{"id":"debd5fca-0a40-4f9f-9f0b-d3e25f80c22c","arxiv_id":"2501.11049","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The proposed generalized entropic quantum speed limit bound is invalid because its key inequality fails for states with small minimum eigenvalues, as demonstrated by the paper's own single-qubit depolarizing example.","lead":"This paper derives a two-parameter family of quantum speed limits based on the alpha-z-Renyi relative entropy, claiming to cover both closed and open quantum systems. The bounds are supposed to unify several known entropic speed limits, but the central inequality is violated by the paper's own depolarizing-channel example.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (4) fails for full-rank states with kmin(ρ0) below exp(-1/(1−α)); at the threshold hα,z vanishes while the LHS Rényi divergence is positive, so the universal QSL family is unsupported.","rationale":"The reader's weakest_assumption identifies the same inversion flaw; my stress test sharpens it to an explicit zero-RHS contradiction, so I agree with the REJECT verdict. The quantitative claim in Eq. (4) is not merely unproven; it is numerically false for the paper's own depolarizing setup. Since the QSL time is defined as a maximum of ratios whose numerators come from Eqs. (4), (6), and (7), an invalid upper bound invalidates the lower-bound time in Eq. (11). I give the paper credit for a competent summary of α-z-Rényi relative entropies and their data-processing regions in Sec. II, and for transparently exposing its formulas in worked examples, but no formal verification or reproducible code is provided to offset the central failure. The additional errors noted by the reader, such as those in Eq. (32), are secondary; the failure of Eq. (4) is decisive on its own.","tokens_in":31319,"tokens_out":5923,"duration_ms":59978,"concrete_test":"Numerically evaluate Eq. (4) for the qubit depolarizing channel with α=1/2, z=1, ρ0=(1/2)(I+rσ_z), r=1−2e^{-2}, and Γτ=1. Compute the exact LHS from D_{1/2,1}(ρ_τ∥ρ_0)=-2 ln Tr(√ρ_τ√ρ_0), where ρ_τ=(1/2)(I+e^{-Γτ}rσ_z), and the RHS from Eq. (4) with h_{1/2,1}(ρ0)=|1+(1/2)ln((1-r)/2)|=0. Since the LHS is strictly positive and the RHS is zero, Eq. (4) is violated. If the authors instead intend to restrict to kmin>exp(-1/(1−α)), rerun the same check with r=3/4 and α=1/2, where the bracket is negative, to show the inequality also fails outside the claimed domain.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (4) is the engine of the paper: all subsequent bounds (6), (7), and the QSL times (11)–(13), (16)–(18), (21)–(23) inherit its proof. That proof breaks at Eq. (B3). The authors bound g_{α,z}(ρ_t,ρ_0)^{-1} by [1+(1−α) ln k_min(ρ_0)]^{-1}, but this inversion is only valid if the bracket is positive, and positivity is neither stated nor shown. Eq. (B5) is only a lower bound on Tr(ρ_t^α ρ_0^{1−α}) and can be negative or zero. The problem is not merely an omitted hypothesis: the auxiliary function defined in Eq. (5) uses the absolute value of the bracket, so for any state with k_min(ρ_0)=exp(-1/(1−α)), h_{α,z}(ρ_0)=0. Then the RHS of Eq. (4) is identically zero, while D_{α,z}(ρ_τ∥ρ_0)>0 for every ρ_τ≠ρ_0 with supp ρ_τ⊆supp ρ_0. Concrete instance: qubit ρ_0=(1/2)(I+rσ_z) with r=1−2e^{-2}≈0.7293, α=1/2, z=1. For depolarizing dynamics ρ_τ=(1/2)(I+e^{-Γτ}rσ_z), Γτ=1, D_{1/2,1}(ρ_τ∥ρ_0)>0, but h_{1/2,1}(ρ_0)=0, so Eq. (4) asserts a positive quantity is bounded above by zero. The same pathology propagates to Eqs. (6), (7), (11), and the unitary/open-system specializations; the claim that the result holds for arbitrary finite-dimensional dynamics, including pure and near-pure states, is therefore unsupported. If the bracket is negative, Eq. (B3) itself is false because the left side is positive and the right side is negative; if it is zero, the bound is not invertible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a two-parameter family of quantum speed limits based on α-z-Rényi relative entropies. The central result is Eq. (4), an upper bound on D_{α,z}(ρ_τ∥ρ_0) in terms of the time-averaged Schatten speed weighted by extremal-eigenvalue factors; from it the authors derive QSL times in Eqs. (11)–(13), specialize