\pi )$</span>. That is, its probability density function (PDF), <span class="mathjax-tex">$g(\theta )$</span>, should satisfy <span class="mathjax-tex">$g(\theta )=1/2\pi $</span>. However, none of the two main methods used today to generate PR-WCPs, namely passive and active, fulfill this condition exactly. In the passive scheme a technique known as gain-switching is used to effectively turn the laser on and off between pulses. However, in these configurations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title=" Liu Y, Chen TY, Wang J, Cai WQ, Wan X, Chen LK et al.. Decoy-state quantum key distribution with polarized photons over 200 km. Opt Express. 2010;18:8587–94. https://doi.org/10.1364/OE.18.008587 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR20" id="ref-link-section-d131444011e658">20</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title=" Yuan Z, Murakami A, Kujiraoka M, Lucamarini M, Tanizawa Y, Sato H et al.. 10-Mb/s quantum key distribution. J Lightwave Technol. 2018;36:3427–33. https://doi.org/10.1109/jlt.2018.2843136 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR22" id="ref-link-section-d131444011e661">22</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title=" Boaron A, Boso G, Rusca D, Vulliez C, Autebert C, Caloz M et al.. Secure quantum key distribution over 421 km of optical fiber. Phys Rev Lett. 2018;121:190502. https://doi.org/10.1103/PhysRevLett.121.190502 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR23" id="ref-link-section-d131444011e664">23</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title=" Yuan ZL, Sharpe AW, Shields AJ. Unconditionally secure one-way quantum key distribution using decoy pulses. Appl Phys Lett. 2007;90:011118. https://doi.org/10.1063/1.2430685 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR35" id="ref-link-section-d131444011e667">35</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title=" Valivarthi R, Zhou Q, John C, Marsili F, Verma VB, Shaw MD et al.. A cost-effective measurement-device-independent quantum key distribution system for quantum networks. Quantum Sci Technol. 2017. 2:04LT01. https://doi.org/10.1088/2058-9565/aa8790 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR38" id="ref-link-section-d131444011e670">38</a>], device imperfections can prevent the phases <i>θ</i> from being uniformly distributed. In the active scheme [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title=" Sibson P, Erven C, Godfrey M, Miki S, Yamashita T, Fujiwara M et al.. Chip-based quantum key distribution. Nat Commun. 2017;8:13984. https://doi.org/10.1038/ncomms13984 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR26" id="ref-link-section-d131444011e677">26</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title=" Bunandar D, Lentine A, Lee C, Cai H, Long CM, Boynton N et al.. Metropolitan quantum key distribution with silicon photonics. Phys Rev X. 2018;8:021009. https://doi.org/10.1103/PhysRevX.8.021009 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR27" id="ref-link-section-d131444011e680">27</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title=" Zhao Y, Qi B, Lo HK. Experimental quantum key distribution with active phase randomization. Appl Phys Lett. 2007;90(4):044106. https://doi.org/10.1063/1.2432296 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR39" id="ref-link-section-d131444011e683">39</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title=" Sun SH, Liang LM. Experimental demonstration of an active phase randomization and monitor module for quantum key distribution. Appl Phys Lett. 2012;101:071107. https://doi.org/10.1063/1.4746402 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR40" id="ref-link-section-d131444011e686">40</a>], an external phase modulator is used to imprint one of <i>N</i> possible random values to the phase of each pulse, such that only a discrete number of phases is selected. Both scenarios violate a crucial assumption of the decoy-state technique.</p><p>The security of QKD with imperfect passive phase randomization has, under certain assumptions, been recently demonstrated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e695">41</a>]. This analysis however, is not applicable to the numerous existing active setups that rely on an external phase modulator for phase randomization .<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup> The security of the latter approach has been analyzed in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e850">42</a>] (see also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title=" Currás-Lorenzo G, Wooltorton L, Twin-Field RM. Quantum key distribution with fully discrete phase randomization. Phys Rev Appl. 2021;15:014016. https://doi.org/10.1103/PhysRevApplied.15.014016 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR43" id="ref-link-section-d131444011e853">43</a>]), but these works restrict themselves to the case in which the discrete random phases are evenly distributed in <span class="mathjax-tex">$[0,2\pi )$</span>, <i>i.e.</i>, they assume that <span class="mathjax-tex">$g(\theta )$</span> satisfies </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g(\theta ) = \frac{1}{N} \sum _{k=0}^{N-1} \delta (\theta -\theta _{k}), $$</span></div><div class="c-article-equation__number"> (1) </div></div><p> where <span class="mathjax-tex">$\delta (x)$</span> represents the Dirac delta function, and <span class="mathjax-tex">$\theta _{k}=2 \pi k/N$</span>, with <i>N</i> being the total number of selected phases. Under this assumption, [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e1071">42</a>] shows that it is possible to approximate the secret key rate achievable in the ideal situation where <span class="mathjax-tex">$g(\theta )=1/2\pi $</span>, with around <span class="mathjax-tex">$N=10$</span> random phases. While this result is remarkable, in practice, inevitable imperfections of the phase modulator and electronic noise might prevent the phases <i>θ</i> from being <i>exactly</i> evenly distributed, thus invalidating the application of the results presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e1143">42</a>] to a real setup.</p><p>The main contributions of this paper are as follows. First, we introduce an analysis that can be applied in the more realistic and practical scenario in which <span class="mathjax-tex">$g(\theta )$</span> is an arbitrary PDF, due to imperfections in the active phase randomization process, and we provide asymptotic secret key rates for this general situation, thus filling an important gap in the literature.</p><p>Second, we show that this analysis can be applied in the scenario in which the PDF <span class="mathjax-tex">$g(\theta )$</span> is not fully characterized. This feature significantly simplifies the applicability of our results to a practical setup, where an accurate characterization of the PDF describing the phase might be challenging.</p><p>Third, we make a noteworthy finding regarding the utilization of basis mismatched events which are typically discarded in QKD security analyses, including that in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e1205">42</a>]. The use of basis mismatched events is already known to provide a key-rate advantage in the presence of bit-and-basis encoding flaws [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title=" Tamaki K, Curty M, Kato G, Lo HK, Azuma K. Loss-tolerant quantum cryptography with imperfect sources. Phys Rev A. 2014;90:052314. https://doi.org/10.1103/PhysRevA.90.052314 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR44" id="ref-link-section-d131444011e1208">44</a>] but here, we show that they can also be advantageous in the presence of imperfections due to a faulty phase randomization process. We believe that this additional result is highly nontrivial as intuitively the decoy state method has no relation with the state preparation flaw in encoding the bit information.</p><p>Fourth, when considering the ideal discrete-phase-randomization case described by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), our analysis delivers considerably higher secret key rates than those provided by the seminal work in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e1217">42</a>], or to put it in other words, it requires to spend fewer random bits for phase selection to achieve an equivalent performance.</p><p>As a side remark, we note that our results are also useful for other quantum communication schemes that go beyond QKD and employ laser sources, as they often rely on decoy-states with active phase randomization.</p><p>Finally, it is worth mentioning that, although, for simplicity, in our derivations we consider collective attacks, our analysis can be lifted to general attacks by applying the extension of the quantum de Finetti theorem [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title=" Renner R, Cirac JI. de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography. Phys Rev Lett. 2009;102:110504. https://doi.org/10.1103/PhysRevLett.102.110504 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR45" id="ref-link-section-d131444011e1227">45</a>] to infinitely-dimensional systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title=" Renner R. Symmetry of large physical systems implies independence of subsystems. Nat Phys. 2007;3(9):645–9. https://doi.org/10.1038/nphys684 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR46" id="ref-link-section-d131444011e1230">46</a>]. Because of this, the asymptotic key rates that we derive in this paper are also valid against general attacks.</p><p>The paper is organized as follows. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec3">2.1</a>, we describe the quantum states emitted by Alice when <i>θ</i> follows an arbitrary PDF, <span class="mathjax-tex">$g(\theta )$</span>. Then, in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec4">2.2</a> we introduce the decoy-state protocol considered, together with its asymptotic secret key rate formula. Next, in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec5">2.3</a>, we present the parameter estimation technique based on SDP, as well as on the use of basis mismatched events, to calculate the different parameters required to evaluate the secret key rate. Then, in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec10">3</a> we simulate the achievable secret key rate for various functions <span class="mathjax-tex">$g(\theta )$</span> of practical interest, both for the cases in which this function is fully (or only partially) characterized. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec15">4</a> concludes the paper with a summary. The paper includes as well some Appendixes with additional calculations.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Methods</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Phase randomization with an arbitrary <span class="mathjax-tex">$g(\theta )$</span> </h3><p>In this section, we describe the quantum states emitted by Alice when each of them has a phase <i>θ</i> that follows an arbitrary PDF, <span class="mathjax-tex">$g(\theta )$</span>.</p><p>In particular, a WCP of intensity <i>μ</i> and phase <i>θ</i> can be written in terms of the Fock basis as </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\sqrt{\mu}e^{i \theta}\bigr\rangle =e^{-\frac{\mu}{2}} \sum _{n=0}^{ \infty}\frac{ (\sqrt{\mu}e^{i \theta} )^{n}}{\sqrt{n !}}|n \rangle , $$</span></div><div class="c-article-equation__number"> (2) </div></div><p> where <span class="mathjax-tex">$|n\rangle $</span> represents a Fock state with <i>n</i> photons.</p><p>If Alice selects the phase <i>θ</i> of each generated signal independently and at random according to <span class="mathjax-tex">$g(\theta )$</span>, its state is simply given by </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \rho ^{\mu}_{[g(\theta )]}= \int _{0}^{2 \pi} g(\theta ){\hat{P}}\bigl(\bigl| \sqrt{ \mu} e^{i \theta}\bigr\rangle \bigr)\,d\theta , $$</span></div><div class="c-article-equation__number"> (3) </div></div><p> with <span class="mathjax-tex">${\hat{P}}(|\phi \rangle )=|\phi \rangle \langle \phi |$</span>.</p><p>Any quantum state can always be diagonalised in a certain orthonormal basis. For the states given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ3">3</a>), we shall denote the elements of such basis by <span class="mathjax-tex">$|\psi _{n, \mu , g(\theta )}\rangle $</span>, since, in general, they might depend on both the intensity <i>μ</i> and the function <span class="mathjax-tex">$g(\theta )$</span>. Here, the subscript <i>n</i> simply identifies the different elements of the basis, which are not necessarily the Fock states. This means, in particular, that we can rewrite the states given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ3">3</a>) as follows </p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \rho ^{\mu}_{[g(\theta )]}=\sum _{n=0}^{\infty} p_{n| \mu , g(\theta )}{ \hat{P}}\bigl(|\psi _{n, \mu , g(\theta )}\rangle \bigr), $$</span></div><div class="c-article-equation__number"> (4) </div></div><p> where the coefficients <span class="mathjax-tex">$p_{n| \mu , g(\theta )}\geq{}0$</span> satisfy <span class="mathjax-tex">$\sum_{n=0}^{\infty} p_{n| \mu , g(\theta )}=1$</span>. That is, these coefficients can be interpreted as the probability with which, in a certain time instance, Alice emits the state <span class="mathjax-tex">$|\psi _{n, \mu , g(\theta )}\rangle $</span>, given that she chose the intensity <i>μ</i> and <i>θ</i> follows the PDF <span class="mathjax-tex">$g(\theta )$</span>.</p><p>For instance, in the ideal scenario where <span class="mathjax-tex">$g(\theta )$</span> is uniformly random in <span class="mathjax-tex">$[0,2\pi )$</span>, the emitted signals are a Poisson mixture of Fock states given by </p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \rho ^{\mu}_{[\frac{1}{2\pi}]}= \frac{1}{2\pi} \int _{0}^{2 \pi}{ \hat{P}}\bigl(\bigl|\sqrt{\mu} e^{i \theta}\bigr\rangle \bigr)\,d\theta=e^{-\mu}\sum_{n=0}^{\infty } \frac{\mu ^{n}}{n !} {\hat{P}}\bigl(|n \rangle \bigr), $$</span></div><div class="c-article-equation__number"> (5) </div></div><p><i>i.e.</i> <span class="mathjax-tex">$p_{n| \mu , 1/2\pi}=e^{-\mu}\mu ^{n}/(n!)$</span> and <span class="mathjax-tex">$|\psi _{n, \mu , 1/2\pi}\rangle =|n\rangle $</span>.</p><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">2.2 </span>Protocol description and key generation rate</h3><p>For concreteness, we shall assume that Alice and Bob implement a decoy-state BB84 scheme with three different intensity settings <span class="mathjax-tex">$\{s, \nu , \omega \}$</span> in each basis, with <span class="mathjax-tex">$s>\nu >\omega \geq{}0$</span>. Moreover, we consider that they generate secret key only from those events in which both of them select the <i>Z</i> basis and Alice chooses the signal intensity setting <i>s</i>. This is the most typical configuration of the decoy-state BB84 protocol. We remark, however, that the analysis below could be straightforwardly adapted to other protocol configurations, or to other combinations of intensity settings.