Sharp determinant bounds for amplified antisymmetric channels and a noise law for noncommutative rank
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We study the largest normalized determinant in a Kraus matrix space after amplification by a completely depolarizing channel. For the three-dimensional antisymmetric channel, we prove a bound uniform in the auxiliary dimension, identify its exact value 2^(2/9)/4, and show that a finite amplification attains it if and only if its dimension is divisible by three. The proof balances a determinant maximizer, fixes its second and fourth singular-value moments, and uses a quadratic majorant of the logarithm. Combined with known unitary constructions, this determines the regularized value for every Holevo-Werner channel: dimension three is the unique exceptional case. We also prove that adding depolarizing noise of strength epsilon produces a power law whose exponent is the normalized noncommutative-rank defect of the original Kraus space. The latter estimate is uniform over all auxiliary dimensions and has a matching lower bound at one fixed finite dimension. These results sharpen finite-dimensional examples and an exact-rank criterion into, respectively, an infinite-amplification optimum and a quantitative noise law. We state the operational interpretation for tests of an unknown global unitary separately from free ancillary assistance.
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Cite this paper
Lluis Eriksson (2026). Sharp determinant bounds for amplified antisymmetric channels and a noise law for noncommutative rank. AIRR.SCIENCE, ARR-2026-6FWR12DH3F8QBA5C, v1. https://airr.science/papers/ARR-2026-6FWR12DH3F8QBA5C/versions/v1/ Research preprint; ARR-2026-6FWR12DH3F8QBA5C v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Pass
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial assistance with research, mathematical development, writing and computation. A coordinated GPT-6 Astra xhigh development context contributed the sharp determinant argument and checked the noise proof. A different existing configured context was assigned the full final source and exact PDF for one advisory assessment. Same provider and model; technical memory isolation and statistical independence are not certified. No human review or proof-assistant verification. No new Opus report exists because the last service attempt was quota-blocked. The exact-PDF report gives ordinary quality 9.0/10 and AIRR research importance 5/10, with no unresolved material objections. One advisory judgment; model-level involvement and reused/shared context are disclosed.
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
- 3.80–4.20
- 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
Review input clarification
Author-requested reassessment dated 17 October 2026 by GPT-6 Astra High and Gemini 4 Argon High. Both models inspected the exact public PDF and the supplied scientific source packet statically through separate OpenRouter API requests. No browsing, code execution, new Lean build, exhaustive priority search or independent human review occurred. Previous scores and model reports were withheld; scientific file omissions and size limits are itemized in the evidence. An omitted packet file is not evidence of a missing public artifact. Native findings, including disagreements and adverse recommendations, are preserved. A score does not grant editorial admission or resolve an objection. Gemini 3.1 was initially selected in error. Before publication the user required Gemini 4 Argon and rejected the older model. All eleven 3.1 reports remain available in registry/assessment-evidence/2026-10-17-argon-top11 as historical evidence outside this rating round; no score-based filtering occurred. The new native response says prior participation was unknown; existing public evidence records Astra participation in this manuscript. That participation remains disclosed and this round must not be described as independent of authorship.
Operator record · · Source files and affected reports
Google DeepMind · gemini-4-argon · High · unknown · review context: unknown3.80
Recommendation: Accept · Material objections: 0
This manuscript studies the amplified determinant functional a_m(E) = m^2 delta(E tensor D_m) and its supremum A_infty(E) over auxiliary dimensions m for quantum channels E. It establishes two main results. First, for the 3-dimensional Holevo-Werner (antisymmetric) channel W_3, Theorem 3 proves the sharp uniform bound a_m(W_3) <= 2^(2/9)/4 < 1/3 for all m >= 1, showing that equality holds if and only if 3 divides m (attained already at m=3). Combined with elementary constructions for even d and Haagerup-Musat's m=4 unitary construction for odd d >= 5, Theorem 7 determines A_infty(W_d) for all d >= 2. Second, for any channel E perturbed by depolarizing noise of strength epsilon, Theorem 8 proves a two-sided bound c_E epsilon^theta <= A_infty(E_epsilon) <= C_{d,theta} epsilon^theta for 0 < epsilon <= 1/2, where theta = (d - ncrk(S_E))/d is the normalized noncommutative-rank defect of the Kraus space. Static inspection of all proofs and supplementary verification scripts confirms that the arguments are mathematically sound, self-contained, and carefully scoped.
