{"id":"bc4df178-d04e-41b4-a63e-7e931d8e80b4","arxiv_id":"2505.19904","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For H = γH0 + V with spectral gap η and bounded V, the eternal leakage between coarse-grained spectral components is bounded by (1 - 4π∥V∥/(γη))^{-1/2} - 1.","lead":"This paper proves that a gapped quantum system hit by a small perturbation can only leak a bounded amount between separated energy regions, and the bound never grows with time. The result covers infinite-dimensional systems with continuous or unbounded spectra as long as the perturbation stays bounded, and it quantifies how well standard effective Hamiltonian methods approximate the true dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central bound is sound and the technical premises are standard and satisfied.","rationale":"The paper's central claim is the eternal, time-independent leakage bound of Theorem 7, with explicit constants depending only on \\|V\\| and the coarse-grained gap \\eta. I re-examined the two points most likely to hide a flaw. First, the existence and L1-norm bound of the function f used in Lemma 8 and Proposition 2 is exactly the Bhatia result cited by the authors; although the paper's proof of Lemma 8 assumes such an f, it does not pretend to prove existence, and the external theorem is standard. The subsequent Catalan-series estimate has the correct combinatorics and yields the stated threshold \\gamma>4\\pi\\|V\\|/\\eta. Second, the extension to unbounded H0 in Theorem 7 is justified by the truncation (62). The truncated operator is bounded, its spectrum is a subset of the original spectrum, and its coarse gap is at least \\eta; choosing E_n outside a bounded component \\sigma_k preserves P_k as a spectral projection. The strong dynamical convergence in Appendix G follows from the standard resolvent argument: pointwise convergence on Dom H plus the uniform resolvent bound gives strong resolvent convergence, hence strong convergence of the unitary groups. I do not see a step that would make the bound fail, and the limitations (bounded V, bounded spectral components, loose constants) are explicitly acknowledged rather than hidden. The absence of machine-checked proofs and reproduction code for the numerics lowers confidence but does not threaten the central theorem. The reader's weakest assumption is therefore not, in my judgment, an actual objection: it is a standard and satisfied technical premise. The correct verdict remains ACCEPT.","tokens_in":20583,"tokens_out":60760,"duration_ms":646249,"concrete_test":"One verification step worth running: in a model with one bounded band and one unbounded continuum, explicitly compute the truncated Hamiltonian (62) for a sequence n, verify that the spectral gap \\eta_n of the truncated operator satisfies \\eta_n\\ge\\eta, and confirm numerically that \\lim_{n\\to\\infty}\\varepsilon(\\|V\\|/(\\gamma\\eta_n))=\\varepsilon(\\|V\\|/(\\gamma\\eta)). Since the unbounded extension in Theorem 7 is the only step carried out in words rather than by a quantitative estimate, this check would settle whether the limiting argument is free of hidden spectral artifacts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim of Theorem 7 is supported by a parameter-free derivation. The one technical premise the reader singled out—the existence of an L1 function f with \\hat f(s)=1/s for |s|\\ge\\eta and inf \\int|f|=\\pi/(2\\eta)—is standard (Bhatia [15], Theorems VII.2.5 and VII.2.15), and the paper invokes it explicitly rather than proving it from scratch. The proof of Lemma 8 correctly reduces the Sylvester equation to this Fourier condition; the Catalan-series argument in Proposition 3 then yields the condition \\gamma>4\\pi\\|V\\|/\\eta. The unbounded extension in Theorem 7 is also sound: the truncation (62) has \\eta_n\\ge\\eta, P_k can be preserved by choosing E_n outside \\sigma_k, and Appendix G supplies the needed strong dynamical convergence. No internal inconsistency or unsupported step of comparable weight was found.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the leakage between coarse-grained spectral components of a Hamiltonian H = γH0 + V, where H0 has a spectral gap η and V is a bounded perturbation. It constructs two effective generators: a Bloch-type generator H_Bloch and a Hermitian Schrieffer–Wolff generator H_SW, and proves time-independent bounds on the distance between the true evolution and the effective evolutions. The main result, Theorem 7, states that for γ > 4π||V||/η the leakage out of any bounded spectral component obeys L_k(t) ≤ (1 − 4π||V||/(γη))^{-1/2} − 1 ≤ 9π||V||/(γη) for all t ∈ R, including for unbounded self-adjoint H0 via a truncation argument. The paper also applies these bounds to a tight-binding chain, a harmonic chain with infinite bands, and a transmon architecture.","tokens_in":20709,"tokens_out":20314,"duration_ms":197416,"significance":"If correct, Theorem 7 is a substantial and useful result: it gives an eternal, time-independent bound on band-to-band transitions that depends only on the perturbation norm and the