</p><p>In each round of the protocol, Alice probabilistically chooses a bit value <span class="mathjax-tex">$b\in \{0,1\}$</span> with probability <span class="mathjax-tex">$p_{b}=1/2$</span>, a basis <span class="mathjax-tex">$\alpha \in \{Z,X\}$</span> with probability <span class="mathjax-tex">$p_{\alpha}$</span>, an intensity value <span class="mathjax-tex">$\mu \in \{s, \nu , \omega \}$</span> with probability <span class="mathjax-tex">$p_{\mu}$</span>, and a random phase <i>θ</i> according to the PDF given by <span class="mathjax-tex">$g(\theta )$</span>. Then, she generates a WCP of intensity <i>μ</i> and phase <i>θ</i>, <span class="mathjax-tex">$|\sqrt{\mu}e^{i \theta}\rangle $</span>, and applies an operation that encodes her bit and basis choices <i>b</i> and <i>α</i> into the pulse. From Eve’s perspective, these states are described by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ4">4</a>) due to her ignorance about the selected phase <i>θ</i>. On the receiving side, Bob measures each arriving signal using a basis <span class="mathjax-tex">$\alpha \in \{Z,X\}$</span>, which he selects with probability <span class="mathjax-tex">$p_{\alpha}$</span>. We shall assume the basis independent detection efficiency condition throughout the paper. That is, the probability that Bob obtains a conclusive measurement outcome does not depend on his basis choice.</p><p>Once the quantum communication phase of the protocol ends, Alice and Bob broadcast (via an authenticated classical channel) both the intensity and basis settings selected for each detected signal. The results related to those detected signals in which both of them used the <i>Z</i> basis with intensity setting <i>s</i> constitute the sifted key. For the detected rounds in which Bob chose the <i>X</i> basis, Alice reveals her bit values <i>b</i> and Bob announces his corresponding measurement outcomes. This data is used for parameter estimation, <i>i.e.</i>, to determine the relevant quantities needed to evaluate the secret key rate formula. Finally, Alice and Bob apply error correction and privacy amplification to the sifted key to obtain a final secret key, following the standard post-processing procedure in QKD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title=" Xu F, Ma X, Zhang Q, Lo HK, Pan JW. Secure quantum key distribution with realistic devices. Rev Mod Phys. 2020;92:025002. https://doi.org/10.1103/RevModPhys.92.025002 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR1" id="ref-link-section-d131444011e3021">1</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title=" Lo HK, Curty M, Tamaki K. Secure quantum key distribution. Nat Photonics. 2014;8(8):595–604. https://doi.org/10.1038/nphoton.2014.149 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR3" id="ref-link-section-d131444011e3024">3</a>]. For a more detailed description of the protocol steps of a decoy-state BB84 scheme, we refer the reader to <i>e.g.</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title=" Lim CCW, Curty M, Walenta N, Xu F, Concise ZH. Security bounds for practical decoy-state quantum key distribution. Phys Rev A. 2014;89:022307. https://doi.org/10.1103/physreva.89.022307 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR16" id="ref-link-section-d131444011e3030">16</a>].</p><p>In the ideal scenario where <span class="mathjax-tex">$g(\theta )=1/2\pi $</span>, Alice’s state preparation process is equivalent to emitting Fock states <span class="mathjax-tex">$|n\rangle $</span> with a Poisson distribution of mean equal to the intensity setting <i>μ</i> selected, as shown by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ5">5</a>). In this situation, both the single-photon and vacuum pulses with the intensity setting <i>s</i> contribute to secret bits [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title=" Lo HK. Getting something out of nothing. Quantum Inf Comput. 2005;5:413–8. https://doi.org/10.26421/QIC5.45-10 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR47" id="ref-link-section-d131444011e3109">47</a>]. The multi-photon signals are insecure due to the PNS attack. Similarly, when <i>θ</i> follows an arbitrary PDF, <span class="mathjax-tex">$g(\theta )$</span>, and Alice chooses the intensity setting <i>μ</i>, from Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ4">4</a>) we have that her state preparation process is equivalent to generating pure states <span class="mathjax-tex">$|\psi _{n, \mu , g(\theta )}\rangle $</span> with probability <span class="mathjax-tex">$p_{n| \mu , g(\theta )}$</span>. The closer the function <span class="mathjax-tex">$g(\theta )$</span> is to a uniform distribution, the closer the signals (probabilities) <span class="mathjax-tex">$|\psi _{n, \mu , g(\theta )}\rangle $</span> (<span class="mathjax-tex">$p_{n| \mu , g(\theta )}$</span>) are to the Fock states <span class="mathjax-tex">$|n\rangle $</span> (probabilities <span class="mathjax-tex">$e^{-\mu}\mu ^{n}/n!$</span>). In this scenario, Alice and Bob can in principle distill secret bits from any <span class="mathjax-tex">$|\psi _{n, \mu , g(\theta )}\rangle $</span> with <span class="mathjax-tex">$\mu =s$</span>, though the main contribution would mainly arise from those with indexes <span class="mathjax-tex">$n=0,1$</span>, which are the ones closer to vacuum and single-photon pulses. These are the contributions that we consider below. Indeed, for the examples studied in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec10">3</a>, we have tested numerically that the improvement in key rate that can be obtained when considering <span class="mathjax-tex">$n>1$</span> is negligible.</p><p>This means that, in this imperfect state preparation scenario, the asymptotic secret key rate formula for the decoy-state BB84 protocol considered can be written as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title=" Lo HK, Ma X, Decoy CK. State quantum key distribution. Phys Rev Lett. 2005;94(23):230504. https://doi.org/10.1103/physrevlett.94.230504 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR15" id="ref-link-section-d131444011e3549">15</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title=" Lo HK. Getting something out of nothing. Quantum Inf Comput. 2005;5:413–8. https://doi.org/10.26421/QIC5.45-10 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR47" id="ref-link-section-d131444011e3552">47</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title=" Gottesman D, Lo HK, Lütkenhaus N, Preskill J. Security of quantum key distribution with imperfect devices. Quantum Inf Comput. 2004;4:325–60. https://doi.org/10.26421/QIC4.5-1 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR48" id="ref-link-section-d131444011e3555">48</a>] </p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} R \geq & p_{Z}^{2}p_{s} \Biggl\{ \sum_{n=0}^{\infty} p_{n| s, g(\theta )}Y_{n, s, g(\theta )}^{Z} \bigl[1-h (e_{n, s, g(\theta )} ) \bigr] -f Q_{s, g(\theta )}^{Z} h \bigl(E_{s, g(\theta )}^{Z} \bigr) \Biggr\} \\ \geq & p_{Z}^{2}p_{s} \Biggl\{ \sum _{n=0}^{1} p_{n| s, g(\theta )}^{ \text{L}}Y_{n, s, g(\theta )}^{Z, \text{L}} \bigl[1-h \bigl(e_{n, s, g( \theta )}^{\mathrm{U}} \bigr) \bigr] -f Q_{s, g(\theta )}^{Z} h \bigl(E_{s, g(\theta )}^{Z} \bigr) \Biggr\} , \end{aligned}$$ </span></div><div class="c-article-equation__number"> (6) </div></div><p> where <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> denotes the yield associated to the state <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span> encoded (and measured) in the <i>Z</i> basis, <i>i.e.</i>, the probability that Bob observes a detection click in his measurement apparatus conditioned on Alice and Bob selecting the <i>Z</i> basis and Alice preparing the state <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span>; the parameter <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> represents the phase error rate of these latter signals; <span class="mathjax-tex">$h(x)=-x\log _{2}{(x)}-(1-x)\log _{2}{(1-x)}$</span> is the binary Shannon entropy function; the quantity <i>f</i> is the efficiency of the error correction protocol; <span class="mathjax-tex">$Q_{s, g(\theta )}^{Z}$</span> is the overall gain of the signals emitted conditioned on Alice selecting the intensity <i>s</i> and Alice and Bob choosing the <i>Z</i> basis, <i>i.e.</i>, the probability that Bob observes a detection click conditioned on Alice sending him such signals; and <span class="mathjax-tex">$E_{s, g(\theta )}^{Z}$</span> is the overall quantum bit error rate (QBER) associated to these latter signals. Moreover, in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ6">6</a>), the superscript L (U) refers to a (an) lower (upper) bound.</p><p>The quantities <span class="mathjax-tex">$Q_{s, g(\theta )}^{Z}$</span> and <span class="mathjax-tex">$E_{s, g(\theta )}^{Z}$</span> are directly observed in the experiment. In principle, the probabilities <span class="mathjax-tex">$p_{n| s, g(\theta )}$</span> could also be known, and depend on the state preparation process. However, in practice it might be difficult to find their value analytically. Instead, in the next section we present a simple method to obtain a lower bound, <span class="mathjax-tex">$p_{n| s, g(\theta )}^{\mathrm{L}}$</span>, on these quantities. There, we also explain how to estimate the parameters <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z, \mathrm{L}}$</span> and <span class="mathjax-tex">$e_{n, s, g(\theta )}^{\mathrm{U}}$</span>, with <span class="mathjax-tex">$n=0,1$</span>, which are needed to evaluate Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ6">6</a>).</p><h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">2.3 </span>Parameter estimation</h3><p>The parameter estimation procedure presented here is an adaptation of the one very recently introduced in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e4741">41</a>] in the context of phase correlations in a passive randomization setup. For simplicity, below we introduce the main results and refer the reader to Appendixes <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec16">A</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec17">B</a> for the detailed derivations.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec6"><span class="c-article-section__title-number">2.3.1 </span>Lower bound on the yields <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> </h4><p>In Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec16">A</a> it is shown that a lower bound on the yields <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> can be obtained by solving the following SDP:<sup><a href="#Fn2"><span class="u-visually-hidden">Footnote </span>2</a></sup></p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \min_{J_{Z}} &\ \operatorname{Tr} \bigl[{\hat{P}}\bigl(|\psi _{n, s, g( \theta )}\rangle \bigr)J_{Z} \bigr] \\ \text{subject to} &\ \operatorname{Tr} \bigl[\rho ^{\mu}_{[g(\theta )]} J_{Z} \bigr]=Q_{\mu , g(\theta )}^{Z}, \quad \forall \mu \in \{s, \nu , \omega \} \\ &\ 0\leq J_{Z} \leq \mathbb{I}. \end{aligned} $$</span></div><div class="c-article-equation__number"> (7) </div></div><p> The states <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span> and <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> are known in principle but inaccessible and depend on the intensity setting selected by Alice and on the function <span class="mathjax-tex">$g(\theta )$</span>. Also, as already mentioned, the gains <span class="mathjax-tex">$Q_{\mu , g(\theta )}^{Z}$</span> are directly observed experimentally in a realization of the protocol. That is, the only unknown in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>) is the positive semi-definite operator <span class="mathjax-tex">$J_{Z}$</span> over which the minimization takes place. Let <span class="mathjax-tex">$J_{Z}^{*}$</span> denote the solution to the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>). Then, we find that </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Y_{n, s, g(\theta )}^{Z}\geq \operatorname{Tr} \bigl[{\hat{P}}\bigl(|\psi _{n, s, g(\theta )}\rangle \bigr)J_{Z}^{*} \bigr]:= Y_{n, s, g(\theta )}^{Z, { \mathrm{L}}}. $$</span></div><div class="c-article-equation__number"> (8) </div></div><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec7"><span class="c-article-section__title-number">2.3.2 </span>Upper bound on the phase-error rates <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> </h4><p>The phase-error rates, <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span>, are defined by means of a virtual protocol [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title=" Koashi M. Simple security proof of quantum key distribution based on complementarity. New J Phys. 2009;8:045018. https://doi.org/10.1088/1367-2630/11/4/045018 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR49" id="ref-link-section-d131444011e5659">49</a>]. For this, we shall consider the standard assumption in which the efficiency of Bob’s measurement is independent of his basis choice. Then, for those rounds in which both Alice and Bob select the <i>Z</i> basis and Alice generates the <i>n</i>-th eigenstate <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span>, we can equivalently describe her state preparation process as follows. First, she prepares the following bipartite entangled state </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\Psi ^{Z}_{n, s, g(\theta )}\bigr\rangle = \frac{1}{\sqrt{2}} \bigl(|0_{Z} \rangle _{A} \hat{V}_{0_{Z}}+|1_{Z}\rangle _{A} \hat{V}_{1_{Z}}\bigr)|\psi _{n, s, g(\theta )}\rangle , $$</span></div><div class="c-article-equation__number"> (9) </div></div><p> where <span class="mathjax-tex">$\hat{V}_{b_{\alpha}}$</span>, with <span class="mathjax-tex">$b=0,1$</span> and <span class="mathjax-tex">$\alpha \in \{Z,X\}$</span>, denotes the encoding operation corresponding to the <i>α</i> basis and the bit value <i>b</i>. Although our analysis is valid for any <span class="mathjax-tex">$\{\hat{V}_{b_{\alpha}}\}$</span>, for simplicity, in our simulations, we assume that these operators, are ideal BB84 encoding operators, given by <span class="mathjax-tex">$\hat{V}_{0_{Z}}|n\rangle =|n\rangle |0\rangle , \hat{V}_{1_{Z}}|n \rangle =|0\rangle |n\rangle $</span>, </p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{gathered} \hat{V}_{0_{X}}|n\rangle =\sum _{k} \frac{1}{\sqrt{2^{n}}} \sqrt{ \begin{pmatrix} n \\ k \end{pmatrix}}|k \rangle |n-k\rangle , \\ \hat{V}_{1_{X}}|n\rangle =\sum_{k}(-1)^{k} \frac{1}{\sqrt{2^{n}}} \sqrt{\begin{pmatrix} n \\ k \end{pmatrix}}|k\rangle |n-k\rangle . \end{gathered} $$</span></div><div class="c-article-equation__number"> (10) </div></div><p> We note that these operators are independent of the physical degree of freedom used for the encoding. For example, in a time-bin encoding setup, the first ket would represent the early time bin, and the second ket would represent the late time bin; while in a polarization-encoding setup, the first ket would represent the horizontally-polarized mode, and the second ket would represent the vertically-polarized mode.</p><p>Next, she measures her ancilla system <i>A</i> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ9">9</a>) in the orthonormal basis <span class="mathjax-tex">$\{|0_{Z}\rangle , |1_{Z}\rangle \}$</span> to learn the bit value encoded, and sends the other system to Bob, who measures it in the <i>Z</i> basis.