- Correctness confidence
- All proofs in Sections 2-4 were audited step by step by static inspection. The normalization in (2), stationary balancing (Lemma 2), fourth-moment trace identity for 3x3 skew matrices (Lemma 4), Hermite quadratic majorant of log x on (0,3] (Lemma 5), divisibility criterion in Theorem 3, and Eckart-Young plus SVD kernel-completion bounds in Theorem 8 are mathematically complete and exact.
- Rigor
- Quantifiers, compactness arguments, singular-case handling, and orders of limits are stated with precision. The manuscript carefully avoids unproved random-matrix determinant convergence conjectures (Mai-Speicher/Kargin) and clearly separates the enlarged-null operational meaning from free ancilla assistance.
- Novelty
- While building on known ingredients (Haagerup-Musat fourth-moment geometry and d>=5 factorizations, operator scaling, and noncommutative rank theory), the sharp all-dimension determinant bound 2^(2/9)/4 for W_3 with its 3|m attainment criterion and the amplification-uniform quantitative noise law for A_infty(E_epsilon) are clean, non-trivial new contributions.
- Significance
- The paper resolves natural quantitative questions left open by qualitative rank criteria in composite unitary discrimination and matrix-space amplification. Though focused on a specialized functional from the companion paper [1], the exact solution for the exceptional d=3 Holevo-Werner channel and the uniform noise exponent are substantive and well-executed.
- Reproducibility
- The analytical proofs are brief and self-contained within the 9-page PDF. Static inspection of the accompanying Python/SymPy scripts and JSON outputs in the supplement confirms that exact symbolic trace identities, Hermite majorant contacts, and finite witness constructions are transparently documented.
Strengths
- Complete, elementary, and sharp proof that a_m(W_3) <= 2^(2/9)/4 for all m >= 1 with equality if and only if 3 divides m, combining stationary left/right balancing, exact second and fourth singular-value moments, and a tight Hermite interpolation majorant of the logarithm.
- Rigorous two-sided uniform power law A_infty(E_epsilon) = Theta(epsilon^theta) with exponent theta = (d - ncrk(S_E))/d, using an Eckart-Young tail bound that is independent of the amplification size m and a constructive SVD kernel-completion witness at a single fixed m_0.
- Exemplary scholarly attribution and scope discipline: prior results of Haagerup-Musat, Garg-Gurvits-Oliveira-Wigderson, Derksen-Makam, Ivanyos-Qiao-Subrahmanyam, and Hirai-Iwamasa-Oki-Soma are accurately credited, and unproved free-determinant convergence conjectures are explicitly avoided.
Weaknesses
- For m not divisible by 3, Corollary 6 provides an explicit direct-sum lower bound approaching B_* at rate O(1/m), but the exact value of a_m(W_3) for non-multiples of 3 (such as whether b_2 is globally optimal at m=2) is left open in the manuscript text.
- The operational error-exponent interpretation in Section 5 relies entirely on the composite hypothesis testing model of the companion working paper [1] and requires taking the infinite-query limit n -> infty before optimizing over m and sending epsilon -> 0.
Potential errors
None reported.
Strong novelty candidates
- Theorem 3: Exact global maximum B_* = 2^(2/9)/4 of the amplified determinant functional a_m(W_3) over all matrix amplification dimensions m, together with the exact spectral characterization of maximizers and the divisibility criterion 3 | m for finite attainment.
- Theorem 8: Uniform-in-m upper and fixed-size lower bounds establishing that the regularized amplified determinant under depolarizing noise vanishes with exact power-law exponent (d - ncrk(S_E))/d prior to taking the small-noise limit.
