{ "schemaVersion": "1.6", "slug": "qutrit-complementary-discrimination", "title": "Sharp qutrit discrimination and unavoidable correlation overcounting for complementary measurements", "shortTitle": "Complementary measurements: a sharp 8/3 for qutrits, and no safe choice from n = 4", "url": "https://evidencepress.org/releases/qutrit-complementary-discrimination/", "oneLine": "Two complementary measurements on a qutrit can score at most 8/3 (out of a trivial 4) when telling two orthogonal states apart, proved by an exact certificate with every best case identified; and for a qubit paired with a system of dimension n ≥ 4, every choice of complementary measurements is fooled by some unentangled state (previously known only from n = 164).", "abstract": "Two parties share a quantum system and each can measure in one of two complementary bases, as with position and momentum. In 2014 Schneeloch, Broadbent and Howell proposed that the classical correlations seen in the two resulting outcome tables never add up to more than the total quantum correlation of the shared state, the complementary quantum correlation (CQC) inequality. In 2026 it was refuted in most dimensions by public code repositories (O'Connor; AlperTheKing), by Wang, Wang and Chen and by Qian; it holds for two qubits, and the qubit–qutrit case is open. This candidate settles two follow-up questions. First, for a qutrit and any pair of complementary bases, the summed triangular discrimination of two orthogonal states is at most 8/3 (the trivial ceiling is 4), answering a question of Qian; every equality case is classified, and for the computational and Fourier bases they are tied to the nine Hesse SIC rays. Consequences: choosing one of the two measurements at random recovers at most two thirds of the quantum Fisher information in a two-state estimation task, and Qian's two-ray qubit–qutrit family obeys CQC with margin at least 0.1005 nats. The proof is computer-assisted: an exact rational sum-of-squares certificate checked by a standalone verifier. Second, for a qubit paired with a qudit of dimension n ≥ 4, every prescribed pair of complementary measurements admits a separable state violating CQC, with a uniform certified margin below −9/20,000,000 nats; previously this every-pair statement was known only for n ≥ 164. A related construction covers every pair when both dimensions are at least three. The full qubit–qutrit question and the sharp constant in dimension 5 remain open. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-04", "dateModified": "2026-11-04", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23933731", "doiUrl": "https://doi.org/10.5281/zenodo.23933731", "conceptDoi": "10.5281/zenodo.23933730", "pdfUrl": "https://github.com/ipitchford/qutrit-complementary-discrimination/releases/download/v0.1.0-candidate/qutrit-complementary-discrimination-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23933731", "repoUrl": "https://github.com/ipitchford/qutrit-complementary-discrimination", "releaseUrl": "https://github.com/ipitchford/qutrit-complementary-discrimination/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/qutrit-complementary-discrimination/index.md", "bibtexUrl": "https://evidencepress.org/releases/qutrit-complementary-discrimination/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/qutrit-complementary-discrimination.mp3", "imageUrl": "https://evidencepress.org/assets/og/qutrit-complementary-discrimination.png?v=b76002ff94", "coverArtUrl": "https://evidencepress.org/assets/art/qutrit-complementary-discrimination.svg?v=33cd5c4a8f", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/qutrit-complementary-discrimination.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/qutrit-complementary-discrimination.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted quantum-information candidate. The qutrit theorem is a computer-assisted exact-arithmetic proof, not a proof-assistant formalisation; the correlation theorems have written proofs with exactly certified margins, which are tiny and structural rather than experimental. Internal AI-assisted review in five roles, a separate-model adversarial review, one revision round, a re-review and a separate-model verification audit. External specialist review, independent reproduction and formal verification have not been established." }, "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/23933731", "note": "The versioned research package, manuscript and exact certificate are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "The exact SOS verifier, negative controls, certified-margin script and numerical cross-checks were rerun with Python -I -B, with and without -O, from a fresh copy of the approved package; all passed (verifier about 22 s per mode; ten corrupted runs rejected). The frozen archive was bound to that replay and the verifier and margins rerun from it under -O; GitHub CI reruns the exact checks on Linux." }, { "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 has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No independent reimplementation has been 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 formalisation; the certificate is checked by exact rational arithmetic." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review 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." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "The verifier pins SymPy 1.14.0 and python-flint 0.9.0 (requirements-exact.txt); the margin script needs only the standard library; cross-checks pin NumPy 2.5.4; not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted quantum-information research and computation (exact certificate search and verification)", "AI-assisted internal review in five roles (Claude Code subagents); adversarial review and verification audit by a separate model (OpenAI Codex); clean replay; re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external 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 final manuscript (revision of 3 November 2026 with DOI insertion, 19 pages), its verification scripts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 4 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "mutually unbiased bases", "complementary quantum correlations", "CQC conjecture", "triangular discrimination", "qutrit", "sum-of-squares certificate", "separable states", "quantum mutual information", "Fisher information", "SIC-POVM" ], "keyResults": [ "Sharp qutrit discrimination (Theorem 2.1, computer-assisted proof): for every complementary (mutually unbiased) pair of qutrit bases and orthogonal states u, v, the summed triangular discrimination is at most 8/3, against a trivial ceiling of 4; this answers Qian's question (arXiv:2608.14806v2, Section 11(4)).", "Every best case (Theorem 2.1): equality holds exactly when the two states span a plane through one vector of each basis and their difference is ±√(3/2) times the difference of those projectors; for the computational and Fourier bases these planes are tied to the nine Hesse SIC rays.", "Measurement efficiency (Corollary 2.2): choosing one of two complementary qutrit settings at random recovers at most 2/3 of the quantum Fisher information when estimating the weight of a two-state mixture; the two-ray qubit–qutrit CQC gap rises from 0.0120 to at least 0.1005 nats (Corollary 3.3).", "No safe choice from n = 4 (Theorem 4.1, proof with exact margins): for a qubit with a qudit of dimension n ≥ 4 and every prescribed pair of complementary measurements, some separable state violates CQC, with a uniform margin below −9/20,000,000 nats (−1.058 × 10⁻⁴ at n = 4); previously every-pair failure was known only for n ≥ 164.", "Both dimensions at least three (Theorem 4.7): every prescribed pair admits a separable violation with a dimension-free margin, extending an earlier, unchecked every-pair statement for equal dimensions by O'Connor. The full qubit–qutrit question remains open." ], "reviews": [], "evidencePackage": "Theorem 2.1: a human-readable frame reduction plus an exact rational sum-of-squares certificate (660,609-byte JSON, four Gram blocks of sizes 43, 41, 40, 41) checked by a standalone verifier that rebuilds the degree-12 target polynomial from the six measurement vectors, checks all 1,820 coefficients, proves strict Gram positivity by exact LDL^T elimination and a rational norm bound, and checks every equality identity; five kinds of corrupted certificate are rejected with and without Python -O. Theorems 4.1 and 4.7: analytic proofs with exact rational margin bounds checked by a standard-library Fraction script; NumPy cross-checks are supplementary. Clean replay from a fresh copy, with and without -O, before freezing; GitHub CI reruns the exact checks on Linux.