{"id":"a442574f-4d7b-4eb4-960e-c3f4884aa807","arxiv_id":"2608.01124","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For α>2, every quantum relative entropy extending the classical Rényi relative entropy is upper bounded by D_{α,α-1}, which is therefore the maximal quantum Rényi relative entropy.","lead":"This paper proves that for alpha>2 the largest quantum extension of the classical Rényi relative entropy is the alpha-z Rényi divergence with z=alpha-1, completing a classification that was open for all alpha above 2. It also yields exact conditions for when a Gibbs-preserving operation with a catalyst can create quantum coherence from an incoherent state.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is internally consistent and the external DPI citation is standard.","rationale":"The reader's verdict ACCEPT is justified. I attempted to find a flaw in the proof of the maximal-extension statement. The delicate part is Lemma 4, where the authors construct an explicit preparation map. I verified each step: the pinching inequality gives σ^{-1}≤mP(σ^{-1}); inversion yields \\tilde{σ}≤σ; the block structure and feasible point satisfy the constraints; the cost computation and the comparison to Q_{α,α−1} via Schatten contractivity under pinching are correct. The regularization in Prop. 5 works because log|spec(ρ^n)|=O(log n). The lower bound is exactly the DPI of D_{α,α−1}, which is a known published result. The application section is more compressed, but the reduction to the maximal extensions on classical inputs is logically sound, and the support-reduction lemma handles the catalyst support issue. The only truly load-bearing assumption is the imported DPI; this is the same point the reader flagged, but it is a standard citation rather than a defect. Therefore the verdict should remain unchanged.","tokens_in":23628,"tokens_out":49329,"duration_ms":394359,"concrete_test":"Independently verify the external DPI by numerically testing D_{3,2}(E(ρ)∥E(σ)) ≤ D_{3,2}(ρ∥σ) for random qubit states ρ,σ with supp(ρ)⊆supp(σ) and random CPTP maps E, sampling α in (2, 10]; if any violation is found, Proposition 2's lower bound and hence Theorem 1's proof would need revision. Separately, recompute Lemma 4's bound for a random 3×3 pair to confirm the constructed feasible point satisfies the constraints and the cost equals m^{α−1}Σ_j r_j^α Tr(σ_j^{α−1}).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1, proven by showing the regularized prepared Rényi relative entropy equals D_{α,α−1} for α>2. The lower bound (Prop. 2, Eq. 39) is the only step that relies on an unproven external input: the data-processing inequality for D_{α,α−1} for α≥2, cited from Zhang [43]. If that inequality failed on any pair with supp(ρ)⊆supp(σ), the lower bound would collapse and Theorem 1 would not follow from this proof. I checked the internal upper-bound construction (Lemma 4 and Prop. 5), including the pinching inequality, the operator-antimonotone step, the feasible-point verification, and the log|spec| regularization; all are algebraically consistent. The support-reduction arguments in Lemmas 14–15 also hold. The reliance on [43] is a normal citation of a published theorem, not an internal gap; the paper even states the exact 'if and only if α≥2' condition. No internal inconsistency, circularity, or unsupported parameter choice was found.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper completes the identification of the maximal quantum extension of the classical Rényi relative entropy. For α>2 it proves that every quantum relative entropy reducing to the classical Rényi relative entropy D_α on commuting states is bounded above by the α-z Rényi relative entropy D_{α,α−1} with z=α−1. The proof is via the prepared Rényi divergence: Proposition 2 gives the lower bound on the regularized prepared divergence from the data-processing inequality of D_{α,α−1}, while Lemma 4 and Proposition 5 construct an explicit preparation whose cost matches D_{α,α−1} up to a logarithmic spectral-size correction that vanishes under regularization. The paper then applies this result to large-sample and catalytic conversion of classical-to-quantum dichotomies, obtaining a rate formula and a characterization of catalytic coherence generation by Gibbs-preserving operations.","tokens_in":23816,"tokens_out":16725,"duration_ms":146630,"significance":"If the result holds, it closes a long-standing gap and provides the maximal element among quantum Rényi relative entropies for the entire range α≥0. The proof is constructive: the preparation map in Lemma 4 is explicit and asymptotically optimal, and the upper-bound argument is elementary and checkable. The applications show that the maximal divergences determine transformation rates and catalytic conversion criteria. The main caveat is