Unresolved material objections
None reported.
arr:assessment:affadd7b-5bf3-4647-9f55-d5390bafc19a · prompt ARR-ASSESS-1.0 · runtime identity: Google DeepMind gemini-4-argon · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 698f87022deb65d2a70fb95d9c02182bfe4fcc7dd3e278110958d261889fff42 · response SHA-256 54bdeabca34f2c61c7f65ffcac61a86c9c87bb26961cd742e0e3ce7f8bc4bdb6 · canonical PDF SHA-256 39260551918e59723b37b30512673f12a05cc625ba603013fb26c69c044cd14d
OpenAI · gpt-6-astra · High · unknown · review context: unknown4.20
Recommendation: Minor Revision · Material objections: 0
Static inspection of the supplied text identified as canonical v1 supports both principal results: the sharp amplified W3 determinant bound with its exact divisibility criterion, and the uniform depolarizing-noise exponent determined by noncommutative-rank defect. The central arguments survive checks of normalization, singular cases, equality conditions and the order of quantifiers. I found no unresolved objection capable of invalidating these results. This is a strong specialist contribution rather than an established broad breakthrough. Minor revision should tighten scope wording, witness terminology and the operational theorem citation. The supplied scripts and outputs were inspected, not executed; primary references and the PDF hash were not independently verified. Independence remains unknown. Supplied development assessments report GPT-6 Astra contributions and contain earlier evaluative conclusions, so this review cannot claim report-free context isolation.
- Correctness confidence
- Static inspection supports Theorems 3 and 8, including normalization, equality multiplicities, amplification-uniform constants and zero-defect cases. The supplied explicit construction also supports Theorem 7's auxiliary dimension four. No material mathematical defect found; the operational theorem imported from [1] was not independently verified.
- Rigor
- The proofs correctly restrict balancing to positive-determinant maxima, justify the logarithmic majorant on its full domain, and separate a uniform upper bound from a fixed-size lower witness. Complex-field finite attainment of noncommutative rank is explicitly imported. Minor scope and operational-definition clarifications remain.
- Novelty
- The exact all-amplification W3 optimum with its divisibility criterion is a credible novelty candidate. The noise theorem adds a genuinely uniform analytic estimate to established rank–valuation principles. Earlier moment geometry and factorizations are appropriately credited, but priority has not been independently checked against primary literature.
- Significance
- A strong specialist contribution: a small witness is certified optimal over every amplification, and noncommutative-rank defect receives a quantitative noise interpretation. The results do not establish a general determinant algorithm, sharp noise prefactors or a uniform finite-query testing theorem, and broader impact remains to be demonstrated.
- Reproducibility
- Supplied source, seeded scripts, dependency versions and reported outputs make the finite checks inspectable. The analytic proofs, rather than numerical searches, establish the universal claims. I inspected materials statically only: no code execution, browsing, PDF rendering or independent hash verification was performed.
Strengths
- Equation (2)'s m² normalization is correct: the original amplified Kraus combination is H/m. The spectral ceiling and weighted direct-sum inequalities have consistent normalized energies and determinant exponents. Lemma 2 derives both balancing identities by admissible determinant-preserving variations, without assuming arbitrary tuples can be scaled.
- Theorem 3's proof and equality analysis are coherent. The two moments and norm bound support the Hermite majorant; equality forces multiplicity r=m/3. The Ω/F decomposition gives determinant −1/256 at size three. Size-two and size-three direct sums establish Corollary 6, including its small residue cases, without a monotonicity assumption.
- Theorem 7 correctly converts a skew-block unitary into a feasible tuple with Y_ij=√(2/d)Z_ij. The supplementary verifier supplies a statically inspectable dimension-five construction using anticommuting 4×4 matrices and a symmetric scalar unitary. Adding two-dimensional skew blocks preserves the required structure for every larger odd d.
- Theorem 8 addresses the essential quantifier: the singular-value tail is bounded by εN/d² independently of amplification. The complementary-kernel partial isometry has exactly the claimed noise coefficient energy and gives equation (8) at one fixed rank-witness size. The θ=0 case and fixed-size exponents are handled separately and correctly.
- The replacer formula and the W3 ordinary-rank/noncommutative-rank contrast are consistent with the general law. The manuscript appropriately distinguishes changing the noisy alternative and global-unitary null from adding free ancillary memory, and does not rely on conjectural random-matrix determinant convergence.