them to unitary and nonunitary dynamics in Eqs. (16)–(23), and illustrate them for single- and two-qubit systems. The paper claims validity for arbitrary finite-dimensional dynamics, including pure, mixed, separable, and entangled states.","tokens_in":31818,"tokens_out":8225,"duration_ms":75145,"significance":"If correct, the result would unify several entropic speed limits and extend the Petz-Rényi bounds of Ref. [56] to open systems using only minimal spectral information. The exposition is clear, and the analytical examples are explicit enough that the derivations can be checked directly. Unfortunately, the central inequality is false, and the proposed family of QSLs is therefore unsupported.","major_comments":[{"comment":"The derivation of the key bound inverts the lower bound (B5), Tr(ρ_t^α ρ_0^{1−α}) ≥ 1+(1−α) ln k_min(ρ_0). This inversion is valid only when the right-hand side is strictly positive. For any state with k_min(ρ_0) < exp(−1/(1−α)), the right-hand side is negative, so [g_{α,z}(ρ_t,ρ_0)]^{−1}, which is positive, cannot be bounded above by a negative number; for exact equality the bound is singular. This unstated positivity condition is not included in the theorem statements, and it is precisely the regime that the absolute value in Eq. (5) is meant to handle.","section":"Appendix B.1, Eq. (B3)"},{"comment":"Equation (4) fails numerically for the paper's own depolarizing example. Take α=1/2, z=1, and r=3/4, for which k_min(ρ_0)=1/8 < e^{−2}. Using Eq. (31), D_{1/2,1}(ρ_∞∥ρ_0) = −2 ln[(√(1−r)+√(1+r))/2] ≈ 0.1855, while the right-hand side of Eq. (4) evaluates to h_{1/2,1}(ρ_0) ∫_0^∞ [k_min(ρ_t)]^{−1/2} ∥dρ_t/dt∥_1 dt ≈ 0.0525. Thus Eq. (4) asserts that a positive divergence is bounded above by a smaller positive number. Since Eqs. (6), (7), and the QSL times (11)–(13), (16)–(23) are all derived from Eq. (4) or from the same inversion step, the main results of the paper collapse.","section":"Eq. (4) and Sec. V.A.2"},{"comment":"Appendix B begins by assuming that ρ_0 and ρ_t are full-rank, invertible density matrices, but the abstract and Sec. IV claim the speed limits hold for pure states and for arbitrary finite-dimensional dynamics. For pure states k_min=0, the factor [k_min(ρ_t)]^{α−1} diverges and the bound (B5) is not defined; no limiting argument is supplied. The domain of validity of the results is therefore narrower than claimed, independently of the algebraic error above.","section":"Appendix B and Sec. IV"}],"minor_comments":[{"comment":"The text states that D_{1/2,z}(ρ_τ∥ρ_0) ≈ 0 for all τ ≥ 0, but Eq. (31) gives a finite asymptotic value, e.g., ≈ 0.185 for r=3/4; the numerical statement should be corrected.","section":"Sec. V.A.2, discussion of Fig. 2"},{"comment":"Equation (8) restricts the result to 1/2 < z ≤ 1, while the stated data-processing-inequality region for α=1/2 includes z=1/2; please clarify whether the bound holds at the endpoint.","section":"Eq. (8)"},{"comment":"The use of the Araki-Lieb-Thirring inequality would be easier to verify if the text explicitly listed the operators and parameters; as written, it is not immediately obvious that the choice A=ρ_0^{(1−α)/2}, B=ρ_t^α, q=z, r=1/z satisfies the hypotheses of the inequality for all z in the stated range.","section":"Eq. (B4)"},{"comment":"Reference [107] contains a typo: 'Bathia' should be 'Bhatia'.","section":"References"}],"recommendation":"reject","confidential_remarks":"The numerical counterexample in Major Comment 2 is decisive and lies inside the advertised parameter region for the paper's own depolarizing channel. The central inequality is false, so the manuscript cannot be accepted in its present form; revision would require replacing the core proof and re-examining all derived bounds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central bound (4) is invalid as stated, and the proof breaks at Eq. (B3). The two-parameter family of entropic QSL times is therefore unsupported. That said, the idea is a reasonable next step in a line the authors know well, and the paper is clearly written; it needs a real fix before it can be used.