</p><p>In this situation, the phase-error rate <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> corresponds to the bit error rate that Alice and Bob would observe if Alice (Bob) instead performed an <i>X</i> basis measurement on the ancilla system <i>A</i> (arriving signal). If Alice performs a <i>X</i> basis measurement on her system <i>A</i>, this is equivalent to emitting the states </p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\lambda ^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )}\bigr\rangle \propto \bigl|\bar{\lambda}^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )} \bigr\rangle ={ }_{A} \bigl\langle \Delta _{X} |\Psi ^{Z}_{n, s, g(\theta )} \bigr\rangle =\frac{1}{2} \bigl[\hat{V}_{0_{Z}}+(-1)^{\Delta } \hat{V}_{1_{Z}} \bigr]|\psi _{n, s, g(\theta )}\rangle , $$</span></div><div class="c-article-equation__number"> (11) </div></div><p> with probability <span class="mathjax-tex">$p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}=\||\bar{\lambda}^{ \mathrm{virtual}}_{\Delta , n, s, g(\theta )}\rangle \|^{2}$</span>, where <span class="mathjax-tex">$\Delta \in \{0,1\}$</span> and <span class="mathjax-tex">$|\Delta _{X}\rangle = [|0_{Z}\rangle +(-1)^{\Delta}|1_{Z} \rangle ] / \sqrt{2}$</span>. Let <span class="mathjax-tex">$Y_{\Delta , n, s, g(\theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}}$</span> denote the probability that Bob obtains the measurement outcome <span class="mathjax-tex">$(\Delta \oplus 1)_{X}$</span> when he performs an <i>X</i> basis measurement on the arriving signal conditioned on Alice emitting the state <span class="mathjax-tex">$|\lambda ^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )}\rangle $</span>. That is, this event corresponds to a phase error. Then, the phase error rate <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> can be written as </p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ e_{n, s, g(\theta )}=\frac{1}{Y_{n, s, g(\theta )}^{Z}}\sum _{\Delta =0}^{1} p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{\Delta , n, s, g( \theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}}. $$</span></div><div class="c-article-equation__number"> (12) </div></div><p>In Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec16">A</a>, it is shown that an upper bound on the quantity <span class="mathjax-tex">$p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{\Delta , n, s, g( \theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}}$</span> can be obtained by solving the following SDP: </p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \max_{L_{(\Delta \oplus 1)_{X}}} &\ \operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl| \bar{\lambda}^{\text{virtual}}_{\Delta , n, s, g(\theta )} \bigr\rangle \bigr) L_{( \Delta \oplus 1)_{X}} \bigr] \\ \text{subject to } &\ \operatorname{Tr} \bigl[\hat{V}_{b_{\alpha}} \rho ^{\mu}_{[g(\theta )]} \hat{V}_{b_{\alpha}}^{\dagger} L_{(\Delta \oplus 1)_{X}} \bigr]=Q_{\mu , g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}, \\ &\ \forall \mu \in \{s, \nu , \omega \}, \forall b\in \{0,1\}, \forall \alpha \in \{Z,X\} \\ &\ 0 \leq L_{(\Delta \oplus 1)_{X}} \leq \mathbb{I}, \end{aligned} $$</span></div><div class="c-article-equation__number"> (13) </div></div><p> where <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> is given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ4">4</a>), and <span class="mathjax-tex">$Q_{\mu , g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}$</span> denotes the probability that Bob observes the result <span class="mathjax-tex">$(\Delta \oplus 1)_{X}$</span> with his <i>X</i> basis measurement given that Alice chose the intensity setting <i>μ</i>, the basis <i>α</i>, the bit value <i>b</i>, and the phases <i>θ</i> follow the PDF <span class="mathjax-tex">$g(\theta )$</span>. We note that Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) includes constraints provided by basis mismatched events [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title=" Tamaki K, Curty M, Kato G, Lo HK, Azuma K. Loss-tolerant quantum cryptography with imperfect sources. Phys Rev A. 2014;90:052314. https://doi.org/10.1103/PhysRevA.90.052314 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR44" id="ref-link-section-d131444011e8256">44</a>] in which Alice prepares the signals in the <i>Z</i> basis and Bob measures them in the <i>X</i> basis, which may result in a tighter estimation. This is because, in general, <span class="mathjax-tex">$|\lambda _{\Delta , n, s, g(\theta )}^{\text{virtual}}\rangle \neq \hat{V}_{\Delta _{X}}|\psi _{n, s, g(\theta )}\rangle $</span>, and <span class="mathjax-tex">${\hat{P}}(|\lambda ^{\text{virtual}}_{\Delta , n,s, g(\theta )})$</span> may be better approximated by an operator-form linear combination of both <i>Z</i>-encoded and <i>X</i>-encoded states, rather than just the latter.</p><p>Importantly, the states <span class="mathjax-tex">$|\bar{\lambda}^{\text{virtual}}_{\Delta , n, s, g(\theta )}\rangle $</span> and <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span>, as well as the operators <span class="mathjax-tex">$\hat{V}_{b_{\alpha}}$</span>, are known and depend on Alice’s state preparation process. The gains <span class="mathjax-tex">$Q_{\mu , g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}$</span> are directly observed in a realization of the protocol. That is, the only unknown in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) is the positive semi-definite operator <i>L</i> over which the maximization takes place.</p><p>Let <span class="mathjax-tex">$L_{(\Delta \oplus 1)_{X}}^{*}$</span> denote the solution to the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>). Then, we have that </p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{\Delta , n, s, g( \theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}}\leq \operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\bar{\lambda}^{\text{virtual}}_{ \Delta , n, s, g(\theta )} \bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{*} \bigr]. $$</span></div><div class="c-article-equation__number"> (14) </div></div><p> That is, </p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ e_{n, s, g(\theta )}\leq \frac{1}{Y_{n, s, g(\theta )}^{Z, {\mathrm{L}}}}\sum _{\Delta =0}^{1} \operatorname{Tr} \bigl[{\hat{P}} \bigl(\bigl|\bar{\lambda}^{\text{virtual}}_{ \Delta , n, s, g(\theta )}\bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{*} \bigr]:=e_{n, s, g(\theta )}^{\mathrm{U}}. $$</span></div><div class="c-article-equation__number"> (15) </div></div><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec8"><span class="c-article-section__title-number">2.3.3 </span>Solving Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) numerically</h4><p>Solving numerically the SDPs presented above is difficult for two main reasons. Firstly, they are infinitely dimensional, because the states <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> are infinite-dimensional. Secondly, this also renders the calculation of the eigendecomposition of <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ4">4</a>) a difficult task. To overcome these two limitations, we follow a technique recently introduced in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title=" Shlok N. Decoy-state quantum key distribution with arbitrary phase mixtures and phase correlations. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR50" id="ref-link-section-d131444011e9287">50</a>] (see also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title=" Upadhyaya T, Himbeeck T, Lin J, Lütkenhaus N. Dimension reduction in quantum key distribution for continuous- and discrete-variable protocols. PRX Quantum. 2021;2:020325. https://doi.org/10.1103/PRXQuantum.2.020325 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR51" id="ref-link-section-d131444011e9290">51</a>]), which consists in projecting the states <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> onto a finite-dimensional subspace that contains up to <i>M</i> photons. We shall denote the projected states as </p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \rho ^{\mu}_{[g(\theta )], M}= \frac{\Pi _{M} \rho ^{\mu}_{[g(\theta )]} \Pi _{M}}{\operatorname{Tr} [\Pi _{M} \rho ^{\mu}_{[g(\theta )]} \Pi _{M} ]}, $$</span></div><div class="c-article-equation__number"> (16) </div></div><p> where <span class="mathjax-tex">$\Pi _{M}=\sum_{n=0}^{M}|{n}\rangle \langle{n}|$</span> denotes the projector onto the <i>M</i>-photon subspace, being <span class="mathjax-tex">$|{n}\rangle $</span> a Fock state. In doing so, now the eigendecomposition of <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )], M}$</span> can be easily obtained numerically. For later convenience, we will denote the eigendecomposition of the numerator of the right hand side of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ16">16</a>) as </p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \Pi _{M}\rho ^{\mu}_{[g(\theta )]}\Pi _{M}=\sum_{n=0}^{M} q_{n| \mu , g(\theta )}{\hat{P}}\bigl(|\varphi _{n, \mu , g(\theta )}\rangle \bigr). $$</span></div><div class="c-article-equation__number"> (17) </div></div><p>Importantly, this technique also allows to transform the infinite-dimensional SDPs given by Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) onto finite-dimensional SDPs that can be solved numerically. The resulting SDPs and their derivation are provided in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec17">B</a>.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec9"><span class="c-article-section__title-number">2.3.4 </span>Lower bound on the probabilities <span class="mathjax-tex">$p_{n| s, g(\theta )}$</span> </h4><p>As explained in the previous subsection, because the states <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> are infinite-dimensional, it might be difficult to calculate their eigendecomposition, and thus the probabilities <span class="mathjax-tex">$p_{n| s, g(\theta )}$</span>. Instead, here we provide a lower bound on these probabilities based on the eigendecomposition given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ17">17</a>). In particular, in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec17">B</a> it is shown that </p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ p_{n| s, g(\theta )}\geq q_{n| s, g(\theta )}-\epsilon _{s}:=p_{n| s, g( \theta )}^{\text{L}} $$</span></div><div class="c-article-equation__number"> (18) </div></div><p> with <span class="mathjax-tex">$\epsilon _{s}=2 \sqrt{1-\operatorname{Tr} [\Pi _{M} \rho ^{s}_{[g( \theta )]} \Pi _{M} ]}$</span>.</p></div></div></section><section data-title="Results"><div class="c-article-section" id="Sec10-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec10"><span class="c-article-section__title-number">3 </span>Results</h2><div class="c-article-section__content" id="Sec10-content"><p>In this section, we now evaluate the secret key rate obtainable for various examples of functions <span class="mathjax-tex">$g(\theta )$</span>. For illustration purposes, we consider three main scenarios, depending on whether or not the function <span class="mathjax-tex">$g(\theta )$</span> is fully characterized. Also, for the simulations, we consider a simple channel model whose transmission efficiency is given by <span class="mathjax-tex">$10^{-\frac{\gamma}{10}}$</span>, where <i>γ</i> (measured in dB) represents the overall system loss, <i>i.e.</i>, it also includes the effect of the finite detection efficiency of Bob’s detectors. Moreover, for simplicity, we disregard any misalignment effect, and assume that the only source of errors are the dark counts of Bob’s detectors, whose probability is set to <span class="mathjax-tex">$p_{d}=10^{-8}$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title=" Boaron A, Boso G, Rusca D, Vulliez C, Autebert C, Caloz M et al.. Secure quantum key distribution over 421 km of optical fiber. Phys Rev Lett. 2018;121:190502. https://doi.org/10.1103/PhysRevLett.121.190502 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR23" id="ref-link-section-d131444011e10300">23</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title=" Yin HL, Chen TY, Yu ZW, Liu H, You LX, Zhou YH et al.. Measurement-device-independent quantum key distribution over a 404 km optical fiber. Phys Rev Lett. 2016;117:190501. https://doi.org/10.1103/PhysRevLett.117.190501 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR52" id="ref-link-section-d131444011e10303">52</a>]. In addition, as already mentioned, we consider that the BB84 encoding operators are ideal even though the analysis presented here is applicable if this condition is not met, and we take an error correction efficiency <span class="mathjax-tex">$f=1.16$</span>.</p><p>To obtain the bounds <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z, \mathrm{L}}$</span> and <span class="mathjax-tex">$e_{n, s, g(\theta )}^{\mathrm{U}}$</span> we use the finite-dimensional versions of the SDPs above, which are presented in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec17">B</a>. Note that, the resulting secret key rate is an increasing function of <i>M</i>. However, the time required to numerically solve such SDPs grows rapidly with this parameter. For this reason, we have set a sufficiently large <i>M</i> so that an increase in this parameter would result in a negligible improvement of the secret key rate as tested numerically. The effect that the parameter <i>M</i> has in the secret key rate, is studied in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec21">D</a>.</p><h3 class="c-article__sub-heading" id="Sec11"><span class="c-article-section__title-number">3.1 </span>Fully-characterized <span class="mathjax-tex">$g(\theta )$</span> </h3><p>Here, we consider the scenario in which the function <span class="mathjax-tex">$g(\theta )$</span> is completely characterized, and we evaluate two specific examples of practical interest. The first example corresponds to the scenario given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), which has been considered in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e10506">42</a>], while the second example can be interpreted as a noisy version of the first one.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec12"><span class="c-article-section__title-number">3.1.1 </span>Ideal discrete phase randomization</h4><p>The results are shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig1">1</a> for different values of the total number of random phases <i>N</i> selected by Alice. In particular, the solid lines in the figure have been obtained using the parameter estimation procedure presented in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec5">2.3</a> based on SDP and the use of basis mismatched events. If we discard these latter events, the obtainable key rate decreases, as illustrated by the dashed-dot lines. Finally, the dotted lines correspond to the analysis in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e10525">42</a>]. For completeness, this latter approach is summarized in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec22">E</a>. In the first two cases, for simplicity, we set the intensity settings to the possibly sub-optimal values <span class="mathjax-tex">$\omega =0$</span>, <span class="mathjax-tex">$\nu =s/5$</span> and we optimize <i>s</i> as a function of the overall system loss <i>γ</i>, while in the later case we set <span class="mathjax-tex">$\omega =0$</span> and optimize both <i>ν</i> and <i>s</i> as a function of <i>γ</i> (which provides the optimal solution for this approach). Importantly, despite this fact, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig1">1</a> shows that the use of SDP and basis mismatched events significantly improve the secret key rate when compared to the results in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e10625">42</a>]. Furthermore, we find that the improvement of using basis mismatched events is more advantageous when <i>N</i> is small. Indeed, when <span class="mathjax-tex">$N\geq 5$</span>, this enhancement in performance is almost negligible. This is expected as basis mismatched events do not improve the estimation in the case of ideal continuous phase randomization, <i>i.e.