coarse-grained spectral gap, with no dependence on the number, size, or dimensionality of the spectral components. The derivation is parameter-free and the constants are explicit; the extension to unbounded spectra via strong dynamical convergence is a genuine step beyond earlier discrete-spectrum results in [12,13]. The proofs are detailed and the technical premise from Bhatia's matrix analysis is standard. The numerical examples are illustrative rather than exhaustive, and no code is supplied, but the analytical claims are the paper's core contribution.","major_comments":[],"minor_comments":[{"comment":"In the truncation argument, the claim that \"by suitably choosing E_n, P_k can be made to coincide with P_{H0^(n)}(σ_k)\" needs clarification: this works when E_n is chosen in a different spectral component than σ_k, not for an arbitrary E_n. Please state this explicitly and justify that such a choice is always possible (for m=1 the leakage is trivial, and otherwise one can pick a point in another component and take n large enough).","section":"Section V, proof of Theorem 7"},{"comment":"The sentence stating that the first bound on the leakage \"exceeds the maximal meaningful value of 2\" is imprecise: for the leakage L_k(t) the trivial upper bound is 1, not 2. The final linear bound 9π||V||/(γη) is still valid, but the argument should be rephrased: for ||V||/(γη) ≥ 1/(9π) the linear bound is trivial, while for smaller values the inequality ε(x) ≤ 9πx follows from a short elementary estimate.","section":"Section V, after Eq. (70)"},{"comment":"Lemma 8 states the Fourier condition for |s| > η, whereas Proposition 2 and Eq. (28) use |s| ≥ η. The two are effectively equivalent for admissible L1 functions because the Fourier transform is continuous, but the statements should be aligned to avoid confusion.","section":"Appendix A, Lemma 8"},{"comment":"The phrase \"Following standard arguments (cf. Appendix B)\" appears to cite the wrong appendix for the inequalities (38)–(40); these are proved in Appendix D. Please correct the cross-reference.","section":"Theorem 4 proof"},{"comment":"The numerical simulations in the examples are described without code or full parameter specifications (e.g., the realizations {r_i} in the tight-binding chain). Since the paper's main results are analytic, this is not blocking, but adding a short reproducibility note would improve the presentation.","section":"Section VI"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid theoretical paper whose central bound appears correct and whose unbounded-spectrum extension is a meaningful improvement over earlier work. The issues I found are local and easily fixed; I recommend minor revision rather than acceptance in the current form because the proof of Theorem 7 would benefit from an explicit statement about the choice of E_n."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a rigorous, parameter-free extension of eternal adiabaticity to coarse-grained spectral decompositions, including unbounded unperturbed Hamiltonians. I read the proof carefully and the central bound holds up; the stress-test note is right that there is no load-bearing gap. The paper deserves serious refereeing.\n\nWhat is actually new: prior eternal bounds [12,13] were fine-grained (discrete eigenspaces). Here the authors adapt the Bloch and Schrieffer–Wolff constructions to arbitrary coarse-grained decompositions with a finite spectral gap, and prove a time-independent leakage bound depending only on ||V|| and η. The extension to unbounded H0 via truncation (Theorem 7) is handled honestly, with strong dynamical convergence proved in Appendix G. The Catalan-series argument in Proposition 3 is clean, and the bound δ(x) is explicit. The paper also shows the Schrieffer–Wolff version and bounds its distance to the true evolution. That is a genuine advance over the cited fine-grained results.\n\nSoft spots, in proportion: the numerical examples are illustrative and no code is provided, so they are not independently reproducible. That is acknowledged and not central. The restriction to bounded perturbations is stated up front and left as future work, so not a flaw. There is a small imprecision in Section V: they call '2' the maximal meaningful value of the leakage, but L_k(t) as an operator norm cannot exceed 1. It doesn't affect the result, since the linear bound is trivially valid for large scaling. The proof relies on the existence of an L1 function f with \\hat f(s)=1/s for |s|≥η; this is standard from Bhatia and the paper invokes it explicitly, so I don't regard it as a soft spot.\n\nWho should read this: people working on adiabatic quantum computing, effective Hamiltonians, and rigorous perturbation theory. It is a solid theoretical tool, not a paradigm shift.