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Schneeloch, Broadbent and Howell (2014). Uncertainty relation for mutual information. Physical Review A. Proposed the complementary quantum correlation (CQC) inequality studied here.", "url": "https://doi.org/10.1103/physreva.90.062119", "doi": "10.1103/physreva.90.062119" }, { "citation": "Berta, Christandl, Colbeck, Renes and Renner (2010). The uncertainty principle in the presence of quantum memory. Nature Physics. Proved, state-independent entropic bounds that remain valid tools.", "url": "https://doi.org/10.1038/nphys1734", "doi": "10.1038/nphys1734" }, { "citation": "Coles, Berta, Tomamichel and Wehner (2017). Entropic uncertainty relations and their applications. Reviews of Modern Physics. Background on complementary measurements.", "url": "https://doi.org/10.1103/revmodphys.89.015002", "doi": "10.1103/revmodphys.89.015002" }, { "citation": "Wootters and Fields (1989). Optimal state-determination by mutually unbiased measurements. Annals of Physics. Mutually unbiased measurements are optimal for full state determination; Corollary 2.2 quantifies their cost for one known pair of states.", "url": "https://doi.org/10.1016/0003-4916(89)90322-9", "doi": "10.1016/0003-4916(89)90322-9" }, { "citation": "OpenAI GPT-5.6 Pro (Zenodo creator; repository owner E. O'Connor) (2026). Exact qutrit counterexample to the CQC conjecture. Zenodo, version v1.1.1. Early exact qutrit counterexample and an earlier, unchecked every-pair statement for equal dimensions, credited in the paper.", "url": "https://doi.org/10.5281/zenodo.21499722", "doi": "10.5281/zenodo.21499722" }, { "citation": "AlperTheKing (2026). cqc-counterexample: exact qutrit counterexample to the complementary-quantum-correlation conjecture. GitHub repository. Early exact qutrit counterexample.", "url": "https://github.com/AlperTheKing/cqc-counterexample", "doi": null }, { "citation": "Wang, Wang and Chen (2026). When complementary measurements count the same classical bit twice: counterexamples to CQC, ECQC, and complementarity-based certification. arXiv:2608.03828v3. Counterexamples in every dimension at least three.", "url": "https://arxiv.org/abs/2608.03828v3", "doi": "10.48550/arXiv.2608.03828" }, { "citation": "Wang, Wang and Chen (2026). Complementary quantum correlations are universal for qubits. arXiv:2608.04916v1. CQC holds for two qubits.", "url": "https://arxiv.org/abs/2608.04916v1", "doi": "10.48550/arXiv.2608.04916" }, { "citation": "Qian (2026). Separable counterexamples to complementary quantum correlations, and why random search missed them. arXiv:2608.14806v2. Posed the sharp qutrit question (Section 11(4)), gave the frame reduction and the every-pair result for n ≥ 164.", "url": "https://arxiv.org/abs/2608.14806v2", "doi": "10.48550/arXiv.2608.14806" }, { "citation": "Iqbal (2026). On the CQC conjecture: a sufficient condition and an extension. Quantum Information Processing. Earlier sufficient conditions for CQC.", "url": "https://doi.org/10.1007/s11128-026-05258-2", "doi": "10.1007/s11128-026-05258-2" }, { "citation": "Tadej and Życzkowski (2006). A Concise Guide to Complex Hadamard Matrices. Open Systems & Information Dynamics. Classification used to reduce every qutrit complementary pair to the Fourier pair.", "url": "https://doi.org/10.1007/s11080-006-8220-2", "doi": "10.1007/s11080-006-8220-2" }, { "citation": "Stacey (2016). SIC-POVMs and Compatibility among Quantum States. Mathematics. The Hesse SIC geometry of the equality cases.", "url": "https://doi.org/10.3390/math4020036", "doi": "10.3390/math4020036" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:qutrit-complementary-discrimination", "attemptIds": [ "ep-attempt:qutrit-complementary-discrimination-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "The best possible two-setting discrimination score for qutrits (8/3, with every best case) and whether any choice of complementary measurements can make two outcome tables a safe lower bound on total correlation for a qubit with a qudit (no, from n = 4).", "scope": "Two orthogonal equal-weight qutrit states and prescribed complementary pairs; qubit–qudit with n ≥ 4 and both dimensions at least three; the full 2 × 3 problem is excluded." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Section 1 and Table 2 separate what is Qian's (frame reduction, determinant condition, attaining configuration, second-variation method), O'Connor's and Wang, Wang and Chen's from the new certificate, equality classification and every-pair results; Section 5 states what is verified how and what stays open.", "remainingRisks": [ "The bounded literature check is not exhaustive priority certification.", "The qutrit proof depends on a computer-checked certificate, not a formalised or short analytic proof." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a quantum-information specialist rerun the exact verifier on an independent machine and check the frame reduction and the second-variation argument behind Theorem 4.1; a symmetry-reduced analytic proof of the qutrit inequality would remove the dependence on the certificate.", "claimCeiling": "Proved 8/3 bound with complete equality classification by an exact computer-checked certificate; every-pair separable CQC violations for qubit–qudit n ≥ 4 and for both dimensions at least three with exactly certified margins. 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