that the converse bound in Proposition 2 imports the data-processing inequality for D_{α,α−1} at α≥2 from Zhang [43]; this is a standard external result, but it is genuinely load-bearing for Theorem 1.","major_comments":[],"minor_comments":[{"comment":"The lower-bound proof is only a few lines and leans entirely on the data-processing inequality for D_{α,α−1} from [43]; please state the precise theorem and its support assumptions so the reader can verify that it applies to all pairs with supp(ρ)⊆supp(σ), including non-faithful states.","section":"§III, Proposition 2"},{"comment":"In Eq. (66), the use of Lemma 13 is correct but terse; a one-sentence reminder that q=α−1 and that the pinching is with respect to the spectral projectors of ρ would make the comparison with Q_{α,α−1} easier to follow.","section":"§IV, Lemma 4"},{"comment":"The continuity step at the end of Theorem 12 is used to pass from ρ_ε to ρ; since the divergences in D are not all globally continuous on arbitrary pairs, the proof should explicitly state that continuity is used on faithful pairs and that the constructed states are full-rank.","section":"§V.C, Definition 11 and Theorem 12"},{"comment":"The rate formula in Eq. (125) is undefined in the degenerate case p=q and ρ=σ, where ratios 0/0 occur; a convention should be added or the case excluded explicitly.","section":"§V.B, Corollary 10"}],"recommendation":"accept","confidential_remarks":"This is a strong paper and the central argument appears sound. The only point I would ask the editor to weigh is the black-box reliance on the data-processing inequality from Zhang [43] for the converse bound; this is a normal citation of a published theorem, but because it is load-bearing, the author could usefully add a precise statement of the cited result. The AI-assistance disclosure in Section VII is transparent and does not affect the mathematical evaluation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper resolves the open case α>2 for the maximal quantum Rényi relative entropy, proving it equals D_{α,α−1}. That closes the characterization that was only known for α∈[0,2], and the proof is genuinely two-sided: the lower bound comes from the known DPI of D_{α,α−1}, and the upper bound from an explicit preparation map whose cost matches D_{α,α−1} up to a log|spec| term that vanishes under regularization. I checked the pinching step, the operator-antimonotone argument, the feasible-point verification, and the support-reduction lemmas; all are internally consistent. The explicit preparation map in Lemma 4 and Remark 1 is a real contribution, and the catalytic coherence application follows as a corollary of the main theorem plus the semiring framework. The paper is honest about what is imported: the DPI for D_{α,α−1} for α≥2 is cited from Zhang [43] rather than reproved, and the application uses Fritz's Vergleichsstellensatz without expanding every detail. Those are normal citations of published theorems, not hidden gaps. The dependence on the external DPI is the only load-bearing imported input; if that inequality failed, the lower bound would collapse, but it is a standard result, and the stress-test note found no other weaknesses. The AI-assistance statement is disclosed, and the mathematical content is checkable in the text, so I do not treat it as a problem. The main theorem is significant within quantum information theory, and the application to Gibbs-preserving catalysis is a strong payoff. The paper deserves a serious referee: the proofs are detailed, the claims are precise, and the result fills a recognized gap. I would send it to peer review without hesitation, though I would ask the referee to verify the external DPI citation carefully.","headline":"Completes the maximal Rényi relative entropy characterization for α>2 with a clean two-sided proof; the main caveat is inherited external inputs, not internal gaps.","tokens_in":24349,"tokens_out":935,"would_cite":true,"duration_ms":9846,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","94A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $\\alpha>2$, the largest quantum R\\'enyi relative entropy is $D_{\\alpha,\\alpha-1}$.","keywords":["R\\'enyi relative entropy","maximal quantum extension","alpha-z R\\'enyi relative entropy","prepared relative entropy","geometric R\\'enyi divergence","Gibbs-preserving operations","catalytic coherence generation","quantum majorization"],"falsifier":"Take any $\\alpha>2$ and a pair of full-support states $\\rho,\\sigma$, and compute $D_{\\alpha,\\alpha-1}(\\rho\\|\\sigma)$ alongside $D_{\\alpha,\\alpha-1}(\\Phi(\\rho)\\|\\Phi(\\sigma))$ for a trace-preserving channel $\\Phi$ such as a dephasing map; a violation would falsify the imported data-processing inequality and with it the lower bound in Proposition 