</i>, in the limit <span class="mathjax-tex">$N\rightarrow \infty $</span>. On the other hand, when <i>N</i> is small, the eigenstates <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span> for <span class="mathjax-tex">$n=0,1$</span> deviate more from a perfect Fock state, meaning that the virtual states <span class="mathjax-tex">$|\lambda ^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )}\rangle $</span> deviate more from the <i>X</i>-encoded states <span class="mathjax-tex">$\hat{V}_{\Delta _{X}}|\psi _{n, s, g(\theta )}\rangle $</span> and thus basis mismatched events provide a tighter estimation. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Figure 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig1_HTML.jpg?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig1_HTML.jpg" alt="figure 1" loading="lazy" width="685" height="481"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Secret key rate in logarithmic scale versus the overall system loss for the ideal discrete phase-randomization scenario given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), as a function of the total number of random phases <i>N</i> selected by Alice. The solid lines correspond to the parameter estimation procedure based on SDP and basis mismatched events considered in this work, while the dashed-dotted lines represent the same procedure overlooking basis mismatched events. Finally, the dotted lines correspond to the analysis in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e10904">42</a>] using linear programming</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Note that, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig2">2</a>, when <span class="mathjax-tex">$N\geq 6$</span>, the improvement in the secret key rate that can be obtained by further increasing the value of <i>N</i> decelerates. Hence, it seems that a value of around <span class="mathjax-tex">$N=8$</span> might be a good practical compromise, as this configuration requires only three random bits per pulse to select the random phase. As in the previous figure, here we set the intensities to <span class="mathjax-tex">$\{s, s/5, 0\}$</span> and optimize <i>s</i> as a function of the overall system loss to simplify the numerics. This is done for both the ideal PR-WCP scenario and for the different values of <i>N</i> to ensure a fear comparison between both scenarios. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Figure 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig2_HTML.jpg?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig2_HTML.jpg" alt="figure 2" loading="lazy" width="685" height="482"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Secret key rate in logarithmic scale versus the overall system loss for the ideal discrete phase-randomization scenario given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), as a function of the total number of random phases <i>N</i> selected by Alice, when Alice and Bob employ the parameter estimation procedure based on SDP and basis mismatched events considered in this work. Remarkably, as shown in the figure, only eight random phases are enough to deliver a secret key rate already quite close to the ideal scenario of perfect PR-WCPs, where the phase of each pulse is uniformly random in <span class="mathjax-tex">$[0,2\pi )$</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec13"><span class="c-article-section__title-number">3.1.2 </span>Noisy discrete phase randomization</h4><p>Here we consider the situation in which the actual phase encoded by Alice in each emitted pulse follows a certain PDF around the selected discrete value <span class="mathjax-tex">$\theta _{k}=2 \pi k/N$</span>. This might happen due to device imperfections of the phase modulator or the electronics that control it. For concreteness and illustration purposes, we shall assume that this PDF is a truncated Gaussian distribution, though we remark that our analysis can be applied to any given distribution. A truncated Gaussian distribution has the form </p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ f (\theta ; \theta _{k}, \sigma _{k}, \lambda _{k}, \Lambda _{k} )= \frac{\phi (\theta ; \theta _{k}, \sigma _{k}^{2} )}{\Phi (\Lambda _{k}; \theta _{k}, \sigma _{k}^{2} )-\Phi (\lambda _{k}; \theta _{k}, \sigma _{k}^{2} )}, $$</span></div><div class="c-article-equation__number"> (19) </div></div><p> when the phase <i>θ</i> is in the interval <span class="mathjax-tex">$\lambda _{k}<\theta <\Lambda _{k}$</span>, and zero otherwise. The functions <span class="mathjax-tex">$\phi (x; \gamma , \sigma ^{2} )$</span> and <span class="mathjax-tex">$\Phi (x; \gamma , \sigma ^{2} )$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ19">19</a>) are, respectively, given by </p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &\phi (x; y, z )=\frac{1}{\sqrt{2 \pi z}} e^{- \frac{(x-y)^{2}}{2 z}}, \\ &\Phi (x; y, z )= \int _{-\infty}^{x} \frac{1}{\sqrt{2 \pi z}} e^{-\frac{(t-y)^{2}}{2 z}} \,dt . \end{aligned} $$</span></div><div class="c-article-equation__number"> (20) </div></div><p> That is, in this scenario the function <span class="mathjax-tex">$g(\theta )$</span> has the following form </p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g(\theta )= \frac{1}{N}\sum _{k=0}^{N-1} f (\theta ; \theta _{k}, \sigma _{k}, \lambda _{k}, \Lambda _{k} ) $$</span></div><div class="c-article-equation__number"> (21) </div></div><p> for certain parameters <span class="mathjax-tex">$\theta _{k}$</span>, <span class="mathjax-tex">$\sigma _{k}$</span>, <span class="mathjax-tex">$\lambda _{k}$</span> and <span class="mathjax-tex">$\Lambda _{k}$</span>.</p><p>In the limit when the standard deviations <span class="mathjax-tex">$\sigma _{k} \to 0$</span> <span class="stix">∀</span><i>k</i>, Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ21">21</a>) converges to the PDF given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), because in that regime each truncated Gaussian distribution approaches the Dirac delta function. On the other hand, when <span class="mathjax-tex">$\sigma _{k} \to \infty $</span>, and given that the concatenation of the truncation intervals defined by <span class="mathjax-tex">$\lambda _{k}$</span> and <span class="mathjax-tex">$\Lambda _{k}$</span> allow the phase to take any value within the range of <span class="mathjax-tex">$[0,2\pi )$</span> but do not overlap each other, Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ21">21</a>) converges to the PDF of a uniform distribution in <span class="mathjax-tex">$[0,2\pi )$</span>. Importantly, this means that the achievable secret key rate will increase with higher values of <span class="mathjax-tex">$\sigma _{k}$</span>, or, to put it in other words, when the uncertainty about the phase actually imprinted by Alice on each of her prepared signals increases, given that <span class="mathjax-tex">$g(\theta )$</span> is completely characterized.</p><p>The simulation results are shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig3">3</a>, which presents a comparison between the achievable secret key rate for two different values of the standard deviations <span class="mathjax-tex">$\sigma _{k}$</span>, which, for simplicity, are assumed to be equal for all <i>k</i>. As expected, the larger the value of <span class="mathjax-tex">$\sigma _{k}$</span> is, the higher the resulting secret key rate, regardless of the number <i>N</i> of random phases selected by Alice, though the improvement is more relevant when <i>N</i> is small. For simplicity and due to the lack of experimental data, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig3">3</a> assumes that <span class="mathjax-tex">$\lambda _{k}=\theta _{k}-3\sigma _{k}$</span> and <span class="mathjax-tex">$\Lambda _{k}=\theta _{k}+3\sigma _{k}$</span>. Moreover, like in the previous example, we set <span class="mathjax-tex">$\omega =0$</span>, <span class="mathjax-tex">$\nu =s/5$</span> and we optimize <i>s</i> as a function of the overall system loss. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Figure 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig3_HTML.jpg?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig3_HTML.jpg" alt="figure 3" loading="lazy" width="685" height="479"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Secret key rate in logarithmic scale versus the overall system loss when <span class="mathjax-tex">$g(\theta )$</span> follows the PDF given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ21">21</a>), as a function of the total number of random phases <i>N</i> selected by Alice, and for two different values of the standard deviations <span class="mathjax-tex">$\sigma _{k}$</span>, which are assumed to be equal for all <i>k</i></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec14"><span class="c-article-section__title-number">3.2 </span>Partially-characterized <span class="mathjax-tex">$g(\theta )$</span> </h3><p>Here, we now consider the scenario in which only partial information about the function <span class="mathjax-tex">$g(\theta )$</span> is known. In particular, and for illustration purposes, we shall assume that the actual phase encoded by Alice in each emitted pulse could be any phase within a certain interval around the selected discrete value <span class="mathjax-tex">$\theta _{k}=2 \pi k/N$</span>, but its precise PDF <span class="mathjax-tex">$g(\theta )$</span> is unknown. Precisely, let <span class="mathjax-tex">$\delta _{\text{max}}$</span> denote the maximum possible deviation between the actual selected phase <span class="mathjax-tex">$\theta _{k}$</span> and the actual imprinted phase, which we shall denote by <span class="mathjax-tex">$\hat{\theta}_{k}$</span>. That is, we assume that the actual imprinted phase lies in the interval <span class="mathjax-tex">$\hat{\theta}_{k}\in [\theta _{k}-\delta _{\text{max}},\theta _{k}+ \delta _{\text{max}}]$</span>, and we conservatively take the combination of values <span class="mathjax-tex">$\hat{\theta}_{k}$</span> for all <i>k</i> that minimizes the secret key rate following the analysis presented in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec20">C</a>.</p><p>The results are illustrated in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig4">4</a>, as a function of the total number of phases <i>N</i> selected by Alice and the value of the maximum deviation <span class="mathjax-tex">$\delta _{\text{max}}$</span>. Like in the previous examples, for simplicity, we fix <span class="mathjax-tex">$\omega =0$</span>, <span class="mathjax-tex">$\nu =s/5$</span> and we optimize <i>s</i> as a function of the overall system loss. As expected, the larger the value of <span class="mathjax-tex">$\delta _{\text{max}}$</span> is, the lower the resulting secret key rate. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Figure 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig4_HTML.jpg?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig4_HTML.jpg" alt="figure 4" loading="lazy" width="685" height="480"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Secret key rate in logarithmic scale versus the overall system loss when the phases lie in the intervals <span class="mathjax-tex">$\theta _{k}\pm \delta _{\text{max}}$</span> and the function <span class="mathjax-tex">$g(\theta )$</span> is unknown, as a function of the total number of random phases <i>N</i> selected by Alice and the value of <span class="mathjax-tex">$\delta _{\text{max}}$</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Also, from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig4">4</a> we see that for higher values of <span class="mathjax-tex">$\delta _{\text{max}}$</span>, the secret key rate becomes less sensitive to the parameter <i>N</i>. Indeed, when <span class="mathjax-tex">$\delta _{\text{max}}=10^{-1}$</span>, the achievable secret key rate for the cases <span class="mathjax-tex">$N=3, 4, 5$</span> essentially overlap each other, which is the left-most curve. This seems to be due to the fact that a significant increase in <span class="mathjax-tex">$\delta _{\text{max}}$</span> allows in principle for some phases to lie close to each other, or even become identical if this parameter is large enough. Under this situation, the increase of <i>N</i> does not help to improve the performance, as the effective randomness remains almost the same.</p></div></div></section><section data-title="Conclusion"><div class="c-article-section" id="Sec15-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec15"><span class="c-article-section__title-number">4 </span>Conclusion</h2><div class="c-article-section__content" id="Sec15-content"><p>In this paper we have considered the security of decoy-state quantum key distribution (QKD) when the phase of each generated signal is not uniformly random, as requested by the theory, but follows an arbitrary, continuous or discrete, probability density function (PDF). This might happen due to the presence of device imperfections in the phase-randomization process, and/or due to the use of an external phase modulator to imprint the random phases on the generated pulses, which limits the possible selected phases to a finite set.</p><p>Our analysis combines a novel parameter estimation technique, based on semi-definite programming, with the use of basis mismatched events, to tightly estimate the relevant parameters that are needed to evaluate the achievable secret key rate. In doing so, we have shown that decoy-state QKD is rather robust to faulty phase-randomization, particularly when the PDF that governs the random phases is well-characterized. Moreover, our results significantly outperform those of previous works while being also more general, in the sense that they can handle more realistic and practical scenarios.</p><p>This work might be relevant as well to other quantum communication protocols beyond QKD that use laser sources and decoy states.</p></div></div></section> <section data-title="Data availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>Not applicable.</p> </div></div></section><section data-title="Code availability"><div class="c-article-section" id="code-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="code-availability">Code availability</h2><div class="c-article-section__content" id="code-availability-content"> <p>The code is available upon request from the authors.</p> </div></div></section><section data-title="Notes"><div class="c-article-section" id="notes-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="notes">Notes</h2><div class="c-article-section__content" id="notes-content"><ol class="c-article-footnote c-article-footnote--listed"><li class="c-article-footnote--listed__item" id="Fn1" data-counter="1."><div class="c-article-footnote--listed__content"><p>This is because the analysis in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e702">41</a>] requires that there is a known non-zero parameter <i>q</i> such that <span class="mathjax-tex">$g(\theta ) \geq q$</span> for all <span class="mathjax-tex">$\theta \in [0,2\pi )$</span>. In the case of active phase randomization, only a discrete number of phases is selected, and therefore there might be many values <span class="mathjax-tex">$\theta \in [0,2\pi )$</span> such that <span class="mathjax-tex">$g(\theta ) =0$</span>.