Is the Reduced Density Matrix Complete for Local Evolution? A Relational-Context Test Beyond Standard Nonlinear Quantum Mechanics

Is the Reduced Density Matrix Complete for Local Evolution? A Relational-Context Test Beyond Standard Nonlinear Quantum Mechanics | 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 Is the Reduced Density Matrix Complete for Local Evolution? A Relational-Context Test Beyond Standard Nonlinear Quantum Mechanics Andrei Tudor Patrascu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8894096/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract We formulate and analyze an operational test of a tacit assumption frequently made in quantum device modeling: that the reduced density matrix ρS of a subsystem S is a sufficient state variable for predicting its future reduced evolution under fixed declared controls. We formalize this assumption as Local Evolution Completeness (LEC): if two global preparations share the same reduced state ρS (0), then they must yield identical ρS ( t ) under identical device dynamics. We first establish a strict-locality null result: when the applied Hamiltonian factorizes as HS ⊗ IE , LEC holds exactly, independent of correlations with ancillary degrees of freedom. We then prove, via an explicit process-tensor counterexample, that LEC can fail even when all intermediate maps are completely positive and trace-preserving and the global dynamics is strictly unitary. In such cases, no reduced map Φ exists with ρS ( t ) = Φ( ρS (0)) for all embeddings realizing the same ρS (0). To operationalize this distinction, we construct a fully solvable qubit–ancilla model and derive closed-form reduced trajectories, along with a statistical inference framework for detecting preparation-dependent deviations under fixed controls. We provide microscopic realizations of the embedding variable responsible for LEC failure, including correlation-invariant and control-reference mechanisms. The proposed protocol isolates a distinct experimental axis from traditional nonlinear quantum mechanics tests: it probes the predictive sufficiency of reduced states rather than convex linearity of state evolution. A violation of LEC would not imply a modification of global unitary dynamics, but rather demonstrate that single-time reduced density matrices are not sufficient statistics for multi-time prediction in the tested setting. This clarifies the structural role of reduced states within the operational foundations of quantum theory. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 14 Apr, 2026 Editor assigned by journal 18 Feb, 2026 Submission checks completed at journal 18 Feb, 2026 First submitted to journal 16 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-8894096","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625174705,"identity":"5a7fd732-cfc2-4fab-a10b-877037bfcb2f","order_by":0,"name":"Andrei Tudor Patrascu","email":"data:image/png;base64,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","orcid":"","institution":"FAST Foundation","correspondingAuthor":true,"prefix":"","firstName":"Andrei","middleName":"Tudor","lastName":"Patrascu","suffix":""}],"badges":[],"createdAt":"2026-02-16 14:53:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8894096/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8894096/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107419378,"identity":"5c5f48b4-1c2e-464a-8c03-1c7fe65b15ea","added_by":"auto","created_at":"2026-04-21 10:13:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":453306,"visible":true,"origin":"","legend":"","description":"","filename":"quantumnonlin.tex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8894096/v1_covered_1161c323-b044-4c28-85e3-5210aa104035.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eIs the Reduced Density Matrix Complete for Local Evolution? A Relational-Context Test Beyond Standard Nonlinear Quantum Mechanics\u003c/p\u003e","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":"","lastPublishedDoi":"10.21203/rs.3.rs-8894096/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8894096/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe formulate and analyze an operational test of a tacit assumption frequently made in quantum device modeling: that the reduced density matrix \u003cem\u003eρS \u003c/em\u003eof a subsystem \u003cem\u003eS \u003c/em\u003eis a sufficient state variable for predicting its future reduced evolution under fixed declared controls. We formalize this assumption as \u003cem\u003eLocal Evolution Completeness \u003c/em\u003e(LEC): if two global preparations share the same reduced state \u003cem\u003eρS\u003c/em\u003e(0), then they must yield identical \u003cem\u003eρS\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) under identical device dynamics.\u003c/p\u003e\n\u003cp\u003eWe first establish a strict-locality null result: when the applied Hamiltonian factorizes as \u003cem\u003eHS \u003c/em\u003e⊗ \u003cem\u003eIE\u003c/em\u003e, LEC holds exactly, independent of correlations with ancillary degrees of freedom. We then prove, via an explicit process-tensor counterexample, that LEC can fail even when all intermediate maps are completely positive and trace-preserving and the global dynamics is strictly unitary. In such cases, no reduced map Φ exists with \u003cem\u003eρS\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) = Φ(\u003cem\u003eρS\u003c/em\u003e(0)) for all embeddings realizing the same \u003cem\u003eρS\u003c/em\u003e(0).\u003c/p\u003e\n\u003cp\u003eTo operationalize this distinction, we construct a fully solvable qubit–ancilla model and derive closed-form reduced trajectories, along with a statistical inference framework for detecting preparation-dependent deviations under fixed controls. We provide microscopic realizations of the embedding variable responsible for LEC failure, including correlation-invariant and control-reference mechanisms.\u003c/p\u003e\n\u003cp\u003eThe proposed protocol isolates a distinct experimental axis from traditional nonlinear quantum mechanics tests: it probes the predictive sufficiency of reduced states rather than convex linearity of state evolution. A violation of LEC would not imply a modification of global unitary dynamics, but rather demonstrate that single-time reduced density matrices are not sufficient statistics for multi-time prediction in the tested setting. This clarifies the structural role of reduced states within the operational foundations of quantum theory.\u003c/p\u003e","manuscriptTitle":"Is the Reduced Density Matrix Complete for Local Evolution? 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