\n\nRecommendation: send it to a serious referee. The proofs are detailed enough for careful checking, and the result is likely to be cited as the standard coarse-grained eternal bound.","headline":"A genuine and rigorous coarse-grained eternal leakage bound; the central proof is sound and the paper should go to peer review.","tokens_in":21248,"tokens_out":3735,"would_cite":true,"duration_ms":25608,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q15","47A55","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a universal time-independent ceiling on transitions between gapped spectral bands of a perturbed quantum system.","keywords":["eternal adiabaticity","spectral gap","Schrieffer-Wolff transformation","Bloch equations","effective Hamiltonian","leakage bounds","perturbation theory","unbounded Hamiltonians"],"falsifier":"A direct numerical test would simulate a two-band model, say $H_0$ with eigenvalues $0$ and $\\eta$, plus an off-diagonal bounded perturbation of norm $\\lVert V\\rVert$, with $\\gamma$ just above $4\\pi\\lVert V\\rVert/\\eta$, and compute $L_1(t)$ at many long times: exceeding $9\\pi\\lVert V\\rVert/(\\gamma\\eta)$ at any $t$ would refute Theorem 7. A sharper test targets the premise directly by trying to construct a pair $H_0,V$ for which the minimal $L^1$ norm of such an $f$ is strictly larger than $\\pi/(2\\eta)$, which would break Proposition 3.","tokens_in":20396,"feed_emoji":"⚛️","tokens_out":8198,"duration_ms":54861,"temperature":0.7,"pith_summary":"The paper proves that when a quantum system with a gapped energy spectrum is weakly perturbed, the chance of making a transition between distinct coarse-grained spectral components stays small forever. For a Hamiltonian $H = \\gamma H_0 + V$ whose unperturbed part has a spectral gap $\\eta$ between components, it shows that if $\\gamma > 4\\pi\\lVert V\\rVert/\\eta$, then the leakage out of any bounded spectral component obeys $L_k(t) \\le (1 - 4\\pi\\lVert V\\rVert/(\\gamma\\eta))^{-1/2} - 1 \\le 9\\pi\\lVert V\\rVert/(\\gamma\\eta)$ for all times $t$. The estimate depends on $H_0$ only through the gap, so it covers continuous spectra, infinitely many bands, and unbounded unperturbed Hamiltonians. The same machinery gives uniform-in-time bounds on how well the Schrieffer–Wolff and Bloch effective dynamics approximate the true evolution.","feed_headline":"Perturbed quantum systems keep band leakage small forever","feed_subtitle":"A time-independent inequality limits transitions between gapped energy bands using only the perturbation norm and the spectral gap.","key_machinery":"The load-bearing construction is the perturbative solution of the Bloch equations. At each order the unknown operator solves a Sylvester equation $[H_0,X]=Y$, and Lemma 8 gives the solution $X=\\int_{-\\infty}^{\\infty} e^{-itH_0}Y e^{itH_0} f(t)\\,dt$ for any $f\\in L^1(\\mathbb{R})$ whose Fourier transform is $1/s$ on $|s|\\ge \\eta$; the best possible choice has $\\inf_f \\int |f| = \\pi/(2\\eta)$. The resulting recursive series for the Bloch wave operator $\\Omega$ has term norms bounded by Catalan numbers, so the Catalan generating function yields the explicit closeness estimate $\\lVert\\Omega - 1\\rVert \\le \\delta(\\lVert V\\rVert/(\\gamma\\eta)) < 1$, which is what turns into the eternal leakage bounds.","core_discovery":"On the paper's own terms, the central discovery is Theorem 7: for a self-adjoint $H_0$ (possibly unbounded), a coarse-grained spectral decomposition with gap $\\eta = \\inf_{k\\neq l}\\operatorname{dist}(\\sigma_k,\\sigma_l) > 0$, and a bounded self-adjoint perturbation $V$, whenever $\\gamma > 4\\pi\\lVert V\\rVert/\\eta$ the leakage $L_k(t) = \\lVert Q_k e^{-itH}P_k\\rVert$ out of any bounded spectral component is bounded uniformly in time by $(1 - 4\\pi\\lVert V\\rVert/(\\gamma\\eta))^{-1/2} - 1 \\le 9\\pi\\lVert V\\rVert/(\\gamma\\eta)$. The proof works by constructing a block-diagonal effective generator $H_{\\mathrm{Bloch}}$ that is similar to $H$, showing the distance between the two evolutions is small for all $t$, and then passing to unbounded $H_0$ by truncating the spectrum and taking the limit. A Hermitian effective generator $H_{\\mathrm{SW}}$ is also built through a Schrieffer–Wolff rotation, with its own uniform error bound.","pith_inferences":["A natural next step is to allow unbounded perturbations, for which the authors expect state-dependent bounds with a slower-than-$1/\\gamma$ decay or even persistent leakage for arbitrarily weak perturbations.","The same block-diagonal construction could be adapted to open quantum systems, where a Lindblad generator that is block-diagonal in the unperturbed decomposition might yield an eternal bound on population leakage under dissipation.","Because the threshold $\\gamma > 4\\pi\\lVert V\\rVert/\\eta$ comes from the optimal $L^1$ constant, simple few-level models with $\\lVert V\\rVert/\\eta$ close to $1/(4\\pi)$ could probe how tight the bound is; the numerics in the paper only test the weak-coupling scaling.","One could test the bound's independence of system size directly in the tight-binding chain: the paper's numerics show size independence for 50–500 cells, and the theorem predicts the same bound for the infinite chain, which is outside numerical reach but within the theorem's scope."],"forward_implications":["For any bounded spectral component, the leakage is forever bounded by $\\sim 