{"id":"24eaf66a-e9c5-49a3-a180-f0509ebd487a","arxiv_id":"2607.24712","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Under Brandão–Plenio axioms, asymmetric quantum resource testing needs only O(log(1/δ)/D^∞(ρ∥F)) copies, and regularised Rényi resource divergences converge as α→1−.","lead":"The paper gives the first finite-copy guarantees for telling resourceful quantum states from free ones, with explicit sample counts. That turns the asymptotic generalised quantum Stein lemma into practical bounds for entanglement and magic testing and distillation.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No load-bearing objection. The new lemma chain (S8–S10) survives independent re-checking; the only soft spot found is a misstated intermediate norm bound in (S42) that is numerically harmless because (S43)–(S44) carry orders of magnitude of slack.","rationale":"Read in good faith: the central claim is the one-shot achievability bound of Theorem S1, from which the limit-commutation result (Thm. S2) and the Θ(log(1/δ)/D^∞) sample complexity (Thm. S3) follow by routine asymptotic arguments that I also checked (binary data processing upper bound (S64); superadditivity D(ρ^{⊗n}‖F_n) ≥ nD^∞; the ε̃-window choice in S3). The load-bearing technical content is the new one-shot blurring toolkit (Lemmas S7–S10), and my independent re-derivation of its key identities found it internally consistent: the Hahn-polynomial singular values, the Vandermonde grouping that makes Dicke vectors eigenvectors of K(c) with eigenvalue q_r(s), and the Chebyshev degree budget all verify, including the exact constants (κ ≈ 0.1926). The weakest point I could locate is the middle inequality of (S42), which as written does not follow from the crude binomial bound it apparently uses; however, a standard sharper binomial estimate repairs it at the cost of a slightly larger constant, and the downstream condition (S44) is protected by a quadratic-in-√(sn) negative term that absorbs any such constant with enormous margin. This is a referee-level correction, not a threat to the argument. I therefore agree with the reader that the genuine limitation is the restricted (ε, n) window — (d²+2)² ≤ log(1/ε) ≤ n/240 — and that it does not undermine any stated application, since Theorem S2 works at fixed s with n→∞ and Theorem S3 explicitly retreats to a smaller ε̃ inside the window. The AI-assistance disclosure covers Lemmas S8–S10, but those are precisely the lemmas whose algebra I found verifiable by hand, so the medium correctness risk the reader assigned is fair and not grounds to downgrade. Verdict unchanged: ACCEPT at MODERATE confidence, with the (S42) middle-line fix and the 'constants independent of ρ' phrasing in Thm. S3 as requested corrections.","tokens_in":53729,"tokens_out":1586,"duration_ms":560573,"concrete_test":"Numerically verify the two ends of the norm-control chain on a grid over the full validity window: for d ∈ {2,...,8}, c_τ ∈ {1, 1/d}, s ∈ [(d²+2)²/n, 1/240] and n up to 10^4, recompute log‖c‖₂² using the sharper bound log C(k+d²−2,d²−2) ≤ (d²−2)log₂(e(k+d²−2)/(d²−2)), and check (i) the final line of (S42) holds with constant 8 or a corrected constant C, and (ii) the quantity in (S44), ‖c‖₂²·2^{(s−K√s+√{240s}log(1/c_τ))n}, stays ≤ 1/2. If (ii) ever exceeds 1/2, the positive-part bound (S45)–(S46) loses its factor-2 and the correction f_{d,τ} in (S30) must be re-derived; if it stays below with wide margin (expected), the misstated middle line of (S42) is confirmed harmless and the paper's constants stand up to a corrected C.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I probed the step most likely to break: the coefficient-norm control (S41)–(S43) feeding the denominator condition of Lemma S7, since everything downstream — the smoothing estimate (S34), the blurring swap (S48), and Theorems S2/S3 — rests on it. The genuinely new ingredients check out: (i) Lemma S10's eigenvalue identity K(c)|α⟩ = q_r(s)|α⟩ follows correctly from (S158) plus multivariate Vandermonde — I re-derived Σ_{β:j(β)=j} (k choose β)(n−k choose α−β) = (s choose j)(n−s choose k−j) directly; (ii) Lemma S9's singular-value bound σ_ℓ^{−2} ≤ ((k+1)/(n+1))e^{2ℓ²/k} is correct (the last step of (S143) reduces to k ≥ 2ℓ, which