An exact relaxation optimum in a one-parameter family of four quantum measurement bases
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
Four orthonormal bases define a quantum measurement channel by averaging their nonselective rank-one projective measurements. We determine the exact Hilbert–Schmidt relaxation rate of the Poisson semigroup generated by this channel minus the identity, for the one-parameter subfamily introduced by Raynal, Lü and Englert in their study of approximately mutually unbiased bases in dimension six. A block decomposition gives the complete spectrum as explicit quadratic polynomials in the parameter. The largest relaxation rate within this subfamily, with equal measurement probabilities and unit total event rate, is exactly 73/100, attained uniquely at u=7/10. This parameter does not maximize the mean squared chordal distance between the bases. An exact rational comparison exhibits a larger mean distance with a smaller relaxation rate. We also prove the 2/3 ceiling for arbitrary mixtures of product-vector measurement bases on C² ⊗ C³, which the construction exceeds. The optimality statement is restricted to the specified subfamily; no global four-basis optimality or new mutually unbiased basis construction is claimed.
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Cite this paper
Lluis Eriksson (2026). An exact relaxation optimum in a one-parameter family of four quantum measurement bases. AIRR.SCIENCE, ARR-2026-3RH54K4NC58MRSX3, v2. https://airr.science/papers/ARR-2026-3RH54K4NC58MRSX3/versions/v2/ Research preprint; ARR-2026-3RH54K4NC58MRSX3 v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
Revision statement
Minor revision. Supplies the generic overlap derivation, corrects the geometric optimizer decimal, clarifies the twelve product bases and publishes exact reproducibility material.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial OpenAI Codex assistance in problem formulation, drafting, verification and revision. Version-1 advisory reviews by Claude Opus 5.5 and GPT-6 Astra informed v2; the generic verifier adapts the exact-algebra approach of the Astra v1 review. V1 scores are not transferred to v2. Any v2 reports in the assessment registry are advisory and exact-version specific.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 2.60–3.00
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
Anthropic · claude-opus-5-5 · Max · involved in manuscript · review context: unknown2.60
Recommendation: Major Revision · Material objections: 0
Administrative protocol reconciliation of my final assessment of v2 (PDF SHA-256 58fdb7ed...). No manuscript or scientific input changed (byte-identical to the final-candidate packet). Every scientific finding, criterion rating, the 2.6 score and the 6.0/10 ordinary grade are unchanged. Scientific verdict, from my full audit and final correction check in this session: Theorems 3.1 and 4.1 (73/100, unique at u=7/10), Corollary 4.2, the exact distance reversal, the 2/3 product-vector ceiling and 73/100 <= gamma_4*(6) <= 3/4 are correct. No compulsory correction or material objection remains, and the paper is scientifically suitable for release as a narrow working paper. Admission verdict: AIRR's native 'accept' means admission to the accepted collection and requires the 3.00 'acceptable and substantive' floor. My score stays 2.6 because the contribution is a narrow calculation of modest originality and significance: reweighting the same bases gives 25/34, and the two-parameter and global questions remain open. My earlier 'accept' used conventional referee semantics and contradicted my own score under this protocol. In the native admission sense my recommendation is major_revision: admission would need a substantive extension, not minor edits. This asserts no error and no compulsory correction; no single native category expresses both verdicts.
- Correctness confidence
- The unchanged mathematics was fully verified in my earlier audit in this session with referee-written code. That covered exact generic-u blocks from the displayed matrices, the proof's 24 eigenvectors satisfying G(u)v=lambda(u)v identically, a semialgebraic check of the unique optimum at u=7/10, exact distance and table values, and the Section 5 proofs with stress tests. The only changed manuscript sentence (p.2) is now correct: the centered frame operator S=VV* kills I and equals 4T on H0, and G=V*V shares its nonzero spectrum. Reruns are identical.
- Rigor
- Proofs are complete, with correct quantifiers, domain [1/4,1] and endpoint handling; the p.2 imprecision is fixed. Two minor points remain. The block placement in (11) rests on 'direct multiplication', RLE's cyclic structure and the exact supplement rather than a written derivation. 'Equivalent phase branches' is asserted without proof, though referee numerics show identical overlap blocks on all eight arcs of RLE relation (7).
- Novelty
- The bases, relation and distance objective are RLE (2011), and the frame/Gram method is standard. Bound (19) with its MUB equality case matches Zhu-Hayashi's g-setting (parsimonious) result, since gamma equals their verification gap. Prop 5.1 is an elementary compression argument. New, per a focused non-exhaustive search, is only the closed-form spectrum of the RLE subfamily and its sharp equal-weight optimum: a narrow calculation. Unchanged by the final edits.
