{ "schemaVersion": "1.6", "slug": "entropy-continuity-stability", "title": "Sharp entropy continuity and stability under trace and operator constraints", "shortTitle": "Exact distances are not error budgets", "url": "https://evidencepress.org/releases/entropy-continuity-stability/", "oneLine": "Two-distance entropy bounds gain equality criteria, a finite error-budget calculation and quantitative stability around one common optimal pair.", "abstract": "For finite-dimensional quantum states, exact trace and operator distances pose a different optimization problem from upper bounds on those distances. Building on published entropy inequalities and capped-spectrum optimization, this candidate characterizes equality, evaluates the upper-cap modulus by endpoints and quadratic roots, and proves that near-extremizers at fixed positive trace distance are close to one common extremal pair. The square-root deficit exponent is sharp, but the coefficient is not claimed optimal and deteriorates as trace distance tends to zero.", "datePublished": "2026-11-06", "dateModified": "2026-11-06", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23997749", "doiUrl": "https://doi.org/10.5281/zenodo.23997749", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/entropy-continuity-stability/releases/download/v0.1.0-candidate/entropy-continuity-stability-v0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23997749", "repoUrl": "https://github.com/ipitchford/entropy-continuity-stability", "releaseUrl": "https://github.com/ipitchford/entropy-continuity-stability/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/entropy-continuity-stability/index.md", "bibtexUrl": "https://evidencepress.org/releases/entropy-continuity-stability/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/entropy-continuity-stability.mp3", "imageUrl": "https://evidencepress.org/assets/og/entropy-continuity-stability.png?v=7a298856ea", "coverArtUrl": "https://evidencepress.org/assets/art/entropy-continuity-stability.svg?v=3960b37909", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/entropy-continuity-stability.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/entropy-continuity-stability.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Internal checking is not external peer review, unaffiliated reproduction or formal verification; bounded source comparison does not guarantee priority." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23997749", "note": "All three frozen assets are publicly available from GitHub and Zenodo with matching full-download hashes; archive-parity receipt retained." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Six frozen-package gates passed in the retained research receipt, including clean normal and optimized producer-side replay with guard tests. Finite examples and floating diagnostics do not prove the universal theorems." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No unaffiliated independent reimplementation established." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant verification claimed." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted checking is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "The frozen source package, pinned environment and checkers passed clean producer-side normal and optimized replay. No unaffiliated reproduction or formal verification is claimed." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical development and revision", "Internal AI-assisted source and proof checking", "Codex publication preparation" ], "humanRole": "Publication authority only; no human mathematical contribution inferred.", "reviewStatus": "Internal AI-assisted review is not unaffiliated peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the revised manuscript and refs.bib on 6 November 2026. Published GitHub and Zenodo assets verified against the frozen local payload by full public downloads of all three files.", "grandfatheredAtSchemaVersion": null, "keywords": [ "quantum entropy", "continuity bound", "trace distance", "operator norm", "stability", "equality cases" ], "keyResults": [ "All equality cases for the prescribed trace/operator-distance entropy optimum, whose formula and attaining family have published predecessors.", "A continuous upper-cap modulus obtained from finitely many endpoints and quadratic-root candidates; direct substitution of caps into the exact-distance formula can underestimate the entropy change.", "At fixed positive trace distance, both states lie within a square-root deficit bound of one common extremal pair, without a rank-gap assumption.", "The square-root exponent is sharp; no optimal coefficient or uniform stability as trace distance vanishes is claimed.", "Restricted common-marginal conditional-entropy attainment and fixed-difference consequences retain their stated dimensional assumptions." ], "reviews": [], "evidencePackage": "Written proofs, finite arithmetic examples, floating-point diagnostics and targeted semantic guard tests. Computations do not establish the universal theorems.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Sason (2013). Entropy Bounds for Discrete Random Variables via Maximal Coupling. IEEE Transactions on Information Theory 59, 7118–7131.", "url": "https://doi.org/10.1109/TIT.2013.2274515", "doi": "10.1109/TIT.2013.2274515" }, { "citation": "Jabbour and Datta (2024). Tightening continuity bounds for entropies and bounds on quantum capacities. IEEE Journal on Selected Areas in Information Theory 5, 645–658.", "url": "https://doi.org/10.1109/JSAIT.2024.3469929", "doi": "10.1109/JSAIT.2024.3469929" }, { "citation": "Audenaert, Bergh, Datta, Jabbour, Capel and Gondolf (2025). Continuity bounds for quantum entropies arising from a fundamental entropic inequality. IEEE Transactions on Information Theory 71, 7029–7038.", "url": "https://doi.org/10.1109/TIT.2025.3586478", "doi": "10.1109/TIT.2025.3586478" }, { "citation": "Audenaert (2007). A sharp continuity estimate for the von Neumann entropy. Journal of Physics A: Mathematical and Theoretical 40, 8127–8136.", "url": "https://doi.org/10.1088/1751-8113/40/28/S18", "doi": "10.1088/1751-8113/40/28/S18" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:entropy-continuity-stability", "attemptIds": [ "ep-attempt:entropy-continuity-stability-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "stability-margin", "description": "Finite-dimensional entropy cap evaluation and common-pair near-equality stability.", "scope": "Exact trace/operator constraints or upper caps as separately defined; stability conditioned on positive trace distance." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Density-matrix distances determine Jordan spectra; retained entropy deficits control one common extremal pair. Upper caps require optimization over actual distances.", "remainingRisks": [ "Bounded prior-art search does not establish exhaustive priority.", "Floating diagnostics are not rigorous numerical enclosures.", "Stability is not uniform as trace distance tends to zero; conditional-marginal scope is restricted." ] }, "humanJudgmentGates": [ "Publication authority is not scientific endorsement.", "External specialist validation remains open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "priorityAssessment" ], "nextAction": "External source-led checking of equality necessity, finite cap optimization and paired stability, distinct from producer checker replay.", "claimCeiling": "Written proof and scoped producer checks; no external validation, formal verification or absolute originality guarantee." }, "impactClaims": [ { "id": "entropy-continuity-stability-no-impact", "aim": "science", "outcome": "Potential research reuse of entropy error-budget and stability formulas.", "setting": "Finite-dimensional quantum information theory", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched research-workflow comparison.", "estimand": "No acceleration or downstream impact effect estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. 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