holds); (iii) Lemma S8's degree count with α=1/16, β=(15−2π)/16 saturates (S129) exactly and κ=β√(2α)≈0.1926 is right. One genuine misstatement: the middle line of (S42) asserts log(k+1)+(d²−2)log(k+d²−2) ≤ 2√(sn). With the crude bound C(k+d²−2,d²−2) ≤ (k+d²−2)^{d²−2} that the line uses, the LHS at the boundary sn=(d²+2)² is ≈ √17(d²−1)(sn)^{1/4}, which exceeds 2√(sn) already at d=2 (≈19.6 vs 12) and grows ~4d√(sn). A sharper binomial bound C(a,b) ≤ (ea/b)^b gives ≲ 5.5(d²−2) ≤ 5.5√(sn), so the final line of (S42) plausibly survives with constant ~10 instead of 8. Crucially, the error cannot propagate: in (S43) the term −(K−√240 log(1/c_τ))√n·√(sn) with √n ≥ √(240sn) contributes a quadratic ≈ −743(sn), dwarfing any linear-in-√(sn) constant; even constant 100 leaves (S44) satisfied with margin. So the denominator condition of Lemma S7, and with it Theorem S1, stands. The reader's flagged window restriction is real but the paper handles it correctly (Thm. S3 picks ε̃ inside the window; Thm. S2 takes n→∞ at fixed s). One minor inaccuracy: Thm. S3 says the Θ-constants do not depend on ρ, but ε̃ is chosen via D^∞(ρ‖F), so the upper-bound constant does depend on ρ — cosmetic only.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies asymmetric quantum resource testing: discriminating ρ^{⊗n} from an adversarially chosen free state σ_n ∈ F_n, for free-state families satisfying the Brandão–Plenio axioms. Its central result (Theorem 1 / Theorem S1) is the first one-shot achievability form of the generalised quantum Stein's lemma: −log β_ε(ρ^{⊗n}‖F_n) ≥ D(ρ^{⊗n}‖F_n) − An√ε − B√(n log(1/ε)), with explicit constants, in the regime (d²+2)² ≤ log(1/ε) ≤ n/240. From this the authors derive (i) the affirmative solution of the limit-commutation problem lim_{α→1⁻} D_α^∞(ρ‖F) = D^∞(ρ‖F), resolving the open problem of Fang–Hayashi [13] (Theorem 2 / S2), and (ii) the sample complexity N_{ε,δ}(ρ‖F) = Θ(log(1/δ)/D^∞(ρ‖F)) as δ→0 at fixed type-I threshold (Theorem 3 / S3), with matching converse from binary data processing. The technical engine is a finite-n version of Lami's quantum blurring, built from Tikhonov-regularised Kraus approximants on the symmetric subspace, a discrete Chebyshev delta approximant (Lemma S8), and Hahn-polynomial singular-value control (Lemma S9). The SI contains a complete, checkable proof chain (Lemmas S4–S10 → Theorem S1 → Theorems S2, S3) and an alternative, quantitatively weaker derivation from Mazzola–Sutter–Renner techniques (Section S.IV).","tokens_in":37992,"tokens_out":6965,"duration_ms":744009,"significance":"If the results hold — and my checking indicates they do, modulo the local (S42) repair — this is a significant contribution to finite-resource quantum information theory. It provides the first one-shot achievability formulation of the generalised quantum Stein's lemma with fully explicit constants (K_{d,τ}, κ ≈ 0.1926, the 240-regime), converts it into a quantitative sample-complexity statement with matching upper and lower bounds, and resolves the limit-commutation problem left open by Fang–Hayashi, unifying the error-exponent and Stein frameworks. The one-shot blurring toolkit (Tikhonov regularisation in variational form, discrete Chebyshev delta approximants, Hahn-polynomial singular-vector control) is genuinely new and likely reusable elsewhere. The proof chain is shipped completely and verifiably in the SI, and the authors include an honest quantitative comparison showing their bound beats what MSR-type techniques yield (s^{1/3} loss). The main caveats are the restricted ε-window, which limits the immediate practical bite of the finite-n guarantee, and the δ→0 nature of the Θ-statement.","major_comments":[{"comment":"Proof of Theorem S1, Eq. (S42): the middle-line estimate log(k+1) + (d²−2)log(k+d²−2) ≤ 2√(sn) does not follow from the displayed substitutions (S41). Inserting k+1, k+d²−2 ≤ 17√(sn) and using the cited log y ≤ √y bound gives a left side of order √17(d²−2)(sn)^{1/4}, which already at the boundary sn=(d²+2)², d=2 evaluates to ≈20 against a right side of 12, and scales like d³ versus d². The final line of (S42) appears repairable — a sharper binomial/cruder direct bound gives the term as ≲5.5(d²−2) ≤ 5.5√(sn), so the constant 8 becomes O(10) — and the