</p></div></li><li class="c-article-footnote--listed__item" id="Fn2" data-counter="2."><div class="c-article-footnote--listed__content"><p>From this point on, if we have two operators, say <i>A</i> and <i>B</i> by <span class="mathjax-tex">$A\leq B$</span> we mean that <span class="mathjax-tex">$B-A\geq 0$</span>, <i>i.e.</i> that <span class="mathjax-tex">$B-A$</span> is a positive semi-definite operator.</p></div></li></ol></div></div></section><section data-title="Abbreviations"><div class="c-article-section" id="abbreviations-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="abbreviations">Abbreviations</h2><div class="c-article-section__content" id="abbreviations-content"><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>QKD:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>quantum key distribution</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>GLLP:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Gottesman, Lo, Lütkenhaus and Preskill</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>SDP:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Semidefinite programming</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>LP:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>linear programming</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>PR-WCP:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>phase-randomized weak coherent pulse</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn>PDF:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>probability density function</p> </dd></dl></div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1"> Xu F, Ma X, Zhang Q, Lo HK, Pan JW. 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NY J Math. 2017;23:1369–93. </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3723514" aria-label="MathSciNet reference 56">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 56" href="http://scholar.google.com/scholar_lookup?&title=Contractive%20channels%20on%20operator%20algebras&journal=NY%20J%20Math&volume=23&pages=1369-1393&publication_year=2017&author=Farenick%2CD&author=Bures%2CRM"> Google Scholar</a>  </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1140/epjqt/s40507-023-00210-0?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>This work was supported by Cisco Systems Inc., the Galician Regional Government (consolidation of Research Units: AtlantTIC), the Spanish Ministry of Economy and Competitiveness (MINECO), the Fondo Europeo de Desarrollo Regional (FEDER) through the grant No. PID2020-118178RB-C21, MICIN with funding from the European Union NextGenerationEU (PRTR-C17.I1) and the Galician Regional Government with own funding through the “Planes Complementarios de I + D + I con las Comunidades Autónomas” in Quantum Communication, the European Union’s Horizon Europe Framework Programme under the Marie Skłodowska-Curie Grant No. 101072637 (Project QSI) and the project “Quantum Security Networks Partnership” (QSNP, grant agreement No 101114043). X.S. acknowledges support from an FPI predoctoral scholarship granted by the Spanish Ministry of Science and Innovation. G.C.-L. acknowledges support from JSPS Postdoctoral Fellowships for Research in Japan. K.T. acknowledges support from JSPS KAKENHI Grant Number JP18H05237.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Vigo Quantum Communication Center, University of Vigo, Vigo, E-36310, Spain</p><p class="c-article-author-affiliation__authors-list">Xoel Sixto, Guillermo Currás-Lorenzo & Marcos Curty</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Escuela de Ingeniería de Telecomunicación, Department of Signal Theory and Communications, University of Vigo, Vigo, E-36310, Spain</p><p class="c-article-author-affiliation__authors-list">Xoel Sixto, Guillermo Currás-Lorenzo & Marcos Curty</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">atlanTTic Research Center, University of Vigo, Vigo, E-36310, Spain</p><p class="c-article-author-affiliation__authors-list">Xoel Sixto, Guillermo Currás-Lorenzo & Marcos Curty</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Faculty of Engineering, University of Toyama, Gofuku 3190, Toyama, 930-8555, Japan</p><p class="c-article-author-affiliation__authors-list">Guillermo Currás-Lorenzo & Kiyoshi Tamaki</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Xoel-Sixto-Aff1-Aff2-Aff3"><span class="c-article-authors-search__title u-h3 js-search-name">Xoel Sixto</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="https://www.biomedcentral.com/search?query=author%23Xoel%20Sixto" class="c-article-button" data-track="click" data-track-action="author link - 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X.S. performed the calculations and the numerical simulations with inputs from G.C.-L. X.S. wrote the manuscript, and all authors contributed towards improving it and checking the validity of the results.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:xsixto@vqcc.uvigo.es">Xoel Sixto</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar1">Ethics approval and consent to participate</h3> <p>Not applicable.</p> <h3 class="c-article__sub-heading" id="FPar2">Consent for publication</h3> <p>All authors consent to the publication of this manuscript.</p> <h3 class="c-article__sub-heading" id="FPar3">Competing interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><h3 class="c-article__sub-heading">Publisher’s Note</h3><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendices</h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading" id="App1">Appendix A: Derivation of the SDPs given by Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>)</h3><p>In this Appendix, we follow a similar approach to the one in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e13225">41</a>] to derive the infinite-dimensional SDPs presented in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) of the main text, under the assumption of collective attacks. We recall that these infinite-dimensional SDP’s cannot be solved numerically and a further dimension-reduction step is needed (see Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec17">B</a>).</p><p>Let Ω denote a quantum channel (or the action of Eve) that acts independently on each optical pulse emitted by Alice. Also, let us assume that in a certain round, Bob measures the incoming signal with a positive operator valued measure (POVM) that contains the element Π. In this scenario, the probability that Bob obtains the outcome associated with the element Π given that Alice sends him a quantum state <i>σ</i> can be expressed as </p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \operatorname{Tr}\bigl[\Omega (\sigma ) \Pi \bigr]= \operatorname{Tr} \biggl( \sum_{k} A_{k} \sigma A_{k}^{\dagger} \Pi \biggr)= \operatorname{Tr} \biggl(\sigma \sum_{k} A_{k}^{\dagger} \Pi A_{k} \biggr) =\operatorname{Tr}(\sigma H), $$</span></div><div class="c-article-equation__number"> (A.1) </div></div><p> where <span class="mathjax-tex">$\Omega (\sigma )$</span> represents the action of Ω on <i>σ</i>, <span class="mathjax-tex">$\{A_{k} \}$</span> denotes the set of Kraus operators corresponding to the operator-sum representation of the channel Ω, and </p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ 0 \leq H=\sum_{k} A_{k}^{\dagger} \Pi A_{k} \leq \sum_{k} A_{k}^{ \dagger} A_{k}=\mathbb{I}. $$</span></div><div class="c-article-equation__number"> (A.2) </div></div><p>Bob measures the incoming signals in either the <i>Z</i> or the <i>X</i> basis. Let us denote the POVM elements corresponding to each of these two measurements by <span class="mathjax-tex">$\{\Pi _{0_{Z}}, \Pi _{1_{Z}}, \Pi _{f} \}$</span> and <span class="mathjax-tex">$\{\Pi _{0_{X}}, \Pi _{1_{X}}, \Pi _{f} \}$</span>, respectively. That is, <span class="mathjax-tex">$\Pi _{b_{\alpha}}$</span> represents the POVM element associated to the outcome <i>b</i> in the basis <i>α</i>, with <span class="mathjax-tex">$\alpha \in \{Z,X\}$</span>, and <span class="mathjax-tex">$\Pi _{f}$</span> represents the POVM element associated to an inconclusive outcome. Note that here we are implicitly considering the basis-independent detection efficiency assumption, which means that the POVM element <span class="mathjax-tex">$\Pi _{f}$</span> is equal for both basis. Let <span class="mathjax-tex">$\Pi _{d}=\mathbb{I}-\Pi _{f}=\Pi _{0_{Z}}+\Pi _{1_{Z}}=\Pi _{0_{X}}+ \Pi _{1_{X}}$</span> denote the operator associated to a conclusive outcome at Bob’s side. Then, after substituting in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ22">A.1</a>) the state <i>σ</i> with Alice’s emitted state when she chooses the <i>Z</i> basis, </p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \rho ^{\mu , Z}_{[g(\theta )]}=\frac{1}{2} \hat{V}_{0_{Z}} \rho ^{\mu}_{[g( \theta )]} \hat{V}_{0_{Z}}^{\dagger}+\frac{1}{2} \hat{V}_{1_{Z}} \rho ^{\mu}_{[g(\theta )]} \hat{V}_{1_{Z}}^{\dagger}, $$</span></div><div class="c-article-equation__number"> (A.3) </div></div><p> and the operator Π with <span class="mathjax-tex">$\Pi _{d}$</span>, we obtain </p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Q_{\mu , g(\theta )}^{Z}=\operatorname{Tr}\bigl[\Omega \bigl(\rho ^{\mu , Z}_{[g( \theta )]}\bigr) \Pi _{d}\bigr]= \operatorname{Tr}\bigl[\rho ^{\mu , Z}_{[g(\theta )]} H\bigr]= \operatorname{Tr}\bigl[\rho ^{\mu}_{[g(\theta )]} J_{Z}\bigr], $$</span></div><div class="c-article-equation__number"> (A.4) </div></div><p> with <span class="mathjax-tex">$H=\sum_{k} A_{k}^{\dagger} \Pi _{d} A_{k}$</span>, and the operator <span class="mathjax-tex">$J_{Z}$</span> satisfying </p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ 0 \leq J_{Z}=\frac{1}{2} \bigl(\hat{V}_{0_{Z}}^{\dagger} H \hat{V}_{0_{Z}}+ \hat{V}_{1_{Z}}^{\dagger} H \hat{V}_{1_{Z}} \bigr) \leq \mathbb{I} . $$</span></div><div class="c-article-equation__number"> (A.5) </div></div><p>Finally, by taking into account that the yield associated to the states <span class="mathjax-tex">$|\psi _{n, s, g(\theta )}\rangle $</span> encoded in the Z basis is given by </p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Y_{n, s, g(\theta )}^{Z}=\operatorname{Tr}\bigl\{ \Omega \bigl[{ \hat{P}}\bigl(\bigl|\psi ^{Z}_{n, s, g(\theta )}\bigr\rangle \bigr)\bigr]\Pi _{d}\bigr\} =\operatorname{Tr}\bigl[{\hat{P}}(|\psi _{n, s, g(\theta )}\rangle ) J_{Z}\bigr], $$</span></div><div class="c-article-equation__number"> (A.6) </div></div><p> with </p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\hat{P}}\bigl(\bigl|\psi ^{Z}_{n, s, g(\theta )}\bigr\rangle \bigr)= \frac{1}{2} \hat{V}_{0_{Z}} {\hat{P}}\bigl(|\psi _{n, s, g(\theta )} \rangle \bigr) \hat{V}_{0_{Z}}^{\dagger} +\frac{1}{2} \hat{V}_{1_{Z}} {\hat{P}}\bigl(|\psi _{n, s, g(\theta )} \rangle \bigr) \hat{V}_{1_{Z}}^{\dagger}, $$</span></div><div class="c-article-equation__number"> (A.7) </div></div><p> we obtain the SDP presented in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>).</p><p>Regarding the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) to estimate the phase error rate, we note that the numerator of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ12">12</a>), can be expressed as </p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{\Delta , n, s, g( \theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}} \\ &\quad =p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )} \operatorname{Tr} \bigl\{ \Omega \bigl[{ \hat{P}}\bigl(\bigl|\lambda ^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )}\bigr\rangle \bigr)\bigr] \Pi _{(\Delta \oplus 1)_{X}} \bigr\} \\ &\quad =\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\bar{\lambda}^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )} \bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}} \bigr], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (A.8) </div></div><p> where <span class="mathjax-tex">$0 \leq L_{(\Delta \oplus 1)_{X}}=\sum_{k} A_{k}^{\dagger} \Pi _{( \Delta \oplus 1)_{X}} A_{k} \leq \mathbb{I}$</span> according to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ22">A.1</a>), and <span class="mathjax-tex">$|\bar{\lambda}^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )} \rangle =\sqrt{p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}}| \lambda ^{\mathrm{virtual}}_{ \Delta , n, s, g(\theta )}\rangle $</span>.</p><p>By using again Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ22">A.1</a>), we have that the gains <span class="mathjax-tex">$Q_{\mu , g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}$</span> can be expressed as </p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Q_{\mu , g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}= \operatorname{Tr} \bigl[ \hat{V}_{b_{\alpha}} \rho _{[g(\theta )]}^{ \mu} \hat{V}_{b_{\alpha}}^{\dagger} L_{(\Delta \oplus 1)_{X}} \bigr]. $$</span></div><div class="c-article-equation__number"> (A.9) </div></div><p>Putting it all together, we find that the SDP presented in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) of the main text, provides an upper bound on <span class="mathjax-tex">$p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{\Delta , n, s, g( \theta )}^{(\Delta \oplus 1)_{X},\mathrm{virtual}}$</span>.</p><h3 class="c-article__sub-heading" id="App1">Appendix B: Finite-dimensional SDPs when <span class="mathjax-tex">$g(\theta )$</span> is fully characterized</h3><h3 class="c-article__sub-heading" id="Sec18"><span class="c-article-section__title-number">2.1 </span>B.1 Lower bound on the yields <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> </h3><p>In this Appendix, we show how to obtain a finite-dimensional relaxation of the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>) to find a lower bound on the yields <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span>. For this, we follow again the approach presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e16280">41</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title=" Shlok N. Decoy-state quantum key distribution with arbitrary phase mixtures and phase correlations. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR50" id="ref-link-section-d131444011e16283">50</a>]. The key idea is rather simple: instead of considering the infinite-dimensional state <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ4">4</a>), we employ a projection <span class="mathjax-tex">$\rho _{[g(\theta )], M}^{\mu}$</span> of this state onto a finite-dimensional subspace with up to <i>M</i> photons (see Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ16">16</a>)), and then we relax the original constraints of the SDP accordingly.</p><p>We begin by briefly introducing some helpful results for this purpose. The first one is a direct consequence of the Cauchy-Schwarz inequality in Hilbert spaces [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title=" Lo HK, Preskill J. Security of quantum key distribution using weak coherent states with nonrandom phases. Quantum Inf Comput. 