Sharp minimax losses for energy-constrained quantum reset channels
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
An unflagged reset channel transmits its input with probability lambda and otherwise replaces it by a fixed state. We study its worst entanglement-assisted capacity when the input and reset have separate mean-energy budgets. A uniform estimate reduces the infinite-dimensional quantum minimax problem to a classical concave-convex entropy game, with absolute error tending to zero independently of the reset budget. For the oscillator, explicit positive discrete certificates solve the fixed energy-ratio limit of that game. The resulting loss curve is parametrized by beta and digamma functions. An asymptotically least-favorable reset family has an atom at zero below a branch junction, whereas the constructed higher-ratio family is diffuse. At every fixed interior ratio, the thermal reset is strictly less harmful than the minimax reset; the limiting loss has square-root onset and quadratic contact with its plateau. At reset-to-input energy ratio at least one the loss is the familiar binary entropy h2(lambda). The proof uses global entropy inequalities and feasible distributions rather than an unproved exchange of limit and optimization. Finite probability-game checks support the formulas but do not supply the proof. No efficient coding theorem, unknown-channel universal code or exact finite-energy capacity is asserted.
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Cite this paper
Lluis Eriksson (2026). Sharp minimax losses for energy-constrained quantum reset channels. AIRR.SCIENCE, ARR-2026-2EXHFGWW0S8MG8QA, v1. https://airr.science/papers/ARR-2026-2EXHFGWW0S8MG8QA/versions/v1/ Research preprint; ARR-2026-2EXHFGWW0S8MG8QA v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial AI assistance in research, proofs, code and writing, directed by the named author. GPT-6 Astra xhigh mathematical audits and private assessment; limitations and adverse outcomes retained. No human peer review, Lean proof, certified worldwide priority or AIRR accepted admission is claimed.
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
- 1
- Range
- 4.50–4.50
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete
Review input clarification
One assessment only: the closed private exact-PDF Astra review53, rendered by the same reviewer into the supported public schema, without re-evaluation or score changes. Importance4.5 and quality5.5 remain distinct. assessed_at is the original archive closure; the AIRR IDs were allocated later and ARR-ASSESS-1.0 names this representation schema, not a claim that the original private task used this public prompt. The first schema rendering exceeded the1600-character summary limit and was never registered; v2 changes only summary length, preserving all other fields. Both responses and provenance files, the original review, verdict and bibliography addendum are preserved. The model family participated in development, the filesystem was shared, and this formatting follow-up reuses its own closed report. No verified technical isolation or external runtime attestation is claimed; structured runtime/context fields are omitted because an honest supported attestation basis is unavailable. This is not a new vote, fresh independent review, two-model admission or editorial acceptance. Native review evidence is available at https://github.com/arr-research/arr-research.github.io/tree/main/registry/assessment-evidence/2026-10-24-reset53 . The notice source_files entry binds the unchanged canonical paper metadata; native response hashes remain separately recorded in the assessment and provenance files.
Operator record · · Source files and affected reports
- metadata.json · SHA-256
49fdace4a0c2ad28d56bb09641a3dd90cbcd56ba84d73b738e66925d6f3dbaa1
openai · gpt-6-astra · involved in manuscript · review context: unknown4.50
Recommendation: Accept · Material objections: 0
This is a format conversion of closed private review 53 and its addendum for the identical PDF, not a new evaluation. The original task began with fresh context; this follow-up only formats that review. The model family participated in manuscript development checks. The filesystem was shared; no technical isolation or external runtime attestation is claimed. Requested configuration: OpenAI gpt-6-astra, xhigh. No material defect was identified under the stated hypotheses. The recommendation remains favorable for specialized publication with nonblocking suggestions (accept), not an editorial decision. Quality remains 5.5/10; AIRR importance remains 4.5/10. millennium_score is the importance rating; overall_stars=5 is its integer display. The main advance is the uniform reduction and explicit fixed-ratio minimax curve with global discrete certificates; the channel family and plateau constant are known. The original manuscript source was read in full; literature reading was selective. Baitenov's full text was unavailable, the search was not exhaustive, and priority remains open. Finite computations are corroboration, not proofs of the infinite limits. assessed_at refers to original package closure; ARR-ASSESS-1.0 identifies this rendering schema.
- Correctness confidence
- The original adversarial review found no material defect under the stated hypotheses after examining the uniform reduction, coherent states, infinite-dimensional entropy arguments, atomic singularity, parameter coverage, both energy constraints, global certificates, thermal gap and endpoint laws. This is model review, not formal certification or a numerical probability of correctness.
- Rigor
- Theorems 2.1 and 3.1 are supported by finite-label conditional-entropy control, Gibbs relative entropy and global discrete inequalities; feasibility and the order of limits are addressed. Two short derivations could be made more explicit, but the original review did not find a missing material premise.
