Bell–CHSH inequality violation from continuous, non-projective measurements

Bell–CHSH inequality violation from continuous, non-projective measurements | 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 Article Bell–CHSH inequality violation from continuous, non-projective measurements Shalender Singh, Santosh Kumar This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8424019/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified from continuous time-series data. Here we develop a general theoretical framework showing that Bell–CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak. Physical sciences/Physics/Quantum physics/Qubits Physical sciences/Physics/Quantum physics/Quantum information Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted 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-8424019","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":564046244,"identity":"6c73eecb-8d10-4084-833d-7dfcb7fe507d","order_by":0,"name":"Shalender Singh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVRIie3RMUvDQBTA8XdkyHKS9QXFfIVXOjgU6lc5yZAlBMeCYIXCudT9ip/CRRwvHNTlugfqkC8g1C1CBUOL2CU9uxW8/3bH/eAdD8DnO8YQmAYIAMI7ALG54k4CW8L1huABBMXP2UGSx0mtP14G5xez9/mqXg9vo3ChoRk9dxJ6m1M5s1n/bFmkSvAU42kh2NQuuwkKMifSXKnTvA8CA6SKU8BkN0lUtjJf0oxVbFtCY7x0EahyMkwagchbIgwSOghV+XX5ILOe4kUKQr/GyuZU7vtLO9hT/SkHCYYLw5r1TRTd217djPYM9tvOOvRf3oN76T6fz/dv+wbO5VWzx3DEbQAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-3677-4043","institution":"Polriqon Inc","correspondingAuthor":true,"prefix":"","firstName":"Shalender","middleName":"","lastName":"Singh","suffix":""},{"id":564046245,"identity":"29bf44d8-6002-427e-b7d5-6f7820362e4d","order_by":1,"name":"Santosh Kumar","email":"","orcid":"","institution":"Polariqon Inc.","correspondingAuthor":false,"prefix":"","firstName":"Santosh","middleName":"","lastName":"Kumar","suffix":""}],"badges":[],"createdAt":"2025-12-22 10:35:53","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8424019/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8424019/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":99795768,"identity":"87b04cb8-53fc-4333-b085-60238eac52a2","added_by":"auto","created_at":"2026-01-08 13:39:40","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":582620,"visible":true,"origin":"","legend":"Article File","description":"","filename":"BellinequalityPolariqonnaturenew3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8424019/v1_covered_26765250-8c9b-4afc-bd86-5ba9f47f9f68.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Bell–CHSH inequality violation from continuous, non-projective measurements","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8424019/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8424019/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified from continuous time-series data. Here we develop a general theoretical framework showing that Bell–CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak.","manuscriptTitle":"Bell–CHSH inequality violation from continuous, non-projective measurements","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-24 05:55:32","doi":"10.21203/rs.3.rs-8424019/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5a89b6d5-13ca-4b04-ab23-14711ce00821","owner":[],"postedDate":"December 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":60059819,"name":"Physical sciences/Physics/Quantum physics/Qubits"},{"id":60059820,"name":"Physical sciences/Physics/Quantum physics/Quantum information"}],"tags":[],"updatedAt":"2026-01-07T13:05:12+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-24 05:55:32","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8424019","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8424019","identity":"rs-8424019","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}