2007;8:431–58. https://doi.org/10.26421/QIC7.5-6-2 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR53" id="ref-link-section-d131444011e16387">53</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title=" Pereira M, Kate G, Mizutani A, Curty M, Tamaki K. Quantum key distribution with correlated sources. Sci Adv. 2020;6:eaaz4487. https://doi.org/10.1126/sciadv.aaz4487 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR54" id="ref-link-section-d131444011e16390">54</a>], which allows to relate the quantities <span class="mathjax-tex">$\operatorname{Tr}[\sigma H]$</span> and <span class="mathjax-tex">$\operatorname{Tr}[\rho H]$</span>, with <span class="mathjax-tex">$0 \leq H \leq \mathbb{I}$</span>, as a function of the fidelity between the states <i>σ</i> and <i>ρ</i>, </p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F(\rho ,\sigma )=\operatorname{Tr} [\sqrt{\sqrt{\sigma} \rho \sqrt{\sigma}} ]^{2}. $$</span></div><div class="c-article-equation__number"> (B.1) </div></div><p> In particular, it states that </p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ G_{-} \bigl( \operatorname{Tr} [\rho H ], F (\sigma , \rho ) \bigr) \leq \operatorname{Tr}[ \sigma H] \leq G_{+} \bigl(\operatorname{Tr} [\rho H ], F ( \sigma , \rho ) \bigr), $$</span></div><div class="c-article-equation__number"> (B.2) </div></div><p> with the functions <span class="mathjax-tex">$G_{\pm}(y, z)$</span> being defined as </p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ G_{-}(y, z)= \textstyle\begin{cases} g_{-}(y, z) & \text{if } y>1-z, \\ 0 & \text{otherwise},\end{cases} $$</span></div><div class="c-article-equation__number"> (B.3) </div></div><p> and </p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ G_{+}(y, z)= \textstyle\begin{cases} g_{+}(y, z) & \text{if } y< z, \\ 1 & \text{otherwise},\end{cases} $$</span></div><div class="c-article-equation__number"> (B.4) </div></div><p> with <span class="mathjax-tex">$g_{\pm}(y, z)=y+(1-z)(1-2 y) \pm 2 \sqrt{z(1-z) y(1-y)}$</span>.</p><p>The remaining results we use, <i>i.e.</i> Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ35">B.5</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ36">B.6</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ37">B.7</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ38">B.8</a>) below, have been derived in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title=" Currás-Lorenzo G, Tamaki K, Curty M. Security of decoy-state quantum key distribution with imperfect phase randomization. Preprint. 2022. arXiv:2210.08183 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR41" id="ref-link-section-d131444011e17078">41</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title=" Shlok N. Decoy-state quantum key distribution with arbitrary phase mixtures and phase correlations. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR50" id="ref-link-section-d131444011e17081">50</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title=" Winter A. Coding theorem and strong converse for quantum channels. IEEE Trans Inf Theory. 1999;45(7):2481–5. https://doi.org/10.1109/18.796385 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR55" id="ref-link-section-d131444011e17084">55</a>]. In particular, we have that </p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} F \bigl(\rho _{[g(\theta )]}^{\mu}, \rho _{[g(\theta )], M}^{\mu} \bigr)&=\operatorname{Tr} \bigl[\Pi _{M} \rho _{[g(\theta )]}^{\mu } \Pi _{M} \bigr] \\ &=\sum_{n=0}^{M} q_{n|\mu ,g(\theta )}:=F^{ \text{proj}}_{\mu ,g(\theta )}, \end{aligned} $$</span></div><div class="c-article-equation__number"> (B.5) </div></div><p> where the coefficients <span class="mathjax-tex">$q_{n|s,g(\theta )}$</span> are given in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ17">17</a>). Also, we have that the quantities <span class="mathjax-tex">$\vert p_{n \mid \mu , g(\theta )}-q_{n \mid \mu ,g(\theta )} \vert $</span> can be upper bounded as </p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \vert p_{n \mid \mu , g(\theta )}-q_{n \mid \mu ,g(\theta )} \vert \leq & 2 \sqrt{1-\operatorname{Tr} \bigl[\Pi _{M} \rho _{[g(\theta )]}^{ \mu }\Pi _{M} \bigr]} \\ =&2 \sqrt{1-F^{\text{proj}}_{\mu ,g(\theta )}}=: \epsilon _{\mu}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (B.6) </div></div><p> Finally, the fidelity <span class="mathjax-tex">$F ({\hat{P}}(|\varphi _{n,\mu ,g(\theta )}\rangle ),{\hat{P}}(| \psi _{n,\mu ,g(\theta )}\rangle ) )=|\langle \varphi _{n,\mu ,g( \theta )}|\psi _{n,\mu ,g(\theta )}\rangle |^{2}$</span> satisfies </p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F \bigl({\hat{P}}\bigl(|\varphi _{n,\mu ,g(\theta )}\rangle \bigr),{\hat{P}}\bigl(| \psi _{n,\mu ,g(\theta )}\rangle \bigr) \bigr) \geq 1- \biggl( \frac{\epsilon _{\mu}}{\delta _{n,\mu}} \biggr)^{2}:=F_{n,\mu ,g(\theta )}^{\text{vec}}, $$</span></div><div class="c-article-equation__number"> (B.7) </div></div><p> with </p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \delta _{0,\mu}=q_{0\mid \mu , g(\theta )}-q_{1\mid \mu , g(\theta )}- \epsilon _{\mu} \\& \begin{aligned}[b] \delta _{n,\mu}&=\min \{q_{n-1 \mid \mu , g(\theta )}-q_{n \mid \mu , g(\theta )}- \epsilon _{\mu}, q_{n \mid \mu , g(\theta )}-q_{n+1 \mid \mu , g(\theta )}-\epsilon _{\mu} \}. \end{aligned} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (B.8) </div></div><p>Then, from Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ8">8</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ32">B.2</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ37">B.7</a>) we have that </p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Y_{n, s, g(\theta )}^{Z, {\mathrm{L}}}=\operatorname{Tr} \bigl[{\hat{P}}(| \psi _{n, s, g(\theta )}\rangle )J_{Z}^{*} \bigr] \geq G_{-} \bigl(\operatorname{Tr} \bigl[{\hat{P}}(|\varphi _{n, s, g( \theta )}\rangle )J_{Z}^{*} \bigr], F_{n,s,g(\theta )}^{\text{vec}} \bigr), $$</span></div><div class="c-article-equation__number"> (B.9) </div></div><p> where <span class="mathjax-tex">$J_{Z}^{*}$</span> is the solution to the SDP presented in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>), and we have used the fact that <span class="mathjax-tex">$G_{-}$</span> is increasing with respect to its second argument. Since <span class="mathjax-tex">$G_{-}(y,z)$</span> is decreasing with respect to its first argument, one can lower bound Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ39">B.9</a>) by finding a lower bound on its first argument.</p><p>From Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ32">B.2</a>), we have that </p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ G_{-} \bigl(Q_{\mu , g(\theta )}^{Z}, F^{\text{proj}}_{\mu ,g(\theta )} \bigr) \leq \operatorname{Tr} \bigl[\rho _{[g(\theta )],M}^{\mu} J_{Z} \bigr] \leq G_{+} \bigl(Q_{\mu , g(\theta )}^{Z}, F^{\text{proj}}_{\mu ,g( \theta )} \bigr), $$</span></div><div class="c-article-equation__number"> (B.10) </div></div><p> with the operator <span class="mathjax-tex">$J_{Z}$</span> defined in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>). Here, since the states <span class="mathjax-tex">$\rho _{[g(\theta )],M}^{\mu}$</span> are finite dimensional, the calculation of <span class="mathjax-tex">$\operatorname{Tr} [\rho _{[g(\theta )],M}^{\mu} J_{Z} ]$</span> can be restricted to operators <span class="mathjax-tex">$J_{Z}$</span> that act on their finite subspace. Putting it all together, we find that a lower bound on <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> can be obtained by solving the following finite-dimensional SDP program </p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \min_{J_{Z}} &\ \operatorname{Tr} \bigl[ {\hat{P}}(|\varphi _{n, s, g( \theta )}\rangle ) J_{Z} \bigr] \\ \text{subject to } &\ G_{-} \bigl(Q_{\mu , g(\theta )}^{Z}, F^{ \text{proj}}_{\mu ,g(\theta )} \bigr) \leq \operatorname{Tr} \bigl[ \rho _{[g(\theta )],M}^{\mu} J_{Z} \bigr] \\ &\hphantom{\ G_{-} \bigl(Q_{\mu , g(\theta )}^{Z}, F^{ \text{proj}}_{\mu ,g(\theta )} \bigr)}\leq G_{+} \bigl(Q_{\mu , g(\theta )}^{Z}, F^{\text{proj}}_{\mu ,g( \theta )} \bigr), \quad \forall \mu \in \{s, \nu , \omega \} \\ &\ 0 \leq J_{Z} \leq \mathbb{I}. \end{aligned} $$</span></div><div class="c-article-equation__number"> (B.11) </div></div><p> That is, we have that </p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \operatorname{Tr} \bigl[{\hat{P}}\bigl(|\varphi _{n, s, g(\theta )} \rangle \bigr)J_{Z}^{*} \bigr]\geq \operatorname{Tr} \bigl[{ \hat{P}}\bigl(|\varphi _{n, s, g( \theta )}\rangle \bigr)J_{Z}^{**} \bigr], $$</span></div><div class="c-article-equation__number"> (B.12) </div></div><p> with <span class="mathjax-tex">$J_{Z}^{**}$</span> being the solution to the SDP in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ41">B.11</a>), and <span class="mathjax-tex">$J_{Z}^{*}$</span> the solution to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>). This holds because the constrains in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ41">B.11</a>) are looser than those in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ7">7</a>).</p><p>Finally, by combining Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ39">B.9</a>) with Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ42">B.12</a>) we have that </p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Y_{n, s, g(\theta )}^{Z, {\mathrm{L}}}\geq G_{-} \bigl(\operatorname{Tr} \bigl[{\hat{P}}\bigl(|\varphi _{n, s, g(\theta )}\rangle \bigr)J_{Z}^{**} \bigr], F_{n,s,g(\theta )}^{\text{vec}} \bigr):={\tilde{Y}}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}. $$</span></div><div class="c-article-equation__number"> (B.13) </div></div><p> The lower bound <span class="mathjax-tex">${\tilde{Y}}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}$</span> is the one we use in our simulations in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec11">3.1</a>.</p><h3 class="c-article__sub-heading" id="Sec19"><span class="c-article-section__title-number">2.2 </span>B.2 Upper bound on the phase-error rates <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> </h3><p>In this Appendix, we show how to estimate an upper bound on <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> by using a finite-dimensional SDP. To do so, let us also define the operator </p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ M_{\mathrm{ph}}:=|0_{X}\rangle \langle 0_{X}| \otimes L_{1_{X}}^{*}+| 1_{X} \rangle \langle 1_{X}| \otimes L_{0_{X}}^{*}, $$</span></div><div class="c-article-equation__number"> (B.14) </div></div><p> where <span class="mathjax-tex">$L_{(\Delta \oplus 1)_{X}}^{*}$</span> denotes the solution to the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>), so that </p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\sum_{\Delta =0}^{1}p^{\mathrm{virtual}}_{\Delta , n, s, g(\theta )}Y_{ \Delta , n, s, g(\theta )}^{ (\Delta \oplus 1)_{X}, \mathrm{virtual}} \\ &\quad \leq \sum_{\Delta =0}^{1}\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl| \bar{\lambda}^{\text{virtual}}_{\Delta , n, s, g(\theta )}\bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{*} \bigr]=\operatorname{Tr} \bigl[{ \hat{P}}\bigl(\bigl|\Psi ^{Z}_{n, s, g(\theta )}\bigr\rangle \bigr) M_{\text{ph}} \bigr]. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (B.15) </div></div><p>Now, let us define the finite-dimensional state </p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\Psi ^{Z, M}_{n, s, g(\theta )}\bigr\rangle = \frac{1}{\sqrt{2}} \bigl(|0_{Z} \rangle _{A} \hat{V}_{0_{Z}}+|1_{Z}\rangle _{A} \hat{V}_{1_{Z}} \bigr)|\varphi _{n, s, g(\theta )}\rangle , $$</span></div><div class="c-article-equation__number"> (B.16) </div></div><p> and the unnormalized states <span class="mathjax-tex">$|\bar{\lambda}^{\mathrm{virtual},M}_{ \Delta , n, s, g(\theta )} \rangle $</span> as </p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\bar{\lambda}^{\mathrm{virtual},M}_{ \Delta , n, s, g(\theta )} \bigr\rangle ={ }_{A}\bigl\langle \Delta _{X} |\Psi ^{Z, M}_{n, s, g(\theta )} \bigr\rangle =\frac{1}{2} \bigl[\hat{V}_{0_{Z}}+(-1)^{\Delta } \hat{V}_{1_{Z}} \bigr]|\varphi _{n, s, g(\theta )}\rangle . $$</span></div><div class="c-article-equation__number"> (B.17) </div></div><p>Then, we have that </p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl\vert \bigl\langle \Psi ^{Z, M}_{n, s, g(\theta )}|\Psi ^{Z}_{n, s, g( \theta )}\bigr\rangle \bigr\vert ^{2}= \bigl\vert \langle \varphi _{n,s,g( \theta )}|\psi _{n,s,g(\theta )}\rangle \bigr\vert ^{2}\geq F^{\mathrm{vec}}_{n,s,g(\theta )}, $$</span></div><div class="c-article-equation__number"> (B.18) </div></div><p> where we have used Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ37">B.7</a>) and the fact that <span class="mathjax-tex">$\hat{V}_{0 Z}^{\dagger} \hat{V}_{0 Z}=\hat{V}_{1 Z}^{\dagger} \hat{V}_{1 Z} = \mathbb{I}$</span>. Now, by applying the Cauchy-Schwarz constraint given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ32">B.2</a>), and taking into account the fact that <span class="mathjax-tex">$G_{+}(y,z)$</span> is a decreasing function with respect to its second argument, we find that </p><div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\Psi ^{Z}_{n, s, g(\theta )} \bigr\rangle \bigr) M_{\text{ph}} \bigr] \leq G_{+} \bigl(\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\Psi ^{Z, M}_{n, s, g( \theta )}\bigr\rangle \bigr) M_{\text{ph}} \bigr], F^{\text{vec}}_{n,s,g( \theta )} \bigr). \end{aligned} $$</span></div><div class="c-article-equation__number"> (B.19) </div></div><p>Importantly, since <span class="mathjax-tex">$G_{+}(y,z)$</span> is an increasing function with respect to its first argument, one can upper bound the previous equation by finding an upper bound on its first argument. Moreover, since the states <span class="mathjax-tex">$|\Psi ^{Z, M}_{n, s, g(\theta )}\rangle $</span> are finite dimensional, one can restrict the optimization search to operators <i>L</i> that act on the corresponding finite subspace. In particular, we have that </p><div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\Psi ^{Z, M}_{n, s, g(\theta )} \bigr\rangle \bigr) M_{\text{ph}} \bigr] \\ &\quad = \sum_{\Delta =0}^{1} \operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\bar{\lambda}^{\text{virtual}, M}_{ \Delta , n, s, g(\theta )}\bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{*} \bigr]\leq{} \sum _{\Delta =0}^{1} \operatorname{Tr} \bigl[{\hat{P}} \bigl(\bigl|\bar{\lambda}^{\text{virtual}, M}_{ \Delta , n, s, g(\theta )}\bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{**} \bigr], \end{aligned} $$</span></div><div class="c-article-equation__number"> (B.20) </div></div><p> where <span class="mathjax-tex">$L_{(\Delta \oplus 1)_{X}}^{**}$</span> is the solution to the finite-dimensional SDP presented below.</p><p>Likewise, the constraints in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ13">13</a>) can be relaxed by using essentially the same techniques discussed in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec18">B.1</a>. In doing so, we find that an upper bound on <span class="mathjax-tex">$\operatorname{Tr} [{\hat{P}}(|\bar{ \lambda}^{\text{virtual}, M}_{ \Delta , n, s, g(\theta )}\rangle ) L_{(\Delta \oplus 1)_{X}} ]$</span> can be found by solving the following SDP </p><div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \max_{L_{(\Delta \oplus 1)_{X}}} &\ \operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl| \bar{ \lambda}^{\text{virtual}, M}_{\Delta , n, s, g(\theta )} \bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}} \bigr] \\ \text{subject to } &\ G_{-} \bigl(Q_{\mu ,g(\theta ), b_{\alpha}}^{( \Delta \oplus 1)_{X}}, F^{\text{proj}}_{\mu ,g(\theta )} \bigr) \leq \operatorname{Tr} \bigl[ \hat{V}_{b_{\alpha}} \rho _{[g(\theta )], \text{M}}^{\mu} \hat{V}_{b_{\alpha}}^{\dagger} L_{(\Delta \oplus 1)_{X}} \bigr] \\ &\hphantom{\ G_{-} \bigl(Q_{\mu ,g(\theta ), b_{\alpha}}^{( \Delta \oplus 1)_{X}}, F^{\text{proj}}_{\mu ,g(\theta )} \bigr)}\leq G_{+} \bigl(Q_{\mu ,g(\theta ), b_{\alpha}}^{(\Delta \oplus 1)_{X}}, F^{\text{proj}}_{\mu ,g(\theta )} \bigr), \\ &\ \forall \mu \in \{s, \nu , \omega \}, \forall b\in \{0,1\}, \forall \alpha \in \{Z,X\} \\ &\ 0 \leq L_{(\Delta \oplus 1)_{X}} \leq \mathbb{I} , \end{aligned} $$</span></div><div class="c-article-equation__number"> (B.21) </div></div><p> where <span class="mathjax-tex">$F^{\text{proj}}_{\mu ,g(\theta )}$</span> is given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ35">B.5</a>).