quadratic-in-(sn) negative term in (S43) dwarfs any such linear-in-√(sn) constant, so no downstream statement should change. Nevertheless this line feeds the coefficient-norm control underlying the denominator condition of Lemma S7, on which Theorems S1–S3 all rest, so the derivation must be corrected explicitly rather than left as written.","section":"S.II.A, proof of Theorem S1"},{"comment":"Theorem 1 (main text) states the one-shot bound for all thresholds 2^{−n/240} ≤ ε ≤ ε0 without telling the reader that consistency with Theorem S1 forces ε0 ≤ 2^{−(d²+2)²}. The proved guarantee therefore only covers type-I thresholds exponentially small in d⁴; for a fixed, operationally natural ε and moderate local dimension d, Theorem 1 is vacuous, and Theorem 3's fixed-ε sample complexity is recovered only by shrinking to an auxiliary ε̃ inside the window ((S78)–(S82)). The window (d²+2)² ≤ log(1/ε) ≤ n/240 should be stated in Theorem 1 itself, and the Discussion's practical framing (certification of entanglement sources, magic-state factories) should be calibrated accordingly — ideally with a short worked regime example (what n, d the bound covers at a given ε). As written, a reader of the main text alone would overestimate the scope of the finite-n guarantee.","section":"III, Theorem 1"}],"minor_comments":[{"comment":"Theorem S3 states that the Θ constants 'only depend on the underlying resource theory and on ε, and not on the state ρ'. However ε̃ in (S78) is chosen using D^∞(ρ‖F), so the additive threshold (S79) — and hence the onset of the Θ regime in δ — is state-dependent. The claim is true for the asymptotic leading constants (4/D^∞ and (1−ε)/(2D^∞)); please make the quantifiers precise.","section":"S.II.B, Theorem S3"},{"comment":"In the proof of Theorem S2, the sentence 'We next identify the limiting hypothesis-testing exponent. Fix 0 < ε ≤ 1/2, fix σ_n ∈ F_n, and let T satisfy Tr ρ^{⊗n}T ≥ 1−ε. Binary data processing for the relative entropy gives' is immediately repeated verbatim before Eq. (S63); delete the duplicated fragment.","section":"S.II.A, proof of Theorem S2"},{"comment":"The Supplementary Information is titled 'Sample complexity of entanglement testing via one-shot quantum blurring', whereas the paper concerns general quantum resource testing; please align the SI title with the manuscript title.","section":"Supplementary Information, title"},{"comment":"The proof-dependency diagram at the start of Section S.II contains a broken cross-reference ('Thms. S3 and ??').","section":"S.II, dependency diagram"},{"comment":"Several typos: 'goverened' (§III), 'a is a penalty term' (§III), 'an thus' (§I), 'approriate' (§I), 'disscussions' (Acknowledgements), 'obtacle' (§IV), and 'even it was true' (twice in §II.B; should be 'even if it was true').","section":"Throughout"},{"comment":"The abstract and introduction's 'first rigorous finite-n bounds' should be qualified in light of the authors' own Note added and Section S.IV, where estimates of the same type (with a worse s^{1/3} correction) are derived from the techniques of [20]; a sentence in the introduction clarifying what is new (the one-shot blurring method and the optimal √s profile) would make the priority claim precise.","section":"Abstract / §I"},{"comment":"Figure S2(b): the caption notes the polynomial exceeds the band by ≈33× on [1,L]; since the guarantee (S110) applies only at integer points, a half-sentence in the caption explaining that this overshoot between enforced zeros is expected would help readers.","section":"S.III, Figure S2"}],"recommendation":"minor_revision","confidential_remarks":"The acknowledgements disclose that GPT 5.5 \"contributed nontrivial insights, which served as basis for Lemmas S8 to S10\" — precisely the genuinely new technical core of the paper. The SI proofs of those lemmas are written out in full and I verified the load-bearing steps independently (Vandermonde aggregation in S10, the singular-value bound in S9, the degree count and constant in S8), so I have no correctness concern beyond the (S42) line flagged in my report; but the editor may wish to ensure a second referee also checks that lemma chain. Second, the \"first\" novelty claims should be read alongside the concurrent manuscript of Mazzola–Sutter–Renner [20], whose techniques, as the authors themselves show in Section S.IV, yield qualitatively similar (quantitatively weaker) estimates; the authors handle this transparently, and I only ask that the main text calibrate its priority language."