\n\nWhat is new: α-z-RRE has not been used in QSL bounds before, and the extension to nonunitary dynamics via the Schatten speed of Kraus operators is a genuine addition. The concrete examples for single qubits and two qubits, including the Markovian/non-Markovian amplitude-damping comparison, are useful and appear computed with care. The paper is also honest about the bounds being loose and about their relationship to Ref. [56].\n\nThe soft spot is load-bearing. Equation (4) is proved by bounding [g_{α,z}(ρ_t,ρ_0)]⁻¹ from above via Eq. (B3). That step uses Eq. (B5) from Ref. [56], which is a lower bound on Tr(ρ_t^α ρ_0^{1−α}) that can be negative. Inverting a negative lower bound is not allowed; you need to know the quantity is positive. In fact, the bracket 1 + (1−α)ln k_min(ρ_0) can be zero, in which case the h_{α,z} defined in Eq. (5) vanishes and the RHS of Eq. (4) is identically zero. The LHS, D_{α,z}(ρ_τ‖ρ_0), is positive for any ρ_τ ≠ ρ_0 with the support condition. The stress-test gives a concrete qubit example: α=1/2, z=1, r=1−2e^{-2}, where h=0 and D>0. And the reader reports that the paper's own depolarizing example violates the bound numerically for α=1/2, z=1, r=3/4; on reading the proof, the issue is exactly the missing positivity. The absolute value in Eq. (5) masks the sign problem; it does not fix it. So the main theorem is false as stated, and all downstream QSL times (11)–(13), (16)–(18), (21)–(23) inherit the problem. This is not a mere missing hypothesis—the bound fails for parameter values the paper explicitly claims.\n\nThe citation pattern is fine: the reliance on Ref. [56] is legitimate, since that lemma is published and external. No circularity, no parameter fitting. This is an honest analytic derivation undermined by a mathematical error.\n\nWho benefits? Someone working on entropic speed limits might read this once to see the construction, but they cannot quote Eq. (4) or the QSL times. The examples could be of interest if the bound is repaired, since they show how to evaluate the quantities involved. A serious referee should ask for major revision or a new proof, with an extra hypothesis on k_min(ρ_0) to keep the bracket positive, and corrected numerical checks.\n\nRecommendation: send it to peer review if the editorial view is that a partially correct construction is salvageable; the error is subtle enough that referee time is justified. But do not accept anything based on Eq. (4) until the proof is fixed.","headline":"The central inequality (4) is invalid as stated—Eq. (B3) inverts a lower bound that can be negative, and the paper's own depolarizing example violates the bound—so the QSL family is unsupported, despite a sensible construction and useful examples.","tokens_in":32338,"tokens_out":3191,"would_cite":false,"duration_ms":33233,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","81P68","94A17"],"pacs":["03.65.-w","03.67.-a"],"model":"deepseek-v4-flash","headline":"This paper derives a two-parameter family of entropic quantum speed limits from the α-z-Rényi relative entropy, claimed to hold for arbitrary finite-dimensional unitary and nonunitary dynamics.","keywords":["quantum speed limits","alpha-z-Renyi relative entropy","Mandelstam-Tamm bounds","open quantum systems","nonunitary dynamics","Schatten norms","Uhlmann fidelity","Petz-Renyi relative entropy"],"falsifier":"Evaluate the right-hand side of Eq. (4) for a single-qubit initial state with $r=3/4$ and $\\alpha=1/2$, where $k_{\\min}(\\rho_0)=1/8$ and $1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]\\approx -0.0397$: the prefactor $h_{\\alpha,z}$ becomes negative while the left-hand side is nonnegative, so the claimed bound cannot hold as written unless the inversion step is repaired.","tokens_in":31132,"feed_emoji":"⏱","tokens_out":6080,"duration_ms":64249,"temperature":0.7,"pith_summary":"The paper tries to establish that the α-z-Rényi relative entropy, a two-parameter family of distinguishability measures that includes Petz-Rényi entropy, sandwiched Rényi entropy, fidelity, and affinity as special cases, can be turned into quantum speed limits: lower bounds on the time a quantum system needs to evolve from one state to another. It claims these bounds apply to finite-dimensional systems undergoing both unitary and nonunitary processes, for