Classical capacities and complementary simulation for qubit Davies channels
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
We extend the thermal block entropy estimate for generalized amplitude damping to the full qubit Davies region, including additional dephasing. The proof combines the triangular inequality of Tang, Zhu, Bai and Wang with a controlled-purification consequence of strong subadditivity: a third vertex completes the fixed-entropy decomposition used for the thermal boundary. The resulting population-resolved bound holds with every finite-dimensional partner. It implies strong superadditivity of the convex roof of output entropy, and hence arbitrary-partner additivity of Holevo information and classical capacity for both the Davies channel and its complementary channel. We give scalar convex-envelope formulas for both capacities. Every bipartite state supported on the two-dimensional Stinespring range has strongly additive entanglement of formation, so its state entanglement cost requires no regularization. A scalar moment program evaluates that formation functional for arbitrary input coherence. The state/channel correspondence, complement duality and existing axial convex-roof geometry are explicitly attributed to prior work. A normalized reverse channel stays within the Davies region, allowing us also to remove regularization from the parallel entanglement cost of the complementary channel. We obtain an elementary rate in a known flat-roof region and exhibit a nonunital example whose simulation cost strictly exceeds the cost of its Choi state. Finite-block and numerical bounds support the results. The triangular theorem remains an external dependency; the original channel's simulation cost, adaptive simulation and quantum capacity are not determined.
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Cite this paper
Lluis Eriksson (2026). Classical capacities and complementary simulation for qubit Davies channels. AIRR.SCIENCE, ARR-2026-5YJX444B6M95ZTG4, v2. https://airr.science/papers/ARR-2026-5YJX444B6M95ZTG4/versions/v2/ Research preprint; ARR-2026-5YJX444B6M95ZTG4 v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Revision statement
Minor revision. Adds the Fang citation and explicit full proofs of previously terse steps; no theorem, hypothesis or figure of v1 changes.
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- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Not assessed
- Lean 4
- Not applicable
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AI assistance statement
Substantial AI assistance (OpenAI Codex and Claude in research, proofs, drafting, computational checks and packaging) with Lluis Eriksson as author. This version was revised by Claude Sonnet 5.5 acting as author-reviewer (not human, not independent of the author). Assessments or scores of the earlier version v1 do not carry over to this version and none exist for this exact PDF. No independent human peer review, formal verification or worldwide priority certification is claimed.
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Model assessments and review context
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- 2
- Range
- 2.70–3.10
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- bounded capability-weighted mean · exact version only
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Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.10, minor_revision) made in an independent Claude Code thread: new session whose working folder held only paper.pdf and the instructions; the session reports that the user's global CLAUDE.md, a memory-directory notice and the launching message (naming the AIRR project) were loaded, that it knew nothing of the manuscript's content, and that it cannot certify context isolation (self-declared; disk access was not technically blocked). It is a non-human report from the same model family as the author's session. review_context and runtime_provenance are omitted because the schema admits no honest basis; the controls file is retained with its sha256 in registry/assessment-evidence/2026-10-22-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled (and the version label corrected from the template's v1 to v2). No other report exists for this version. No cost or tokens recorded.
Operator record · · Source files and affected reports
Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (2.70, major_revision, independence 'not_involved_in_manuscript' as the model reported), non-human, from an independent Claude Code thread: new session with only paper.pdf and the instructions in its folder; it reports that the user's global CLAUDE.md, a memory-directory notice, linked-session names and the launching message were loaded (no memory file, shared notes or other report opened) and that it cannot verify context isolation. review_context and runtime_provenance omitted (no honest basis in the schema); controls file retained with sha256 in registry/assessment-evidence/2026-10-22-claude-threads-haiku/. Model JSON unedited; only placeholder ids filled (and version label v1 corrected to v2). A separate base model from the Sonnet 5.5 thread report on the same version, which it does not replace. Reviewer's two unresolved material objections: (1) Lemma 3 (controlled purification) is said to assume an isometry that requires Tr(X_i^* X_j)=delta_ij, which is not a hypothesis, affecting Theorem 4 at vertex C and the interior case; (2) Corollary 7 is asserted through an unexplained reduction to Theorem 6. The reviewer also reports a probable misprint in equation (20). They remain open, are the reviewer's, and are NOT resolved and NOT refuted here; not checked against the PDF by the author's session. Revisable.