2.","tokens_in":23420,"feed_emoji":"⚛️","tokens_out":9033,"duration_ms":77621,"temperature":0.7,"pith_summary":"The paper completes the search for the maximal quantum extension of the classical R\\'enyi relative entropies. It proves that every quantum relative entropy agreeing with $D_\\alpha$ on classical states is bounded above by the geometric R\\'enyi entropy $\\hat D_\\alpha$ for $\\alpha\\in[0,2]$ and by the $\\alpha$-$z$ R\\'enyi relative entropy $D_{\\alpha,\\alpha-1}$ for $\\alpha\\in(2,\\infty]$, closing the one remaining gap. The proof works by showing that the regularized prepared R\\'enyi relative entropy equals $D_{\\alpha,\\alpha-1}$, with an explicit asymptotically optimal preparation map. This characterization also yields necessary and sufficient conditions for catalytically assisted Gibbs-preserving operations to turn an energy-incoherent state into an energy-coherent one.","feed_headline":"For alpha>2, the largest quantum Renyi entropy is now known","feed_subtitle":"It equals the alpha-z Renyi divergence with z=alpha-1, completing the search and yielding a coherence test.","key_machinery":"The load-bearing object is the prepared R\\'enyi relative entropy $D^P_\\alpha(\\rho\\|\\sigma)$, defined as the infimum of $D_\\alpha(p\\|q)$ over classical distributions $(p,q)$ and preparation channels sending $p$ to $\\rho$ and $q$ to $\\sigma$, together with its regularization over tensor powers. The paper proves that the regularized limit equals $D_{\\alpha,\\alpha-1}$ for $\\alpha>2$ through two mechanisms: a lower bound obtained by applying the data-processing inequality of $D_{\\alpha,\\alpha-1}$, and an upper bound from an explicit preparation map constructed by pinching $\\sigma$ onto the spectral projectors of $\\rho$. The map costs $D_{\\alpha,\\alpha-1}(\\rho\\|\\sigma)+\\log|\\operatorname{spec}(\\rho)|$, and this spectral-size correction grows only logarithmically in the number of copies, so it vanishes after regularization.","core_discovery":"The central claim is Theorem 1: for every pair of quantum states $\\rho,\\sigma$, every additive quantum relative entropy $D_\\alpha$ that reduces to the classical R\\'enyi relative entropy on commuting states satisfies $D_\\alpha(\\rho\\|\\sigma)\\le\\hat D_\\alpha(\\rho\\|\\sigma)$ for $\\alpha\\in[0,2]$ and $D_\\alpha(\\rho\\|\\sigma)\\le D_{\\alpha,\\alpha-1}(\\rho\\|\\sigma)$ for $\\alpha\\in(2,\\infty]$, where $D_{\\alpha,\\alpha-1}$ is the $\\alpha$-$z$ R\\'enyi relative entropy at $z=\\alpha-1$. The matching lower and upper bounds on the regularized prepared divergence, Propositions 2 and 5, identify this quantity exactly and establish additivity in the previously open range. Since $D_{\\alpha,\\alpha-1}$ itself satisfies data processing and additivity for $\\alpha\\ge2$, the theorem closes the maximal-extension problem.","pith_inferences":["The paper does not pursue it, but the same pinching-based preparation map should transfer to other resource theories whose monotone is a prepared divergence, because the spectral correction always vanishes under regularization.","A testable extension would allow the catalyst to be correlated with the system; the paper's criterion covers uncorrelated catalysts with approximate conversion, and it is open whether allowing correlation relaxes the inequalities in Theorem 12.","Because the maximal extension is now fixed, any new candidate quantum R\\'enyi-like divergence for $\\alpha>2$ can be screened simply by checking whether it lies between the sandwiched divergence and $D_{\\alpha,\\alpha-1}$ on a pair of test states."],"forward_implications":["The maximal quantum extension of $D_\\alpha$ is now known for every $\\alpha\\in[0,\\infty]$: the geometric R\\'enyi divergence for $\\alpha\\in[0,2]$ and $D_{\\alpha,\\alpha-1}$ for $\\alpha>2$.","The regularized prepared R\\'enyi divergence is additive for $\\alpha>2$, so it is a genuine quantum relative entropy and can serve as a monotone in quantum resource theories.","The optimal asymptotic rate for transforming a classical dichotomy $(p,q)$ into a quantum dichotomy $(\\rho,\\sigma)$ is the minimum over both argument orderings of the ratios of these maximal divergences, as stated in Corollary 10.","An energy-incoherent state $p$ can be catalytically converted into a coherent state $\\rho$ by a Gibbs-preserving operation exactly when the R\\'enyi inequalities $D(p\\|\\gamma)\\ge D(\\rho\\|\\gamma)$ hold for every divergence in the completed family, including both orderings, as stated in Theorem 12.","At $\\alpha=\\infty$, the bound reduces to the max-relative entropy, so the characterization includes the sharp ordering by $D_{\\max}$ in the limiting case."],"supporting_citations":[{"why":"It defines the family of $\\alpha$-$z$ R\\'enyi relative entropies, including the quantity $D_{\\alpha,\\alpha-1}$ that