Visualization of Multi-Qubit Pure States with Separation of Local and Nonlocal Degrees of Freedom

Visualization of Multi-Qubit Pure States with Separation of Local and Nonlocal Degrees of Freedom | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Visualization of Multi-Qubit Pure States with Separation of Local and Nonlocal Degrees of Freedom Satoru Shoji This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8989960/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Understanding the structure of multi-qubit quantum states is essential for both quantum information research and education, yet intuitive visualization beyond the single-qubit Bloch sphere remains challenging. In this work, we propose a unified geometric framework for visualizing two- and three-qubit pure states in which local degrees of freedom and entanglement degrees of freedom are explicitly separated. For two qubits, we combine Bloch-sphere representations of reduced density operators with a complex concurrence plotted on the complex plane, enabling simultaneous visualization of entanglement strength and phase structure. For three qubits, building on the generalized Schmidt decomposition, we introduce bipartite and GHZ-type tripartite complex concurrences, which, together with local Bloch vectors, provide a complete coordinate representation of the state. Unlike classification-based approaches, our method focuses on representing a given concrete state, revealing how local properties and nonlocal correlations coexist. The framework distinguishes states with identical entanglement magnitudes but different interference structures and provides intuitive insight into the balance between pairwise and genuinely tripartite entanglement. This approach offers both conceptual clarity and potential applications in quantum education and state analysis. Quantum Computing Multipartite entanglement Quantum State Visualization Generalized Schmidt Decomposition Complex Concurrence Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 14 May, 2026 Reviews received at journal 12 May, 2026 Reviewers agreed at journal 15 Apr, 2026 Reviewers invited by journal 15 Apr, 2026 Editor assigned by journal 03 Mar, 2026 Submission checks completed at journal 03 Mar, 2026 First submitted to journal 27 Feb, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8989960","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625053010,"identity":"e7285d5e-478f-4471-a691-0481fc182eea","order_by":0,"name":"Satoru Shoji","email":"data:image/png;base64,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","orcid":"","institution":"Graduate School of Engineering, Tohoku University","correspondingAuthor":true,"prefix":"","firstName":"Satoru","middleName":"","lastName":"Shoji","suffix":""}],"badges":[],"createdAt":"2026-02-27 16:08:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8989960/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8989960/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107592258,"identity":"26cd64af-5a96-47a9-b97f-199a13cbaad7","added_by":"auto","created_at":"2026-04-23 03:40:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1489822,"visible":true,"origin":"","legend":"","description":"","filename":"entanglementpb.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8989960/v1_covered_888f85e0-9ccc-4203-ada1-bef46ba42482.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Visualization of Multi-Qubit Pure States with Separation of Local and Nonlocal Degrees of Freedom","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"quantum-information-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qinp","sideBox":"Learn more about [Quantum Information Processing](http://link.springer.com/journal/11128)","snPcode":"11128","submissionUrl":"https://submission.nature.com/new-submission/11128/3","title":"Quantum Information Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Quantum Computing, Multipartite entanglement, Quantum State Visualization, Generalized Schmidt Decomposition, Complex Concurrence","lastPublishedDoi":"10.21203/rs.3.rs-8989960/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8989960/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUnderstanding the structure of multi-qubit quantum states is essential for both quantum information research and education, yet intuitive visualization beyond the single-qubit Bloch sphere remains challenging. 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