</p><p>Let <span class="mathjax-tex">$L_{(\Delta \oplus 1)_{X},}^{**}$</span> denote the operator that maximizes the SDP given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ51">B.21</a>), then </p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ e_{n, s, g(\theta )}\leq \frac{1}{{\tilde{Y}}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}}G_{+} \Biggl(\sum _{ \Delta =0}^{1}\operatorname{Tr} \bigl[{\hat{P}}\bigl(\bigl|\bar{\lambda}^{ \text{virtual}, M}_{\Delta , n, s, g(\theta )}\bigr\rangle \bigr) L_{(\Delta \oplus 1)_{X}}^{**} \bigr], F^{\text{vec}}_{n,s,g(\theta )} \Biggr) :={\tilde{e}}_{n, s, g(\theta )}^{\mathrm{U}}. $$</span></div></div><p>This is the upper bound that we use in our simulations in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec11">3.1</a>.</p><h3 class="c-article__sub-heading" id="App1">Appendix C: Finite-dimensional SDPs when <span class="mathjax-tex">$g(\theta )$</span> is partially characterized</h3><p>Here, we consider the scenario studied in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec14">3.2</a>, <i>i.e.</i>, when the actual imprinted phases lies in certain intervals <span class="mathjax-tex">$\hat{\theta}_{k}\in [\theta _{k}-\delta _{\text{max}},\theta _{k}+ \delta _{\text{max}}]$</span>, with <span class="mathjax-tex">$\theta _{k}=2 \pi k/N$</span>, and the exact form of <span class="mathjax-tex">$g(\theta )$</span> is unknown.</p><p>A direct solution to this case could be found as follows. First, one defines a dense grid with <i>p</i> discrete values within each interval, and then one follows essentially the approach in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec12">3.1.1</a> for each possible combination of these discrete phases from the different intervals. The secret key rate would then correspond to the worst case scenario, <i>i.e.</i>, the one that minimizes it among all possible combinations. The main drawback of this approach is, however, that the number of SDPs that needs to be solved grows very rapidly, as <span class="mathjax-tex">$\propto p^{N}$</span>.</p><p>Instead, here we introduce a much simpler approach based on a modified version of the SDPs presented in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ41">B.11</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ51">B.21</a>). In particular, let <span class="mathjax-tex">$f(\theta )$</span> denote the PDF associated to the ideal discrete phase randomization scenario given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), and let <span class="mathjax-tex">$\rho ^{\mu}_{[f(\theta )], M}$</span> be the finite-dimensional state obtained by projecting <span class="mathjax-tex">$\rho ^{\mu}_{[f(\theta )]}$</span> onto the subspace that contains up to <i>M</i> photons. Also, let <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> denote the state actually emitted by Alice in the scenario described above, <i>i.e.</i>, when <span class="mathjax-tex">$g(\theta )$</span> is partially characterized. Then, we can bound the fidelity between <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span> and <span class="mathjax-tex">$\rho ^{\mu}_{[f(\theta )], M}$</span> by means of the Bures distance, which is defined as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title=" Farenick D, Bures RM. Contractive channels on operator algebras. NY J Math. 2017;23:1369–93. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR56" id="ref-link-section-d131444011e24211">56</a>] </p><div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ d_{B}(\rho ,\sigma )^{2} = 2\bigl[1- \sqrt{F(\rho ,\sigma )}\bigr], $$</span></div><div class="c-article-equation__number"> (C.1) </div></div><p> for any state <i>ρ</i> and <i>σ</i>. This distance satisfies the triangle inequality [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title=" Farenick D, Bures RM. Contractive channels on operator algebras. NY J Math. 2017;23:1369–93. " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR56" id="ref-link-section-d131444011e24308">56</a>], which means that </p><div id="Equ53" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sqrt{F\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )], M}\bigr)} =&1 - \frac{1}{2} d_{B}\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f( \theta )], M}\bigr)^{2} \\ \geq & 1 - \frac{1}{2} \bigl[d_{B} \bigl(\rho ^{\mu}_{[f(\theta )]}, \rho ^{ \mu}_{[f(\theta )], M} \bigr) \\ &{}+d_{B}\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )]}\bigr) \bigr]^{2}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (C.2) </div></div><p>We now compute the fidelities that correspond to the Bures distances <span class="mathjax-tex">$d_{B} (\rho ^{\mu}_{[f(\theta )]}, \rho ^{\mu}_{[f(\theta )], M})$</span> and <span class="mathjax-tex">$d_{B}(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )]})$</span> so that, via Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ52">C.1</a>), we can obtain the necessary fidelity bound with Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ53">C.2</a>).</p><p>In particular, from Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ35">B.5</a>), we have that <span class="mathjax-tex">$F(\rho ^{\mu}_{[f(\theta )]}, \rho ^{\mu}_{[f(\theta )], M})=F^{ \text{proj}}_{\mu ,f(\theta )}$</span>. The fidelity <span class="mathjax-tex">$F(\rho ^{\mu}_{g(\theta )}, \rho ^{\mu}_{[f(\theta )]})$</span>, on the other hand, can be computed by considering the following purifications of the states <span class="mathjax-tex">$\rho ^{\mu}_{[f(\theta )]}$</span> and <span class="mathjax-tex">$\rho ^{\mu}_{[g(\theta )]}$</span>, respectively, </p><div id="Equ54" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} & \bigl|\psi _{[f(\theta )]}^{\mu ,N} \bigr\rangle =\frac{1}{\sqrt{N}} \sum_{k=0}^{N-1}|k \rangle \bigl\vert \sqrt{\mu} e^{2 \pi k i / N} \bigr\rangle , \\ & \bigl|\psi _{[g(\theta )]}^{\mu ,N}\bigr\rangle =\frac{1}{\sqrt{N}} \sum _{k=0}^{N-1} e^{i \phi _{k}}|k\rangle \bigl\vert \sqrt{\mu} e^{i (2 \pi k / N+ \delta _{k} ) } \bigr\rangle . \end{aligned} $$</span></div><div class="c-article-equation__number"> (C.3) </div></div><p> We find, therefore, that </p><div id="Equ55" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} F\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho _{[f(\theta )]}^{\mu} \bigr) \geq& \bigl| \bigl\langle \psi _{[f(\theta )]}^{\mu ,N}|\psi _{[g(\theta )]}^{\mu ,N} \bigr\rangle \bigr|^{2} \\ =& \Biggl|\sum_{k=0}^{N-1} \frac{1}{N}\bigl\langle \sqrt{\mu} e^{2 \pi k i / N}| \sqrt{\mu} e^{i(2 \pi k / N+\delta _{k})}\bigr\rangle \Biggl|^{2} \\ \geq& \Biggl|\sum_{k=0}^{N-1} \frac{1}{N}\bigl\langle \sqrt{\mu} e^{2 \pi k i / N} |\sqrt{\mu} e^{i(2 \pi k / N+\delta _{\text{max}})}\bigr\rangle \Biggl|^{2} \\ =& \bigl|\bigl\langle \sqrt{\mu}| \sqrt{\mu}e^{i\delta _{\text{max}}}\bigr\rangle \bigr|^{2}, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (C.4) </div></div><p> where in the first inequality we have used the fact that the states on the right hand side are a purification of those on the left hand side; in the first equality we have taken into account that the phases <span class="mathjax-tex">$\phi _{k}$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ54">C.3</a>) can be chosen so that they cancel the phase of the inner product, and in the second inequality we have used the fact that <span class="mathjax-tex">$|\delta _{k}|\leq \delta _{\text{max}}$</span> <span class="stix">∀</span><i>k</i>.</p><p>Since the function <span class="mathjax-tex">$g(\theta )$</span> is unknown, we do not have access to the exact form of the eigenvectors <span class="mathjax-tex">$|{\varphi _{n,s,[g(\theta )]}}\rangle $</span> of <span class="mathjax-tex">$\rho _{[g(\theta )],M}^{s}$</span> which are needed to solve the relevant finite-dimensional SDP, but we can lower bound the value of <span class="mathjax-tex">$\operatorname{Tr}[{\hat{P}}(|{\varphi _{n,s,[g(\theta )]}}\rangle )J_{Z}]$</span>, with <span class="mathjax-tex">$0 \leq J_{Z} \leq \mathbb{I}$</span>, by employing the Cauchy-Schwartz constraint presented in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ32">B.2</a>). Precisely, we have that </p><div id="Equ56" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \operatorname{Tr} \bigl[{\hat{P}}\bigl(|{\varphi _{n,s,[g(\theta )]}}\rangle \bigr)J_{Z} \bigr] \geq G_{-} \bigl(\operatorname{Tr} \bigl[{\hat{P}}\bigl(|{\varphi _{n,s,[f(\theta )]}}\rangle \bigr)J_{Z} \bigr], F\bigl(|{\varphi _{n,s,[g(\theta )]}}\rangle , |{\varphi _{n,s,[f(\theta )]}}\rangle \bigr) \bigr), \end{aligned}$$ </span></div><div class="c-article-equation__number"> (C.5) </div></div><p> where <span class="mathjax-tex">$|{\varphi _{n,s,[f(\theta )]}}\rangle $</span> are the eigenvectors of <span class="mathjax-tex">$\rho _{[f(\theta )],M}^{s}$</span>, and the value of <span class="mathjax-tex">$F(|{\varphi _{n,s,[g(\theta )]}}\rangle , |{\varphi _{n,s,[f(\theta )]}}\rangle )$</span> is calcuated numerically as explained below.</p><p>With these considerations, we can now find a lower bound on the yields <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span>. For this, we first solve the following optimization problem to find the operator <span class="mathjax-tex">$J^{**}_{\text{z}}$</span> that minimizes its objective function </p><div id="Equ57" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \min_{J_{Z}} &\ \operatorname{Tr} \bigl[{\hat{P}}\bigl( |{\varphi _{n,s,[f(\theta )]}}\rangle \bigr)J_{Z} \bigr] \\ \text{subject to} &\ G_{-} \bigl(Q_{\mu , g(\theta )}^{Z}, F\bigl(\rho ^{ \mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )], M} \bigr) \bigr) \\ &\ \quad \leq \operatorname{Tr} \bigl[\rho _{[f(\theta )],M}^{\mu}J_{Z} \bigr]\leq G_{+} \bigl(Q_{\mu , g(\theta )}^{Z}, F\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )], M}\bigr) \bigr), \\ &\ 0 \leq J_{Z} \leq \mathbb{I}. \end{aligned} $$</span></div><div class="c-article-equation__number"> (C.6) </div></div><p>Following Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ56">C.5</a>), we now define </p><div id="Equ58" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \hat{Y}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}:= G_{-} \bigl(\operatorname{Tr} \bigl[{\hat{P}}\bigl(|{\varphi _{n,s,[f(\theta )]}}\rangle \bigr)J^{**}_{Z} \bigr], F\bigl(|{\varphi _{n,s,[g(\theta )]}}\rangle , |{\varphi _{n,s,[f(\theta )]}}\rangle \bigr) \bigr). $$</span></div><div class="c-article-equation__number"> (C.7) </div></div><p> Finally, by using the arguments introduced in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec18">B.1</a>, we obtain that a lower bound on <span class="mathjax-tex">$Y_{n, s, g(\theta )}^{Z}$</span> is given by </p><div id="Equ59" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Y_{n, s, g(\theta )}^{Z} \geq G_{-} \bigl(\hat{Y}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}, F^{\text{vec}}_{n,s,g(\theta )} \bigr):=\tilde{Y}_{n, s, g( \theta )}^{Z, {\mathrm{L}}}. $$</span></div><div class="c-article-equation__number"> (C.8) </div></div><p>Note that, since we do not know which values of <span class="mathjax-tex">$\hat{\theta}_{k}$</span> result in the set of states <span class="mathjax-tex">$|{\varphi _{n,s,[g(\theta )]}}\rangle $</span> that minimizes the key rate, we find the worst case scenario numerically. To do so, we implement a Montecarlo simulation by considering a dense grid of values in <span class="mathjax-tex">$\theta _{k}\pm \delta _{\text{max}}$</span> for every <i>k</i> and we find the combination of <span class="mathjax-tex">$\hat{\theta}_{k}$</span> that minimizes Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ59">C.8</a>) (which includes the fidelity in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ58">C.7</a>)). This allow us to find the desired lower bound with arbitrary precision. Also, note that the number of SDPs that need to be solved grows very rapidly in the case of the direct solution mentioned at the beginning of this section. With this approach, this problem has been circumvented by reducing it to a simple calculation of the fidelities, which makes it computationally much faster, despite possibly providing looser bounds.</p><p>Regarding the estimation of an upper bound on the phase error rate, we follow the same procedure described in Appendix <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec19">B.2</a>. In doing so, we first solve the following finite-dimensional SDP, </p><div id="Equ60" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \max_{L_{(\Delta \oplus 1)_{X}}} &\ \operatorname{Tr} \bigl[\hat{P}\bigl( \bigl|\bar{\lambda}_{\Delta , n,s,[f(\theta )]}^{\text{virtual},M}\bigr\rangle \bigr)L_{( \Delta \oplus 1)_{X}} \bigr] \\ \text{subject to } &\ G_{-} \bigl(Q_{\mu , b_{\alpha}}^{(\Delta \oplus 1)_{X}}, F\bigl(\rho ^{\mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )], M} \bigr) \bigr) \\ &\ \quad \leq \operatorname{Tr} \bigl[\hat{V}_{b_{\alpha}} \rho _{[f( \theta )], M}^{\mu} \hat{V}_{b_{\alpha}}^{\dagger} L_{(\Delta \oplus 1)_{X}} \bigr] \leq G_{+} \bigl(Q_{\mu , b_{\alpha}}^{(\Delta \oplus 1)_{X}}, F\bigl(\rho ^{ \mu}_{[g(\theta )]}, \rho ^{\mu}_{[f(\theta )], M} \bigr) \bigr), \\ &\ 0 \leq L_{(\Delta \oplus 1)_{X}} \leq \mathbb{I}, \end{aligned} $$</span></div><div class="c-article-equation__number"> (C.9) </div></div><p> where <span class="mathjax-tex">$Q_{\mu , b_{\alpha}}^{(\Delta \oplus 1)_{X}}$</span> represents the observed rate at which Bob obtains the result <span class="mathjax-tex">$(\Delta \oplus 1)_{X}$</span> conditioned on Alice choosing the intensity setting <i>μ</i>, the basis <i>α</i>, the bit value <i>b</i> and Bob choosing the <i>X</i> basis. Now, similarly to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ58">C.7</a>), we define </p><div id="Equ61" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \hat{e}^{\mathrm{U}}_{n, s, g(\theta )}:=\sum_{\Delta =0}^{1}G_{+} \bigl( \operatorname{Tr} \bigl[\hat{P}\bigl( \bigl|\bar{\lambda}_{\Delta , n,s,[f(\theta )]} ^{\text{virtual},M}\bigl\rangle \bigr)L^{**}_{( \Delta \oplus 1)_{X}} \bigr], F\bigl( \bigl|\bar{ \lambda}_{\Delta , n,s,[f(\theta )]}^{\text{virtual},M}\bigr\rangle , \bigl|\bar{\lambda}_{\Delta ,n,s,[g(\theta )]}^{\text{virtual},M}\bigr\rangle \bigr) \bigr), $$</span></div><div class="c-article-equation__number"> (C.10) </div></div><p> where <span class="mathjax-tex">$L^{**}_{(\Delta \oplus 1)_{X}}$</span> is the solution to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ60">C.9</a>). This way, we obtain that the phase error rate <span class="mathjax-tex">$e_{n, s, g(\theta )}$</span> is upper bounded by </p><div id="Equ62" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ e_{n, s, g(\theta )}\leq \frac{G_{+} (\hat{e}_{n, s, g(\theta )}^{\mathrm{U}}, F^{\text{vec}}_{n,s,g(\theta )} )}{\tilde{Y}_{n, s, g(\theta )}^{Z, {\mathrm{L}}}} :={\tilde{e}}_{n, s, g(\theta )}^{\mathrm{U}}, $$</span></div><div class="c-article-equation__number"> (C.11) </div></div><p> where again, we use the combination of <span class="mathjax-tex">$\hat{\theta}_{k}$</span> that maximizes Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ62">C.11</a>) to obtain the relevant upper bound.