the paper proves maximal for $\\alpha>2$.","marker":"[1]"},{"why":"It supplies the reverse-test variational formulation of prepared divergences used to set up the optimization in Lemma 3.","marker":"[28]"},{"why":"It establishes the geometric R\\'enyi relative entropy as the prepared divergence for $\\alpha\\in[0,2]$, giving the previously known half of the maximal characterization that the paper extends.","marker":"[29]"},{"why":"It provides the data-processing inequality for $D_{\\alpha,\\alpha-1}$ for all $\\alpha\\ge2$, which is imported to prove the lower-bound half of Theorem 1.","marker":"[43]"},{"why":"It supplies the preordered-semiring comparison theorem that converts entropy inequalities into existence of catalytic and multi-copy transformation channels.","marker":"[12]"},{"why":"It builds the quantum-majorization semiring and its power universals, the structure used in the application to state transformations.","marker":"[17]"},{"why":"It shows that Gibbs-preserving operations can generate coherence, the capability whose catalytic limits the paper characterizes.","marker":"[9]"},{"why":"It classifies the classical monotone homomorphisms and derivations, reducing the semiring conditions for classical input pairs to R\\'enyi-order inequalities.","marker":"[10]"}],"fun_headline_variants":["Maximal quantum Rényi entropy found for alpha>2","Alpha>2: largest Rényi entropy is alpha-z, z=alpha-1","For alpha>2, maximal Rényi entropy is now known","Completing the search: maximal Rényi entropy for alpha>2","Rényi entropy cap: alpha>2 solved with z=alpha-1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the divergence $D_{\\alpha,\\alpha-1}$ satisfies the data-processing inequality for every $\\alpha\\ge2$ on all pairs of full-support states; the paper imports this result from the literature rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Maximal quantum Rényi entropy found for alpha>2","Alpha>2: largest Rényi entropy is alpha-z, z=alpha-1","For alpha>2, maximal Rényi entropy is now known","Completing the search: maximal Rényi entropy for alpha>2","Rényi entropy cap: alpha>2 solved with z=alpha-1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1330,"prompt_tokens":961,"completion_tokens":369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":269}},"tokens_in":577,"tokens_out":369,"duration_ms":3428,"temperature":1.0,"reasoning_tokens":269,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:12:28.676085+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any $\\alpha>2$ and a pair of full-support states $\\rho,\\sigma$, and compute $D_{\\alpha,\\alpha-1}(\\rho\\|\\sigma)$ alongside $D_{\\alpha,\\alpha-1}(\\Phi(\\rho)\\|\\Phi(\\sigma))$ for a trace-preserving channel $\\Phi$ such as a dephasing map; a violation would falsify the imported data-processing inequality and with it the lower bound in Proposition 2.","supporting_citations":[{"cited_title":"The following lemma provides an explicit characterization of such elements","cited_arxiv_id":null,"evidence_quote":"It defines the family of $\\alpha$-$z$ R\\'enyi relative entropies, including the quantity $D_{\\alpha,\\alpha-1}$ that the paper proves maximal for $\\alpha>2$."},{"cited_title":"Optimal sequence of quantum measurements in the sense of Stein’s lemma in quantum hypothesis testing","cited_arxiv_id":null,"evidence_quote":"It establishes the geometric R\\'enyi relative entropy as the prepared divergence for $\\alpha\\in[0,2]$, giving the previously known half of the maximal characterization that the paper extends."},{"cited_title":"From Blackwell dominance in large samples to R´ enyi divergences and back again","cited_arxiv_id":null,"evidence_quote":"It provides the data-processing inequality for $D_{\\alpha,\\alpha-1}$ for all $\\alpha\\ge2$, which is imported to prove the lower-bound half of Theorem 1."},{"cited_title":"α-z-Relative Renyi Entropies","cited_arxiv_id":null,"evidence_quote":"It supplies the preordered-semiring comparison theorem that converts entropy inequalities into existence of catalytic and multi-copy transformation channels."},{"cited_title":"Min-and max-relative entropies and a new entanglement monotone","cited_arxiv_id":null,"evidence_quote":"It builds the quantum-majorization semiring and its power universals, the structure used in the application to state transformations."},{"cited_title":"We call a monotone homomorphismdegenerateifx⊴y⇒Φ(x) = Φ(y)","cited_arxiv_id":null,"evidence_quote":"It shows that Gibbs-preserving operations can generate coherence, the capability whose catalytic limits the paper characterizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It classifies the classical monotone homomorphisms and derivations, reducing the semiring conditions for classical input pairs to R\\'enyi-order inequalities."}],"review_version":1}