mixed and pure, separable and entangled states. The speed-limit time depends only on the smallest and largest eigenvalues of the probe and instantaneous states plus the Schatten speed of the dynamics, so it is cheap to evaluate. If correct, this supplies a unified entropic version of Mandelstam-Tamm bounds and extends earlier Petz-Rényi speed limits to open quantum systems.","feed_headline":"One entropy family recovers fidelity and Rényi quantum speed limits","feed_subtitle":"A two-parameter relative-entropy bound applies to unitary and noisy dynamics, needing only extreme eigenvalues and Schatten speed.","key_machinery":"The load-bearing object is the α-z-relative purity $g_{\\alpha,z}(\\rho,\\varrho)=\\mathrm{Tr}\\big[(\\varrho^{(1-\\alpha)/2z}\\rho^{\\alpha/z}\\varrho^{(1-\\alpha)/2z})^z\\big]$, whose logarithm defines the α-z-Rényi relative entropy. The proof bounds the time derivative of this purity using the Araki-Lieb-Thirring inequality, a trace lower bound $\\mathrm{Tr}(\\rho_t^\\alpha\\rho_0^{1-\\alpha})\\ge 1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]$, and Hölder and operator-norm estimates, producing the prefactor $h_{\\alpha,z}(\\rho_0)$ and a weighted time integral of the Schatten speed $\\|d\\rho_t/dt\\|_1$. The speed-limit time is then obtained by inverting that weighted average.","core_discovery":"The central claim is Eq. (4): for $0<\\alpha<1$ and $1\\ge z\\ge \\max\\{\\alpha,1-\\alpha\\}$, the α-z-Rényi relative entropy is bounded by $D_{\\alpha,z}(\\rho_\\tau\\|\\rho_0) \\le \\frac{\\alpha\\,h_{\\alpha,z}(\\rho_0)}{|1-\\alpha|}\\int_0^\\tau [k_{\\min}(\\rho_t)]^{\\alpha-1}\\|d\\rho_t/dt\\|_1\\,dt$, with $h_{\\alpha,z}(\\rho_0)$ given by a closed expression in the extreme eigenvalues of the initial state. From this bound, its swapped version, and the symmetrized version, the paper derives a generalized speed-limit time $\\tau^{\\mathrm{QSL}}_{\\alpha,z}$ in Eq. (11) that is claimed to lower-bound the actual evolution time. The family is two-parameter, is symmetric under $\\alpha\\to 1-\\alpha$, and specializes to Petz-Rényi speed limits at $z=1$ and to fidelity- and affinity-based bounds at $\\alpha=1/2$. The authors claim validity for pure or mixed, separable or entangled states, and for unitary and nonunitary dynamics.","pith_inferences":["The claimed coverage of pure states is not directly supported by the proof as written, because $k_{\\min}(\\rho_0)=0$ makes the logarithmic factor in $h_{\\alpha,z}$ singular; a separate limiting argument or full-rank regularization would be needed.","The positivity of $1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]$ is required for the inversion step in Eq. (B3), and it fails for sufficiently mixed states, including the paper's own example $r=3/4$, $\\alpha=1/2$; a repaired version would need either a restriction on $k_{\\min}(\\rho_0)$ or a different inversion bound.","In the depolarizing-channel example the speed limit nearly vanishes near $\\alpha=1/2$, suggesting that the practically useful regime of this family may lie away from the symmetric relative-entropy branch.","A natural testable extension is to check whether the same bounding strategy can be pushed beyond the data-processing-inequality region in the $(\\alpha,z)$ plane, which the paper leaves open."],"forward_implications":["For closed-system dynamics, the entropic speed limit scales with the inverse of the energy variance, placing it in the Mandelstam-Tamm class of bounds.","For open-system dynamics, the speed limit is expressed through the Schatten $1$-norm of the rate of change of the Kraus operators, so it can be evaluated from a quantum-channel description without diagonalizing the full Liouvillian.","At $z=1$ the family reduces to Petz-Rényi speed limits, and at $\\alpha=1/2$ it produces fidelity- and affinity-based speed limits, recovering several known results as particular choices of parameters.","The symmetry $\\tau^{\\mathrm{QSL}}_{\\alpha,z}=\\tau^{\\mathrm{QSL}}_{1-\\alpha,z}$ means every bound below $\\alpha=1/2$ has a matching bound above it.","If valid for higher-dimensional systems, the bounds provide a low-cost estimate of how fast distinguishability changes, requiring only extreme