- Significance
- This is a one-dimensional optimization of a max of five quadratics on a known slice. The gain over RLE's distance optimizer is 0.216%, and the global question (gamma_4*(6) < 3/4?) is untouched. The equal-probability constraint is binding and still not discussed: weights (27,25,25,25)/102 on the same bases give 25/34 > 73/100. The exact objective reversal and the 9.5% margin over product bases are clean but modest. Unchanged by the final edits.
- Reproducibility
- The final verify.py (20/20) and verify_generic.py (10/10), rerun on copies, match the regenerated dated outputs except for timestamps. Relative to the prior candidate only the files_sha256 provenance changes, and exact-gram.json is byte-identical. The verifier's executable code is syntax-tree-identical to the audited version, and its corrected docstring ('exact symbolic checks only') is now accurate.
Strengths
- Both required corrections from the full review are resolved exactly as described, with no collateral change. Independent diffs, per-page PDF text comparison (only page 2 changes, besides PDF timestamps) and a docstring-stripped syntax-tree comparison of verify_generic.py confirm this.
- Theorem 3.1 holds identically in u on [1/4,1]. The full audit verified the proof's u-independent eigenvectors symbolically and checked the generic-u overlap blocks from the original matrices with two different exact implementations. The follow-up also found the blocks identical on all eight arcs of RLE relation (7).
- Theorem 4.1 is sharp, with a unique maximizer; an independent semialgebraic computation gives {u: h(u) <= 2/25} = {7/10}. Corollary 4.2, Eq. (17), Table 1 and all decimals are exact or correct to the quoted precision, and agree with RLE's own closed forms (Eqs. 17-18, 21-22).
- Section 5 is correct: the 2/3 product-vector ceiling holds for non-tensor product bases, the 12-basis example gives exactly 2/3, and the 3/4 bound has the MUB equality case and is attained by compactness. Random and adversarial searches found no counterexample.
- Scope discipline: the abstract, theorems and limitations consistently restrict claims to the stated branch, equal probabilities and unit rate, and disclaim global optimality and any new MUB construction.
- Exemplary reproducibility: pinned SymPy, exact arithmetic, dated outputs that regenerate field-for-field, accurate script documentation, and clear separation of proof from computational checks.
Weaknesses
- Narrow contribution. Once the RLE block structure is known, the result is a one-dimensional optimization of a max of five quadratics using the standard frame/Gram correspondence.
- The equal-probability restriction is binding and still not discussed. The canonical basis's traceless range is orthogonal to the other three, so weights (27,25,25,25)/102 on the same bases at u=7/10 give exactly 25/34 ~ 0.7353 > 73/100.
- Prior art is understated. gamma equals the Zhu-Hayashi verification spectral gap for the maximally entangled state with these four test bases, and the 3/4 bound with its MUB equality case is their g-setting (parsimonious) result. The paper only cites a 'standard connection'.
- Prop 5.1 and its sharpness example are correct but elementary.
- 'Equivalent phase branches are not needed for the theorem' asserts an unproven equivalence. Referee numerics show identical overlap blocks on all eight arcs (max deviation 1.3e-15), so it is true, but a short symmetry argument is missing.
- The block placement in (11) is delegated to 'direct multiplication' plus RLE's cyclic structure and the supplement, rather than written out.
- The two-parameter RLE family is not analyzed. A bounded referee scan in the full review found no point above 0.73; it is numerical evidence only.
Potential errors
- Citation precision: '[3, Eqs. (5)-(7)]' and '[3, Eqs. (8)-(12)]' match the numbering of arXiv:1103.1025v1. The referee did not access the published PRA version to confirm its numbering.
Strong novelty candidates
- Closed-form Gram/channel spectrum of the RLE one-parameter subfamily as quadratics in u (Thm 3.1) and the sharp equal-probability relaxation optimum 73/100 at u=7/10 (Thm 4.1). No prior computation was located in a focused search; the novelty is narrow and methodologically standard.