Operator record · · Source files and affected reports
anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown2.70
Recommendation: Major Revision · Material objections: 2
Working paper extending additivity and formula-level results for qubit Davies channels (classical capacity via a block-entropy vertex argument, plus complementary-channel simulation cost). Much of the algebra I could check holds, including the Thm 4 vertex decomposition, Lemma 5 parameters, Cor 11 and the Prop 12 derivative and chord identity. Material gaps remain: the Lemma 3 isometry step is unjustified as written, and the Cor 7 / Thm 10 / Prop 12 additivity steps rest on unspelled reductions and external preprints I could not check. Recommendation: major revision.
- Correctness confidence
- Checked: the Thm 4 vertex weights reproduce (alpha, beta, eta, delta), K and u, and the Lemma 5 parameters satisfy the constraint with u = v(p). Not sound as written: Lemma 3 says V|i> = |i>|X_i>> is an isometry, which holds only if the X_i are Hilbert-Schmidt orthonormal, and this is not assumed. Eq (20) misprints a_p = pb/t (should be pb/(1-t)).
- Rigor
- Explicit algebra passes: Vieta roots and c' <= 1 in Cor 11; J'(1/2) = 0.0425 > 0 and the chord identity (21) at p = 1/2 (both sides 0.48811) in Prop 12. Strict separation and the Cor 7 tensor additivity depend on steps that are asserted, not shown (the reduction to Thm 6 through V).
- Novelty
- The new content is a third vertex added to the Tang et al./OpenAI triangular-entropy argument, plus extension to the nonunital interior and to complementary cost. Much is explicitly attributed to prior work (Hellmund-Uhlmann, Matsumoto-Shimono-Winter, Fang [4]). The main novelty claim depends on the Lemma 3 proof, which needs repair.
- Significance
- Moderate. It gives additivity and capacity formulas for the nonunital interior of the qubit Davies family, a complementary-channel cost formula, and a strict Choi-cost separation. Impact stays within a narrow channel family, and the separation result is conditional on the state-additivity steps.
- Reproducibility
- Scripts, versions and tables are described, but I did not run them and had no code access here. All numerics are floating-point and not interval-certified, and one 75-digit computation is quoted without code. Most formulas are explicit enough to recheck by hand, which I partly did.
Strengths
- Thm 4 vertex decomposition: the weights w_A, w_B, w_C reproduce the original (alpha, beta, eta, delta), the off-diagonal K and u, so the concavity step is consistent.
- Lemma 5 mapping to Thm 4 satisfies alpha+beta+eta+delta = 1 and alpha*delta - beta*eta = kp(1-p) >= K^2, with u = v(p).
- Cor 11 Vieta step: the roots of Q(x) have product k^2 and sum 2k+4ab; c' <= 1 by Cauchy-Schwarz, so c <= k <= c' and D >= 0 on the stated region.
- Prop 12 checks: 324p^2-312p+9 has roots about 0.0298 and 0.9332; J'(1/2) = h(3/10)-h(1/10)-(3/5)ln(3/2) is about 0.0425 > 0; the chord identity (21) matches at p = 1/2.
- The scope is stated openly: external dependencies (Lemma 2, Hughston-Jozsa-Wootters, continuity of E_F) and the floating-point status of the numerics are declared.
Weaknesses
- Lemma 3 (controlled purification): V is claimed to be an isometry, but V^dagger V has Gram matrix <X_i|X_j>, which is the identity only if the X_i are orthonormal; the step S(ABE) = S(rho) therefore fails as written.