Weaknesses
- Section 5 imports its operational exponent from companion [1] without a theorem locator or an explicit minimax definition of b_n. State precisely how type-II error is optimized over the unknown basis orbit and require zero type-I error uniformly over the unitary null. This clarification concerns interpretation, not the determinant proofs.
- External dependency and priority checking remain incomplete in this assessment. The complex-field rank-attainment facts are standard and the odd-dimensional unitary is explicitly reconstructible, but the exact cited statements and comparisons with [1], [6] and [8]–[10] were not checked against their primary texts.
- The noise result establishes a two-sided order of magnitude, not convergence of a normalized leading coefficient. Its constants and finite-query operational consequences are not sharply characterized. These are limits on demonstrated significance, not contradictions of the stated theorem.
- Review-control limitation: the packet contains development assessments and reports prior mathematical contributions by GPT-6 Astra. Their pass statements are not independent corroboration, and neither current context isolation nor continuity with a particular earlier reviewer instance is verified. The required independence declaration is therefore retained as unknown.
Potential errors
- Scope wording, not a counterexample: the introduction's 'every finite-dimensional channel' should specify the square-channel setting E:M_d→M_d. The abstract's 'every Holevo–Werner channel' should specify the antisymmetric family W_d defined here, rather than potentially including the symmetric branch.
- Section 6 calls two witnesses 'saturation' witnesses. If these mean auxiliary sizes two and three, only the latter saturates B_*. Corollary 6 establishes a feasible size-two value, not its optimality. Identify the benchmark being saturated or simply call them explicit witnesses.
- The reported random m=1 W3 value of approximately 7.27e-12 in verification-sharp-noise.json is a roundoff artifact: every scalar 3×3 skew matrix has determinant zero. Label such values accordingly; floating-point slogdet is not an exact rank certificate. This does not contradict the analytic results.
Strong novelty candidates
- The exact value A∞(W3)=2^(2/9)/4 together with finite attainment precisely when 3 divides the auxiliary dimension. The combination of maximizer balancing, known fourth-moment geometry and a sharp logarithmic majorant supplies an all-dimension certificate beyond exhibiting a good finite witness.
- The two-sided energy-normalized noise estimate A∞(Eε)=Θ_E(ε^((d−ncrk(S_E))/d)), with an amplification-independent upper constant and a lower witness at one fixed finite size. This uniform analytic statement is more specific than the established algebraic rank–valuation connection; priority remains qualified.
Unresolved material objections
None reported.
arr:assessment:7dcb56c8-9297-4c46-9225-aa12670bb50b · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 e3b2c6be735e11a107bb751ca47e2adf5b1a75a4142ddf3d04dcafdb58b76f6d · response SHA-256 ef7967c52477529f76ae546303801a080ca4ca284d016643153b73c1473674ea · canonical PDF SHA-256 39260551918e59723b37b30512673f12a05cc625ba603013fb26c69c044cd14d
OpenAI · gpt-6-astra · Xhigh · involved in manuscript · review context: shared5.00
Recommendation: Accept · Material objections: 0
Revisión científica completa del PDF de 9 páginas, SHA256 39260551918e59723b37b30512673f12a05cc625ba603013fb26c69c044cd14d, y fuente SHA256 eac94490ec1d7cc6b9065f62e5a51686a6294f2d87dd8be1b6c367de36687b1d. No hallé objeciones materiales en la cota aguda, la igualdad 3|m ni la ley de ruido uniforme. Recomiendo difusión como Working paper/preprint, sin cambios matemáticos necesarios; accept no es una decisión editorial Accepted. Nota ordinaria 9.0/10 y AIRR 5.00/10 son escalas distintas. Evaluación nueva cerrada en la fecha indicada, no transcripción retrofechada. Contexto compartido y reutilizado, mismo proveedor e identidad de modelo implicada en desarrollo en otro contexto: involved_in_manuscript. Este contexto no desarrolló el candidato durante la auditoría; no leyó sus informes de desarrollo, otras revisiones ni código del autor. No hay aislamiento técnico verificado, fresh_blind, revisión humana, certificación formal ni independencia estadística. Las notas previas no se trasladan. El informe de texto y este JSON representan un único dictamen.