eigenvalues and the generator of the dynamics."],"supporting_citations":[{"why":"Supplies the lower bound $\\mathrm{Tr}(\\rho_t^\\alpha\\rho_0^{1-\\alpha})\\ge 1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]$ on which the inversion of the relative-purity bound depends.","marker":"[56]"},{"why":"Defines the α-z-Rényi relative entropy and the α-z-relative purity, including the properties $g_{\\alpha,z}\\le 1$ and skew symmetry used throughout.","marker":"[59]"},{"why":"Provides the Araki-Lieb-Thirring inequality used to lower-bound the relative purity by a simpler trace expression.","marker":"[108]"},{"why":"Provides the Lieb-Thirring inequality behind the same trace lower bound.","marker":"[109]"},{"why":"Supplies the Schatten-speed concept used to express the instantaneous quantum speed in the bound.","marker":"[88]"},{"why":"Gives the original Mandelstam-Tamm bound that the unitary specialization is claimed to extend.","marker":"[36]"}],"fun_headline_variants":["Two-parameter entropy family sets quantum speed limits","Unified entropic speed limits for noisy and clean dynamics","Entropy family bounds evolution speed for any quantum state","Alpha-z-Rényi entropy recovers and extends speed limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain assumes that the factor $1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]$ is positive so its reciprocal can be used to invert the lower bound on the relative purity, but this positivity is never stated and fails for sufficiently mixed states, including the paper's own $r=3/4$, $\\alpha=1/2$ example.","fun_headline_variants_meta":{"raw":{"variants":["Two-parameter entropy family sets quantum speed limits","Unified entropic speed limits for noisy and clean dynamics","Entropy family bounds evolution speed for any quantum state","Alpha-z-Rényi entropy recovers and extends speed limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000287,"raw_usage":{"total_tokens":1740,"prompt_tokens":1055,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":671,"completion_tokens_details":{"reasoning_tokens":620}},"tokens_in":671,"tokens_out":685,"duration_ms":7808,"temperature":1.0,"reasoning_tokens":620,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:42:57.121902+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the right-hand side of Eq. (4) for a single-qubit initial state with $r=3/4$ and $\\alpha=1/2$, where $k_{\\min}(\\rho_0)=1/8$ and $1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]\\approx -0.0397$: the prefactor $h_{\\alpha,z}$ becomes negative while the left-hand side is nonnegative, so the claimed bound cannot hold as written unless the inversion step is repaired.","supporting_citations":[{"cited_title":"Speed limits on correlations in bipartite quantum systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the lower bound $\\mathrm{Tr}(\\rho_t^\\alpha\\rho_0^{1-\\alpha})\\ge 1+(1-\\alpha)\\ln[k_{\\min}(\\rho_0)]$ on which the inversion of the relative-purity bound depends."},{"cited_title":"Bounding gener- alized relative entropies: Nonasymptotic quantum speed limits,","cited_arxiv_id":null,"evidence_quote":"Defines the α-z-Rényi relative entropy and the α-z-relative purity, including the properties $g_{\\alpha,z}\\le 1$ and skew symmetry used throughout."},{"cited_title":"Unified ( r, s)- Relative Entropy,","cited_arxiv_id":null,"evidence_quote":"Provides the Araki-Lieb-Thirring inequality used to lower-bound the relative purity by a simpler trace expression."},{"cited_title":"Thermodynamic uncertainty relation for quantum entropy production","cited_arxiv_id":"2404.18163","evidence_quote":"Provides the Lieb-Thirring inequality behind the same trace lower bound."},{"cited_title":"Work, Entropy Pro- duction, and Thermodynamics of Information under Pro- tocol Constraints,","cited_arxiv_id":null,"evidence_quote":"Supplies the Schatten-speed concept used to express the instantaneous quantum speed in the bound."},{"cited_title":"Estimating the time evolution of NMR sys- tems via a quantum-speed-limit-like expression,","cited_arxiv_id":null,"evidence_quote":"Gives the original Mandelstam-Tamm bound that the unitary specialization is claimed to extend."}],"review_version":1}