- Novelty
- The probable advance is the uniform reduction and explicit fixed-energy-ratio minimax constant curve, including atomic and diffuse constructions, thermal-adversary separation and endpoint laws. The identity/replacement channel family, binary-branch mechanism and plateau constant have antecedents. Priority is not certified: the search was directed, and Baitenov (2024) was accessible only through its primary editorial abstract.
- Significance
- A substantive specialized solution of an adversarial reset problem, potentially useful as an exact reference case. It does not determine regularized quantum capacity, finite-energy exact capacity, general attenuator behavior or efficient universal codes, and there is no evidence here of field-wide impact or Millennium-level significance.
- Reproducibility
- The original review replayed 28/28 corrected finite cases on Windows with Python 3.12.14, NumPy 2.2.6 and SciPy 1.18.0; maximum constraint residual was 3.675e-14 and reset LP gap 4.181e-11. The failed initial solver case was preserved separately. Additional checks covered 120 complex-state pairs and 30 continuum profiles. These are floating-point checks, not proofs; six large-energy continuum cases, Linux, internal audits and CI were not newly reproduced.
Strengths
- Theorem 2.1 gives a reduction uniform over finite reset budgets, including changing adversaries, coherent states and the infinite-dimensional oscillator setting; dephasing is justified through finite-label continuity rather than an unsupported entropy-continuity assertion.
- Theorem 3.1 and Lemmas 3.2–3.3 combine explicit atomic and diffuse profiles with full-support relative-entropy certificates and feasible discrete approximations. The proof retains both input and reset energy bounds.
- The consequences include the vanishing-deficit criterion M=o(E), the fixed-ratio curve and plateau, and a strict asymptotic gap for thermal reset adversaries when 0<c<1. The two endpoint laws were checked separately in the original review.
- The original investigation distinguished an initial solver success flag from a valid saddle certificate, retained the failed historical case, and separated finite numerical checks from the analytical proof.
Weaknesses
- Nonblocking exposition suggestion A: in Section 5, pages 7–8, before S1 and the connection used in S4, explicitly evaluate the mixed-measure saddle functional as Ψ(f,t*)=2λ ln[B_z(s,3/2)/b]+(q−2λs)ln z+b(2λs−qc), retaining the atomic contribution −qw ln q. This is a short algebraic clarification, not a repair of a theorem.
- Nonblocking exposition suggestion B: at the end of Section 3.1, page 4, show the substitution r=sy and exponential domination of scaled moments in the limit z decreasing to q. State that λ is fixed in (0,1); the review does not claim uniform endpoint control in λ.
- Bibliographic suggestion from the original addendum: cite Giovannetti–Lloyd–Ruskai, quant-ph/0408103v3, Section IV.B, equation (34), for the finite-dimensional identity/replacement family with arbitrary fixed replacement state. Its output-norm multiplicativity question differs from the present energy-constrained entanglement-assisted minimax problem.
- The full text of Baitenov, On Jensen Gap and Capacity of a Shifted Depolarizing Quantum Channel (2024), could not be obtained. Its editorial abstract describes a one-shot capacity and two optimal-ensemble regimes. Detailed overlap remains unverified; this limits novelty confidence without establishing a material mathematical objection.
- The claim is specialized and asymptotic. The review does not establish a general Gaussian-extremality counterexample, an attenuator result, uniqueness of all optimal sequences, efficient coding or a broad operational universality theorem. Literature access was selective, the search was not exhaustive, and no external runtime attestation, human review or formal proof certificate is supplied.
Potential errors
None reported.
Strong novelty candidates
- Theorem 2.1: a uniform high-energy reduction of the adversarial entanglement-assisted reset problem to a diagonal entropy game, with a correction independent of the finite reset budget.
- Theorem 3.1 and Lemmas 3.2–3.3: the explicit constant-loss curve at every fixed positive energy ratio, its atomic/diffuse profile structure, and global discrete certificates that retain feasibility.
- Section 5: strict separation of thermal reset adversaries for 0<c<1 and the square-root and quadratic endpoint laws, subject to the prior-art search and access limits stated above.
Unresolved material objections
None reported.
arr:assessment:f4a549f3-2c34-4cc8-867a-6525ec0b4a44 · prompt ARR-ASSESS-1.0 · response SHA-256 53ba13b952969486f4faeaba9931e9159befcd0318dd2047d832b383c0609a83 · canonical PDF SHA-256 f472a6c9380da16347a66b390d12b967a50d221e77a0c96905fb5f3d4868df7e
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Exact-version Working-paper release authorized by the author and processed through the official operator handoff. The author is the AIRR operator; Codex acts on his explicit instructions. Not admitted to the accepted collection; private model reports do not constitute independent human review or the required two-model admission gate.