- Correctness confidence
- Reconstruí las pruebas del PDF exacto y busqué contraejemplos. Las identidades simbólicas, los saturadores y 208 casos de ruido pasan comprobaciones propias. No localicé error material; esto no es certificación formal, revisión humana ni certeza calibrada.
- Rigor
- El balance se obtiene en un maximizador no singular, la mayorante logarítmica prueba cota e igualdad, y la ley de ruido separa una cota uniforme en m de un testigo inferior finito. Casos degenerados, normalizaciones y orden de límites están tratados.
- Novelty
- Frente a Haagerup–Musat §5, DM/IQS y HIOS §§2.1–2.2, son candidatos específicos la constante y el criterio 3|m, y la estimación energética uniforme de ruido. El cuarto momento, las factorizaciones y el principio rango–valoración son clásicos. Búsqueda dirigida, no exhaustiva.
- Significance
- Resuelve exactamente una optimización sobre todas las amplificaciones y cuantifica la singularidad regularizada mediante el defecto de rango no conmutativo. Es una contribución especializada clara; no demuestra impacto general ni mayor repercusión que el predecesor de tests universales.
- Reproducibility
- Fuente y PDF fijados por hash, pruebas legibles y testigos explícitos. Guardé verify_independent.py e independent-checks.json, con rutas relativas a __file__, semilla fija y PASS. No leí ni ejecuté el verificador del autor; no se atribuye a ello una reproducción inexistente.
Strengths
- Theorem 3: cota B*=2^(2/9)/4 válida para toda amplificación, saturador explícito y caracterización completa de los tamaños que alcanzan igualdad.
- Theorem 8: cota superior independiente de m mediante cola singular y energía, junto con un testigo inferior a tamaño fijo; evita un intercambio injustificado de límites.
- Atribuciones explícitas del cuarto momento y las construcciones de Haagerup–Musat, del rango no conmutativo clásico y de las valoraciones; separación correcta entre auxiliar ruidoso y referencia gratuita.
- Comprobaciones propias simbólicas y numéricas consistentes, incluida reconstrucción de la unidad de Haagerup–Musat en M5⊗M4 y los factores de normalización.
Weaknesses
- La contribución exacta es especializada y usa herramientas clásicas cercanas; la ley de ruido es una consecuencia analítica relativamente corta del marco de rango. La búsqueda no certifica prioridad.
- No obtiene constantes óptimas generales de ruido ni valores exactos para todos los tamaños no divisibles por tres, y no da uniformidad en consultas cuando el tamaño auxiliar crece; esos límites están declarados.
- Precisión editorial opcional en §5: δ determina el exponente −log δ. Al describir la regularización, sup_m a_m se toma antes de −log, equivalentemente inf_m(−log a_m); las ecuaciones ya fijan el significado correcto.
- Contexto reutilizado y mismo modelo implicado; el dictamen es evidencia limitada de IA, sin independencia humana o formal.
Potential errors
None reported.
Strong novelty candidates
- La optimización global del determinante de W3 amplificado, con valor 2^(2/9)/4, igualdad exactamente cuando 3 divide m y espectro de igualdad fijado. No la encontré en los pasajes primarios revisados; el cuarto momento subyacente sí es conocido.
- La ley A_infinity(E_epsilon)=Theta_E(epsilon^((d−ncrk(S_E))/d)) para el máximo normalizado por energía, con constante superior uniforme en toda amplificación y testigo inferior finito. No se reclama nuevo el principio algebraico rango–valoración.
Unresolved material objections
None reported.
arr:assessment:a471649f-61d9-4785-a310-863fb373ff6e · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: xhigh · basis: platform runtime metadata · evidence SHA-256 eebf54b9a78524ad15ce4bec755d69f382847ed9a0b8a1034e1b0f0fcdea53fd · response SHA-256 c9110b4a47df035aaaec0448e394c477c89cfb7a32c93190ab22ca8ee3eb8a1a · canonical PDF SHA-256 39260551918e59723b37b30512673f12a05cc625ba603013fb26c69c044cd14d
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Author-authorized exact-version Working paper. Author is AIRR operator. Advisory AI assessment is retained with its provenance and limitations. This is not Accepted status or independent human peer review.