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news is finite-n control, not another asymptotic restatement. They turn Lami’s blurring into a genuine one-shot tool (Tikhonov + discrete Chebyshev/Hahn delta + product-vector Kraus on the symmetric subspace) and get an explicit achievability bound: −log β_ε(ρ⊗n∥Fn) ≥ D(ρ⊗n∥Fn) − A n√ε − B √(n log 1/ε) inside a concrete window on ε and n. From that they close the α→1− / n→∞ commutation for regularised Rényi resource divergences and prove N_{ε,δ} = Θ(log(1/δ)/D∞) as δ→0. That is new, and it is what people actually need for entanglement/magic certification budgets.\n\nThe SI is complete and honest. Lemmas S4–S10 feed Theorem S1 cleanly; matching upper bounds via binary data processing lock the Stein and sample-complexity statements; they even include a weaker MSR-style bound and say so. Brandão–Plenio axioms and a full-rank free state are used as external structure, not smuggled conclusions. Self-citations to the asymptotic GQSL are tools with stated hypotheses.\n\nSoft spots are real but proportionate. The quantitative bound only lives in a restricted window ((d²+2)² ≤ log(1/ε) ≤ n/240, hand-tuned κ and 240). They recover fixed-ε sample complexity by picking a smaller ε̃ inside the window, which is legitimate, but the constants are conservative and the upper-bound prefactor does depend on ρ via that choice (their “independent of ρ” claim is slightly overstated). A stress-check of the coefficient-norm step (S42) finds a loose intermediate inequality; the slack in (S43)–(S44) is huge, so the denominator condition and Theorem S1 still stand. The chain is long and not machine-checked—moderate, not high, confidence is fair.\n\nThis is for people who work on resource theories, finite-blocklength QI, or distillation/certification resource estimates. It deserves a serious referee. I would bring it to reading group and cite the sample-complexity and limit-commutation statements.","headline":"Solid one-shot upgrade of the GQSL that actually settles Fang–Hayashi and gives the first usable sample-complexity scaling; the long lemma chain holds up.","tokens_in":30926,"tokens_out":566,"would_cite":true,"duration_ms":14029,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","81P18","94A17"],"pacs":["03.67.-a","03.67.Mn","03.65.Ud"],"model":"grok-4.5","headline":"Finite-copy bounds turn the generalised quantum Stein lemma into sample-complexity guarantees for resource testing.","keywords":["quantum resource testing","generalised quantum Stein lemma","sample complexity","one-shot quantum blurring","regularised relative entropy","Rényi divergence continuity","entanglement testing","magic testing"],"falsifier":"Exhibit a concrete Brandão–Plenio free-state family and a state ρ for which, inside the stated window on ε and n, every measurement with type-I error at most ε has type-II error larger than 2^{-D(ρ^{⊗n}∥F_n)} times the claimed penalty factor.","tokens_in":30332,"feed_emoji":"⚛️","tokens_out":1236,"duration_ms":20441,"temperature":0.7,"pith_summary":"The paper supplies the first rigorous finite-n estimates for quantum resource testing: discriminating many copies of a known resourceful state from every free (resourceless) state. Earlier work only controlled the asymptotic decay rate of the false-negative error; here that rate is made quantitative, with explicit correction terms that vanish as the number of copies grows. The resulting sample-complexity formula says that, for any fixed false-positive tolerance, the number of copies needed to drive the false-negative probability below δ scales as log(1/δ) divided by the regularised relative entropy of the resource. The same bounds settle a limit-commutation question for regularised Rényi divergences and, via the Brandão–Plenio correspondence, give concrete copy counts for distillation under asymptotically resource-non-generating operations. A sympathetic reader cares because the estimates turn an abstract asymptotic theorem into a practical recipe for how