- Cor 7 (strong superadditivity and tensor additivity of E_F for Stinespring-range states) is derived from Thm 6 through a 'lift through V' step that is not spelled out.
- Thm 10 and the strict separation in Prop 12 depend on E_F(omega^{tensor n}) = n E_F(omega) from Cor 7 and on the Berta et al. formula (16); neither is verified in the text.
- In the Thm 4 proof, eta and delta denote both the original coordinates and the vertex coordinates (eta_B = u/U0, delta_B = K^2/U0). 'U0 eta = u' holds only for vertex values, which makes the B-vertex check hard to follow.
- The novelty of the triangular step depends on 2026 preprints (Tang et al., OpenAI, Fang [4]) that I could not check; the authors themselves say the third vertex is the only addition.
Potential errors
- Eq (20): a_p = pb/t should read pb/(1-t). As printed, at p = 1/2 the values a_p = 3/8, b_p = 1/8, k_p = 5/8 sum to 9/8, not 1. Prop 12 uses a_p = 1/4, which matches pb/(1-t).
- Lemma 3 proof: 'the map V|i> = |i>|X_i> is an isometry' is false without an orthonormality hypothesis on the X_i.
- Prop 12 Step 3: the strict inequality E_C^par(N) >= G(p1)/ln2 > G(1/2)/ln2 uses E_F(omega^{tensor n}) = n E_F(omega), the claim that Cor 7 must establish.
- Notation: t denotes both time (in the exponents of eq. 1) and output population t(p); the paper flags this but it is easy to misread.
Strong novelty candidates
- The vertex-augmented block-entropy inequality (Thm 4), which covers the interior 0 <= lambda^2 < k. If the Lemma 3 gap is repaired, it extends capacity additivity to the nonunital interior of the Davies family.
- The scalar population-resolved formulas for the complementary channel: the capacity identity (Thm 9) and the parallel simulation cost on the region of Cor 11, including the reversed-channel closure.
Unresolved material objections
- Lemma 3 (controlled purification) assumes that V|i> = |i>|X_i>> is an isometry, which requires Tr(X_i^* X_j) = delta_ij; that is not a hypothesis. Thm 4 at vertex C, and through it Lemma 5, Thm 6 and the interior additivity and capacity identities of Thm 1, depend on this step. As written, the interior case is not established.
- Cor 7 (tensor additivity and strong superadditivity of entanglement of formation for Stinespring-range states) is asserted through an unexplained reduction to Thm 6. The additivity identities of Thm 10 and the strict separation of Prop 12 depend on it, so the complementary cost results are not established by the text.
arr:assessment:7401592d-3263-41a9-86b9-a9ee8f705f2e · prompt ARR-ASSESS-1.0 · response SHA-256 b9a708e55242e796eee2fd39cde3b52309d2b57f91a84c3d904606de62aa9acc · canonical PDF SHA-256 aef993b265fc45b1d69aff0e4d150114dcc3cea22931fea7a2d15316cd7d4651
anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.10
Recommendation: Minor Revision · Material objections: 0
The manuscript extends a 2026 triangular block-entropy inequality to the full qubit Davies region via a third vertex (Lemma 3 plus concavity), derives a population-resolved output-entropy bound, and from it strong Holevo additivity with arbitrary partners for the Davies channel and its complement, scalar convex-envelope capacity formulas, strong EoF additivity on the Stinespring range, and a complementary parallel simulation cost (with a flat-roof closed form and a strict separation from Choi cost). My hand checks of the main algebraic steps found no error. Main limits: reliance on unverified external results, narrow scope, heavy overlap with concurrent GAD work, and uncertified numerics.
- Correctness confidence
- I re-derived by hand the Theorem 4 triangle vertices and weights, the block structure of Lemma 5 (cross terms vanish by phase covariance), the reverse-channel parameters (20), the environment spectrum (15), Cor. 11 Vieta identities and J'(1/2)>0 in Prop. 12. All consistent. Rests on unverified external Lemma 2 (Tang et al.) and Berta et al. (16).