</p><p>The bounds <span class="mathjax-tex">$\tilde{Y}_{n, s, g(\theta )}^{Z, L}$</span> and <span class="mathjax-tex">${\tilde{e}}_{n, s, g(\theta )}^{\mathrm{U}}$</span> are used in the simulations presented in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec14">3.2</a>.</p><p>As shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig4">4</a>, higher values of <span class="mathjax-tex">$\delta _{\text{max}}$</span> result in an almost negligible impact of the parameter <i>N</i> on the secret key rate, as explained in the main text.</p><h3 class="c-article__sub-heading" id="App1">Appendix D: Influence of the parameter M in the secret key rate</h3><p>As stated in the main text, the secret key rate is an increasing function of the size of the finite dimensional SDP, denoted by <i>M</i>. However, the computational time for solving these SDPs significantly increases with higher values of <i>M</i>. In this Appendix we briefly explore how the size of the SDP impacts the secret key rate. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Fig5">5</a> displays the secret key rate for a case involving 8 random phases, showcasing changes as <i>M</i> varies. It is evident from the figure that selecting a small <span class="mathjax-tex">$M<8$</span> results in a considerable drop in performance. Nevertheless, when <span class="mathjax-tex">$M>8$</span> the key rate appears to saturate and the improvement that we can get by enlarging <i>M</i> is negligible. Hence, in the figures presented in the main text, for each <i>N</i>, we choose an <i>M</i> such that, further increases offer only marginal improvements in the secret key rate. This leads us to select different <i>M</i> values depending on <i>N</i>, as less random phases require smaller SDPs to fulfill the condition above. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Figure 5"><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 5</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/5" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig5_HTML.jpg?as=webp"><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjqt%2Fs40507-023-00210-0/MediaObjects/40507_2023_210_Fig5_HTML.jpg" alt="figure 5" loading="lazy" width="685" height="482"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p>Secret key rate as a function of the size of the SDP, represented by the parameter <i>M</i> when the number of random phases is fixed to <span class="mathjax-tex">$N=8$</span>. As one can see, selecting a small <i>M</i> causes a significant drop in performance. Increasing M requires exponentially more time to solve the SDPs</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/10.1140/epjqt/s40507-023-00210-0/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="App1">Appendix E: Parameter estimation procedure based on linear programming</h3><p>For completeness, in this Appendix we summarize the parameter estimation technique presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e29760">42</a>], using linear programming, to evaluate the case of perfect discrete phase randomization for the protocol described in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Sec4">2.2</a>.</p><p>In particular, given that the PDF follows Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ1">1</a>), which we will denote as <span class="mathjax-tex">$f(\theta )$</span> as in the previous Appendix and <span class="mathjax-tex">$N \geq 1$</span>, a purification of Alice’s emitted states can be expressed as </p><div id="Equ63" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\psi ^{\mu ,N}_{[f(\theta )]} \bigr\rangle =\sum_{k=0}^{N-1}|k\rangle _{A}\bigl| \sqrt{\mu} e^{2 k \pi i / N}\bigr\rangle =\sum_{j=0}^{N-1}|j\rangle _{A} \bigl\vert \beta ^{\mu}_{j} \bigr\rangle , $$</span></div><div class="c-article-equation__number"> (E.1) </div></div><p> where the second equality corresponds to the Schmidt decomposition. Note that in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ63">E.1</a>) we consider unnormalized states, which we will do throughout this Appendix for convenience. The states <span class="mathjax-tex">$|{j}\rangle _{A}$</span> can be interpreted as a quantum coin with <i>N</i> random outputs, while the states <span class="mathjax-tex">$|\beta ^{\mu}_{j}\rangle $</span> are given by </p><div id="Equ64" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl|\beta ^{\mu}_{j}\bigr\rangle =\sum _{k=0}^{N-1} e^{-2 k j \pi i / N}\bigl|e^{2 k \pi i / N} \sqrt{\mu}\bigr\rangle . $$</span></div><div class="c-article-equation__number"> (E.2) </div></div><p> By using Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ2">2</a>), these latter states can be rewritten as </p><div id="Equ65" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl\vert \beta ^{\mu}_{j} \bigr\rangle =\sum _{l=0}^{\infty} \frac{(\sqrt{\mu})^{l N+j}}{\sqrt{(l N+j) !}}|l N+j\rangle . $$</span></div><div class="c-article-equation__number"> (E.3) </div></div><p> Indeed, it is easy to show that when <i>N</i> is large, <span class="mathjax-tex">$|\beta ^{\mu}_{j}\rangle $</span> approaches a Fock state with <i>j</i> photons.</p><p>If Alice measures her ancilla system <i>A</i> from the state <span class="mathjax-tex">$|\psi ^{\mu ,N}_{[f(\theta )]}\rangle $</span> in the basis <span class="mathjax-tex">$\{|{j}\rangle _{A}\}$</span>, she obtains the result <i>j</i> with probability <span class="mathjax-tex">$P^{\mu}_{j}$</span> given by </p><div id="Equ66" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} P^{\mu}_{j} =& \frac{ \langle \beta ^{\mu}_{j} \mid \beta ^{\mu}_{j} \rangle}{\sum_{j=0}^{N-1} \langle \beta ^{\mu}_{j} \mid \beta ^{\mu}_{j} \rangle} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (E.4) </div></div><div id="Equ67" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} =&\sum_{l=0}^{\infty} \frac{\mu ^{l N+j} e^{-\mu}}{(l N+j) !}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (E.5) </div></div><p>Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title=" Cao Z, Zhang Z, Lo HK, Ma X. Discrete-phase-randomized coherent state source and its application in quantum key distribution. New J Phys. 2015;17(5):053014. https://doi.org/10.1088/1367-2630/17/5/053014 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR42" id="ref-link-section-d131444011e30701">42</a>] employs the Gottesman, Lo, Lütkenhaus and Preskill (GLLP) security analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title=" Gottesman D, Lo HK, Lütkenhaus N, Preskill J. Security of quantum key distribution with imperfect devices. Quantum Inf Comput. 2004;4:325–60. https://doi.org/10.26421/QIC4.5-1 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR48" id="ref-link-section-d131444011e30704">48</a>], which needs to determine the basis dependence <span class="mathjax-tex">$\Delta ^{\mu}_{j}$</span> of the source, which is closely related to the fidelity <span class="mathjax-tex">$F^{\mu}_{j}$</span> between the states in the <i>X</i> and <i>Z</i> basis. Precisely, let us define </p><div id="Equ68" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \Delta ^{\mu}_{j}=\frac{1-F^{\mu}_{j}}{2 Y^{Z}_{j,\mu , f(\theta )}}, $$</span></div><div class="c-article-equation__number"> (E.6) </div></div><p> where <span class="mathjax-tex">$Y^{Z}_{j,\mu , f(\theta )}$</span> refers to the yield that corresponds to the states <span class="mathjax-tex">$|\beta ^{\mu}_{j}\rangle $</span> encoded in the Z basis, and the fidelity <span class="mathjax-tex">$F^{\mu}_{j}$</span> can be bounded by </p><div id="Equ69" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &F^{\mu}_{j}\geq \biggl\vert \frac{\sum_{l=0}^{\infty} \frac{\mu ^{l N+j}}{(l N+j) !} 2^{-\frac{l N+j}{2}} (\cos \frac{l N+j}{4} \pi +\sin \frac{l N+j}{4} \pi )}{\sum_{l=0}^{\infty} \frac{\mu ^{l N+j}}{(l N+j) !}} \biggr\vert . \end{aligned} $$</span></div><div class="c-article-equation__number"> (E.7) </div></div><p>Moreover, since <span class="mathjax-tex">$\vert \beta _{j}^{\mu} \rangle \neq \vert \beta _{j}^{\gamma} \rangle $</span> when <span class="mathjax-tex">$\mu \neq \gamma $</span>, one can relate the yields and bit error rates associated to different intensity settings as follows [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title=" Gottesman D, Lo HK, Lütkenhaus N, Preskill J. Security of quantum key distribution with imperfect devices. Quantum Inf Comput. 2004;4:325–60. https://doi.org/10.26421/QIC4.5-1 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR48" id="ref-link-section-d131444011e31268">48</a>] </p><div id="Equ70" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} & \vert Y_{j,\mu , f(\theta )}-Y_{j,\gamma , f(\theta )} \vert \leq \sqrt{1-F_{\mu \gamma}^{2}}, \\ & \bigl\vert e^{b}_{j,\mu , f(\theta )} Y_{j,\mu , f(\theta )}-e^{b}_{j, \gamma , f(\theta )} Y_{j,\gamma , f(\theta )} \bigr\vert \leq \sqrt{1-F_{ \mu \gamma}^{2}}, \end{aligned} $$</span></div><div class="c-article-equation__number"> (E.8) </div></div><p> where <span class="mathjax-tex">$e^{b}_{j,\mu , f(\theta )}$</span> denotes the bit error rate corresponding to the states <span class="mathjax-tex">$|\beta _{j}^{\mu}\rangle $</span>, <i>i.e.</i>, the probability that Alice and Bob obtain different results when they use the same basis and Alice emits the state <span class="mathjax-tex">$|\beta _{j}^{\mu}\rangle $</span>. The parameter <span class="mathjax-tex">$F_{\mu \gamma}$</span>, on the other hand, is given by </p><div id="Equ71" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F_{\mu \gamma} := \frac{\sum_{l=0}^{\infty} \frac{(\mu \gamma )^{l N / 2}}{(l N) !}}{\sqrt{\sum_{l=0}^{\infty} \frac{\mu ^{l N}}{(l N) !} \sum_{l=0}^{\infty} \frac{\gamma ^{l N}}{(l N) !}}}. $$</span></div><div class="c-article-equation__number"> (E.9) </div></div><p>The phase error rate <span class="mathjax-tex">$e_{j,\mu ,g(\theta )}$</span> in the <i>Z</i> basis can be upper bounded by means of the bit error rate <span class="mathjax-tex">$e_{j,\mu , g(\theta )}^{b}$</span> in the <i>X</i> basis and the basis dependence parameter <span class="mathjax-tex">$\Delta ^{\mu}_{j}$</span> as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title=" Lo HK, Preskill J. Security of quantum key distribution using weak coherent states with nonrandom phases. Quantum Inf Comput. 2007;8:431–58. https://doi.org/10.26421/QIC7.5-6-2 . " href="/articles/10.1140/epjqt/s40507-023-00210-0#ref-CR53" id="ref-link-section-d131444011e32012">53</a>] </p><div id="Equ72" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} e_{j,\mu ,f(\theta )} \leq& e_{j,\mu , f(\theta )}^{b, X}+4 \Delta ^{ \mu}_{j} \bigl(1-\Delta ^{\mu}_{j} \bigr) \bigl(1-2 e^{b, X}_{j,\mu , f(\theta )} \bigr) \\ &{}+4 \bigl(1-2 \Delta ^{\mu}_{j} \bigr) \sqrt{\Delta ^{\mu}_{j} \bigl(1- \Delta ^{\mu}_{j} \bigr) e^{b, X}_{j,\mu , f(\theta )} \bigl(1-e^{b, X}_{j, \mu , f(\theta )} \bigr)}, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (E.10) </div></div><p> where we have included the superscript <i>X</i> in the bit error rate to emphasize that it refers to that in the <i>X</i> basis.</p><p>Putting it all together, we have that a lower bound on the yields <span class="mathjax-tex">$Y^{Z}_{j,s, f(\theta )}$</span> encoded in the Z basis can be estimated with the following linear program </p><div id="Equ73" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \begin{aligned} \mathrm{min} &\ Y^{Z}_{j,s, f(\theta )} \\ \text{subject to} &\ \bigl\vert Y^{Z}_{j,\mu , f(\theta )}-Y^{Z}_{j,\gamma , f(\theta )} \bigr\vert \leq \sqrt{1-F_{\mu \gamma}^{2}}, \\ &\ \forall \mu , \gamma \in \{s, \nu , \omega \}, \mu \neq \gamma , \\ &\ Q_{\mu , f(\theta )}^{Z}=\sum_{j=0}^{N-1} P_{j}^{\mu }Y^{Z}_{j, \mu , f(\theta )},\quad \forall \mu \in \{s, \nu , \omega \}. \end{aligned} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (E.11) </div></div><p>Similarly, an upper bound on the bit error rate <span class="mathjax-tex">$e_{j,\mu , g(\theta )}^{b, X}$</span> can be calculated with the following linear program </p><div id="Equ74" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \begin{aligned} \mathrm{max} &\ \xi ^{X}_{j,s, f(\theta )} \\ \text{subject to} &\ \bigl\vert \xi ^{X}_{j,\mu , f(\theta )}- \xi ^{X}_{j,\gamma , f(\theta )} \bigr\vert \leq \sqrt{1-F_{\mu \gamma}^{2}}, \\ &\ \forall \mu , \gamma \in \{s, \nu , \omega \}, \mu \neq \gamma , \\ &\ E^{X}_{\mu , f(\theta )} Q^{X}_{\mu , f(\theta )}=\sum _{j=0}^{N-1} P_{j}^{\mu } \xi ^{X}_{j,\mu , f(\theta )},\quad \forall \mu \in \{s, \nu , \omega \}, \end{aligned} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (E.12) </div></div><p> where <span class="mathjax-tex">$\xi ^{X}_{j,s, f(\theta )}=e^{b, X}_{j,s, f(\theta )} Y^{X}_{j,s, f( \theta )}$</span>. In particular, let <span class="mathjax-tex">$\xi ^{X*}_{j,s, f(\theta )}$</span> denote the solution to the linear program above, then we have that </p><div id="Equ75" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ e^{b, X}_{j,s, f(\theta )}\leq \frac{\xi ^{X*}_{j,s, f(\theta )}}{Y^{X, {\mathrm{L}}}_{j,s, f(\theta )}}:=e^{b, X, {\mathrm{U}}}_{j,s, f(\theta )}, $$</span></div><div class="c-article-equation__number"> (E.13) </div></div><p> where <span class="mathjax-tex">$Y^{X, {\mathrm{L}}}_{j,s, f(\theta )}$</span> represents a lower bound on the yield <span class="mathjax-tex">$Y^{X}_{j,s, f(\theta )}$</span> in the <i>X</i> basis. This quantity can be calculated with the linear program given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ73">E.11</a>) by simply replacing the superscript <i>Z</i> with <i>X</i>.</p><p>Finally, one can calculate the phase error rate <span class="mathjax-tex">$e_{j,\mu ,f(\theta )}$</span> in the <i>Z</i> basis by means of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ72">E.10</a>), after replacing <span class="mathjax-tex">$e_{j,\mu , f(\theta )}^{b, X}$</span> with its upper bound and <span class="mathjax-tex">$\Delta ^{\mu}_{j}$</span> with the upper bound obtained after replacing a lower bound for the yield in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ68">E.6</a>). Importantly, with this approach there is no need to make a projection onto a finite dimensional subspace. This means that when evaluating the secret key rate formula given by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ6">6</a>), the probabilities <span class="mathjax-tex">$p^{L}_{n\mid s, f(\theta )}$</span> are directly given by <span class="mathjax-tex">$P^{\mu}_{j}$</span> as defined in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1140/epjqt/s40507-023-00210-0#Equ66">E.4</a>).</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Sixto, X., Currás-Lorenzo, G., Tamaki, K. <i>et al.</i> Secret key rate bounds for quantum key distribution with faulty active phase randomization. <i>EPJ Quantum Technol.</i> <b>10</b>, 53 (2023). https://doi.org/10.1140/epjqt/s40507-023-00210-0</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1140/epjqt/s40507-023-00210-0?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-09-26">26 September 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-12-05">05 December 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-12-15">15 December 2023</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1140/epjqt/s40507-023-00210-0</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box 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