many copies an experiment or a distillation protocol actually needs.","feed_headline":"How many copies detect quantum resources","feed_subtitle":"First finite-n bounds turn the Stein rate into an explicit sample-complexity formula","key_machinery":"One-shot quantum blurring: a controlled replacement of a carefully chosen number of tensor factors by a free full-rank state, combined with a discrete polynomial approximant to the Kronecker delta and a variational Tikhonov regularisation that converts a large purification overlap into an operator inequality on the blurred state.","core_discovery":"Under the Brandão–Plenio axioms there exist constants such that, for all sufficiently large n and all type-I thresholds in an explicit window 2^{-n/240}⩽ε⩽ε₀, the hypothesis-testing relative entropy satisfies (1/n)D_H^ε(ρ^{⊗n}∥F_n)≥(1/n)D(ρ^{⊗n}∥F_n) minus an explicit penalty of order √( (1/n)log(1/ε) ) plus √ε. Consequently the sample complexity of asymmetric resource testing is Θ(log(1/δ)/D^∞(ρ∥F)) as δ\to0, and the regularised Rényi relative entropies of the resource are continuous from below at α=1.","pith_inferences":["The explicit constants (blurring width ~√(s n), discrete-delta degree, universal κ≈0.19) can be numerically optimised for small fixed dimension, potentially yielding tighter laboratory sample-size prescriptions than the universal worst-case bounds.","Because the argument never uses more than the Brandão–Plenio axioms, any future resource theory that satisfies those axioms automatically inherits the same sample-complexity formula without further proof.","The one-shot blurring-plus-Tikhonov toolkit is likely reusable for other composite hypothesis-testing problems whose free sets are only known to be convex, permutation-invariant and full-rank."],"forward_implications":["For fixed false-positive tolerance, O(log(1/δ)/D^∞(ρ∥F)) copies suffice to drive the false-negative probability below any δ\to0.","Regularised Petz and sandwiched Rényi relative entropies of any Brandão–Plenio resource converge to the ordinary regularised relative entropy as α\to1 from below.","Distilling k ebits under non-entangling operations requires only O(k/E(ρ)) copies of a bipartite state whose regularised relative entropy of entanglement is E(ρ).","The same finite-n estimates apply verbatim to magic-state testing and to any other resource theory obeying the five Brandão–Plenio axioms.","Exponential decay of both error types is simultaneously achievable with type-II exponent arbitrarily close to the Stein exponent."],"fun_headline_variants":["First finite-n bounds for quantum resource testing","Sample complexity of asymmetric quantum resource tests","Explicit copy counts to detect quantum resources","Finite-copy guarantees for resource discrimination","Regularised Rényi resource entropies converge at α=1"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The quantitative finite-n guarantee is proved only inside a restricted window on the type-I error and block length; outside that window the bound is not established, and the sample-complexity claim is recovered by choosing a smaller fixed tolerance that still lies inside the window.","fun_headline_variants_meta":{"raw":{"variants":["First finite-n bounds for quantum resource testing","Sample complexity of asymmetric quantum resource tests","Explicit copy counts to detect quantum resources","Finite-copy guarantees for resource discrimination","Regularised Rényi resource entropies converge at α=1"]},"model":"grok-4.5","effort":"low","cost_usd":0.004218,"raw_usage":{"total_tokens":1336,"prompt_tokens":895,"num_sources_used":0,"completion_tokens":53,"cost_in_usd_ticks":42184000,"prompt_tokens_details":{"text_tokens":895,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":388,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":895,"tokens_out":53,"duration_ms":6561,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T07:03:31.660522+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete Brandão–Plenio free-state family and a state ρ for which, inside the stated window on ε and n, every measurement with type-I error at most ε has type-II error larger than 2^{-D(ρ^{⊗n}∥F_n)} times the claimed penalty factor.","supporting_citations":[],"review_version":1}