- Rigor
- Statements are precise, antecedents and scope limits are explicit, external dependencies are listed (Lemma 2, E_F continuity, HJW, eq. 16). Weaker spots: Cor. 13 induction, the mixed-input step of Thm 1 and the deficit identity (23) are terse; numerics are explicitly not interval-certified.
- Novelty
- Genuine but incremental: a third vertex extends the 2026 pure-output and GAD block inequalities to the full Davies region, then standard MSW/Holevo/Berta machinery yields additivity, complement capacity and a Choi-cost separation. Overlap with concurrent GAD work [3,4] is disclosed; I could not check priority (no web use).
- Significance
- Arbitrary-partner Holevo additivity and exact capacities for a physically relevant two-parameter-plus-dephasing nonunital qubit family, plus a complementary-channel simulation cost strictly above Choi cost, are of solid but specialised interest. Qubit-only, memoryless, original channel's simulation cost left open.
- Reproducibility
- Appendix A lists scripts, versions and compared outputs, and a few exact values (J'(1/2)=ln(2^6 3^9/7^7)/10, D=-55/192) I confirmed by hand. I had no access to the supplement or code, so I executed nothing; floating-point checks are error-finding aids, not certificates.
Strengths
- Third-vertex argument (Thm 4) is short and checks out: vertices, weights and the bound at A, B, C verify.
- Lemma 5 hidden assumption (vanishing population-coherence cross terms) holds for Davies maps by phase covariance.
- Reverse-channel closure (20) keeps the Davies region; I verified a_p, b_p and lambda_p by direct computation.
- Prop. 12 separation is proved in exact arithmetic and depends only on lower bounds plus (16).
- Candid scoping: overlaps, concurrent work, external dependencies and non-certified numerics are stated.
Weaknesses
- Lemma 2 (Tang et al.), E_F continuity and Berta et al. Theorem 12 are imported unverified; the whole additivity chain depends on Lemma 2.
- Qubit Davies only, memoryless, with the original channel's simulation cost, adaptive cost and quantum capacity open.
- Cor. 13 and eq. (23) are given with minimal proof; Thm 1 mixed-input step is a one-line appeal to concavity.
- Numerical tables are floating-point only; supplement not inspectable from the PDF.
- Provenance: AI-drafted and AI-reviewed with prior-involved reviewers; no human peer review.
Potential errors
- Section 8 strict interior example uses lambda^2=1/2 while Prop. 12 uses lambda^2=1/10 for the same (a,b); easy to conflate, not an error.
- Cor. 13 induction over conditional remaining states with nonidentical channels is only sketched; the step combining populations by convexity of each f_i needs explicit statement.
- Berta et al. formula (16) is applied to a channel whose Stinespring range need not be full; the cited theorem's exact hypotheses (maximization over inputs, diamond-norm convergence) should be quoted verbatim.
Strong novelty candidates
- Strong convex-roof superadditivity (Thm 6) with arbitrary partner for all Davies maps and their complements, giving arbitrary-partner Holevo additivity.
- Closure of the normalized reverse channel inside the Davies region, yielding an exact scalar formula for complementary parallel simulation cost and a strict separation from Choi-state cost.
Unresolved material objections
None reported.
arr:assessment:76ea43df-0c85-46c4-88a2-a448fb997e5c · prompt ARR-ASSESS-1.0 · response SHA-256 36f98b54a49e2ea21237e5a291eca518fa365961b9a34cdcd96864afdff93185 · canonical PDF SHA-256 aef993b265fc45b1d69aff0e4d150114dcc3cea22931fea7a2d15316cd7d4651
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Editorial disclosure
Working-paper version v2 supersedes v1. Adds the Fang citation and explicit full proofs of previously terse steps; no theorem, hypothesis or figure of v1 changes. Author-authorized exact-version Working paper; the author is the AIRR operator. It remains outside the AIRR accepted collection.