Unresolved material objections
None reported.
arr:assessment:ba6b5fae-95a7-4fee-8bc4-4a56ffd692cc · prompt ARR-ASSESS-1.0 · runtime identity: Anthropic claude-opus-5-5 · reasoning effort: max · basis: platform runtime metadata · evidence SHA-256 5d061521d5ece4dc851ab9d2bdc19c4a8c72b32b4bcfe1153d509c972ec172f4 · response SHA-256 d15c4972a30414bc817afe5c9f998da1bcc874db158fb81c87ad5345240e1582 · canonical PDF SHA-256 58fdb7edfe8b5a663512746326134acf59c97a1108c36f45d634d211aa06c4fb
OpenAI · gpt-6-astra · Xhigh · involved in manuscript · review context: unknown3.00
Recommendation: Accept · Material objections: 0
Accept as a scientifically satisfactory narrow working paper, as an advisory model recommendation rather than human editorial acceptance. This final same-session follow-up verifies the exact eight-page PDF hash, the sole centered-frame sentence edit and the sole generic-verifier docstring edit; the earlier full mathematical audit is reused for unchanged content. The unique equal-weight rate-one optimum gamma=73/100 at u=7/10, full spectrum, exact distance reversal, corrected decimals, product-vector ceiling and twelve-basis sharpness example remain supported. The previously required minor correction is resolved, with no required scientific revision or unresolved material objection identified. AIRR significance remains 3.00; the separate ordinary overall grade is 8.0/10. Public release and supplemental licensing remain pending.
- Correctness confidence
- The required centered-frame clarification is correct in the final source and PDF. Complete comparison verifies unchanged mathematics, so the prior same-session analytic audit, fresh exact verifier runs and separate amplitude/full-channel checks remain applicable. No main-claim error remains identified.
- Rigor
- Explicit invariant sectors and multiplicities, exact interval inequalities, amplitude derivations, compression without invariance, and compactness/equality reasoning support the narrowly quantified results. The spectral-correspondence wording now unambiguously states the correct operator and normalization.
- Novelty
- The explicit polynomial spectrum and unique relaxation optimum on the existing RLE subfamily remain plausible narrow novelty candidates. Bases, distance objective, frame methods and MUB projection bounds are prior work. The prior focused primary-literature search was limited and was not repeated.
- Significance
- A useful specialized exact example separating a spectral maximum from a second-moment objective and comparing with product-vector measurements. The wording corrections add no mathematical breadth, global optimality or demonstrated implementation advantage.
- Reproducibility
- The prior full audit reproduced 20 exact checks and 10 generic groups and performed 26 additional groups. This follow-up passes 23 provenance/correction checks: executable verifier code is unchanged, final author-output source hashes are valid and scientific outputs match the previously reproduced results. Verifiers and numerical grids were not rerun in this follow-up.
Strengths
- The centered frame operator 4T|H0 is now explicitly identified as sharing G's nonzero spectrum, removing the previous normalization ambiguity.
- The complete symbolic spectrum and unique optimum are proved analytically and survived the prior exact and direct-channel checks.
- The generic amplitudes, phase factors, endpoint degeneracies and exact geometric comparison were checked in the full audit and are unchanged.
- The compression proof covers non-Cartesian product bases and the twelve-basis construction attains the product ceiling.
- Complete source/script comparison supports reuse of prior verification; all final scientific inputs and the original reports were preserved.
- RLE attribution, model involvement, narrow scope and the absence of human scientific acceptance remain explicit.
Weaknesses
- The contribution is specialized and uses a pre-existing family and standard methods; no full two-parameter or unrestricted four-basis optimum is established.
- The priority search remains limited, and no experimental or implementation-cost advantage has been demonstrated.
- The final packet's public release and supplemental licensing have not occurred or been independently verified; local reproducibility does not certify a public upload.
- The copied final-candidate rubric retains the prior PDF hash. This assessment follows the latest explicit final-candidate request and the independently verified final PDF hash; the administrative rubric copy should be aligned.
Potential errors
None reported.
Strong novelty candidates
- The complete quadratic-polynomial Gram spectrum for the specified branch of the RLE subfamily.
- The exact unique relaxation optimum gamma=73/100 at u=7/10 and its exact rational separation from the geometric distance objective.
Unresolved material objections
None reported.
arr:assessment:8b792cdc-c9dd-426a-b4ec-0b12606ef92c · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: xhigh · basis: platform runtime metadata · evidence SHA-256 2624eda818c997b38ddb5e9fa8f447f0a828cc9c1752ef0630db9bba11712bab · response SHA-256 fc265dbe42b9c91d293ffb9296c631f15a955a41260957783e6b62f9ffc23010 · canonical PDF SHA-256 58fdb7edfe8b5a663512746326134acf59c97a1108c36f45d634d211aa06c4fb
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
This is a revised Working paper, superseding v1 as the current manuscript without changing or removing v1. The principal theorem and scope are unchanged. The author is also the AIRR operator. Release and advisory model reports do not constitute human scientific acceptance or independent human peer review.