9\\pi\\lVert V\\rVert/(\\gamma\\eta)$, so in the weak-perturbation limit transitions out of a band are suppressed linearly in the perturbation strength at all times.","The bound applies to unbounded and continuous spectra as long as the coarse-grained gap is positive and the perturbation is bounded, covering crystal band models, harmonic chains, and transmon-like cosine potentials.","The effective evolutions generated by the Bloch and Schrieffer–Wolff Hamiltonians stay uniformly close to the true evolution, putting quantitative error control under standard effective-Hamiltonian methods.","Because the estimate depends on $H_0$ only through $\\eta$, it is independent of the number of bands, the dimension of each band, and the system size, including infinite chains.","The leakage decays at least as $O(1/\\gamma)$ as the unperturbed part strengthens, matching the leading-order behaviour seen in the paper's numerical examples."],"supporting_citations":[{"why":"supplies Lemma 8: the Sylvester-equation solution via an $L^1$ kernel whose Fourier transform is $1/s$ on $|s|\\ge\\eta$, and the optimal constant $\\pi/(2\\eta)$ used in the norm estimates.","marker":"[15]"},{"why":"provides the eternal-adiabaticity formalism and the defining requirements (block-diagonal, similar, close-to-identity effective generator) plus the identity relating leakage to the distance between evolutions.","marker":"[12]"},{"why":"defines the Schrieffer–Wolff transformation as a direct rotation between unperturbed and perturbed subspaces, used to construct the Hermitian effective generator.","marker":"[6]"},{"why":"introduces the Bloch wave-operator equations that the paper adapts to coarse-grained spectral decompositions.","marker":"[7]"},{"why":"gives the recursive wave-operator method for solving the Bloch equations that the perturbative ansatz follows.","marker":"[9]"},{"why":"supplies the general theory of effective Hamiltonians behind the three requirements the constructed generator must satisfy.","marker":"[14]"},{"why":"is the fine-grained large-time stability result that this paper extends to coarse-grained and unbounded spectra.","marker":"[13]"}],"fun_headline_variants":["Eternal bound on quantum spectral leakage under perturbation","Uniform bound on leakage between spectral bands","Leakage stays small forever under perturbation","Time-independent leakage bound for gapped quantum systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain of bounds depends on the existence of an integrable function $f$ whose Fourier transform is exactly $1/s$ for every frequency $|s|\\ge\\eta$, with a finite and optimally small $L^1$ norm; if that single analytic fact failed, the recursive Bloch solution and all subsequent estimates would not get off the ground.","fun_headline_variants_meta":{"raw":{"variants":["Eternal bound on quantum spectral leakage under perturbation","Uniform bound on leakage between spectral bands","Leakage stays small forever under perturbation","Time-independent leakage bound for gapped quantum systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001037,"raw_usage":{"total_tokens":4355,"prompt_tokens":925,"completion_tokens":3430,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":3374}},"tokens_in":541,"tokens_out":3430,"duration_ms":20581,"temperature":1.0,"reasoning_tokens":3374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:04:29.423081+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical test would simulate a two-band model, say $H_0$ with eigenvalues $0$ and $\\eta$, plus an off-diagonal bounded perturbation of norm $\\lVert V\\rVert$, with $\\gamma$ just above $4\\pi\\lVert V\\rVert/\\eta$, and compute $L_1(t)$ at many long times: exceeding $9\\pi\\lVert V\\rVert/(\\gamma\\eta)$ at any $t$ would refute Theorem 7. A sharper test targets the premise directly by trying to construct a pair $H_0,V$ for which the minimal $L^1$ norm of such an $f$ is strictly larger than $\\pi/(2\\eta)$, which would break Proposition 3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies Lemma 8: the Sylvester-equation solution via an $L^1$ kernel whose Fourier transform is $1/s$ on $|s|\\ge\\eta$, and the optimal constant $\\pi/(2\\eta)$ used in the norm estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the eternal-adiabaticity formalism and the defining requirements (block-diagonal, similar, close-to-identity effective generator) plus the identity relating leakage to the distance between evolutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the Schrieffer–Wolff transformation as a direct rotation between unperturbed and perturbed subspaces, used to construct the Hermitian effective generator."},{"cited_title":"Bravyi, D","cited_arxiv_id":null,"evidence_quote":"introduces the Bloch wave-operator equations that the paper adapts to coarse-grained spectral decompositions."},{"cited_title":"Lindgren, The Rayleigh-Schr¨ odinger perturbation and the linked-diagram theorem for a multi-configurational model space, J","cited_arxiv_id":null,"evidence_quote":"gives the recursive wave-operator method for solving the Bloch equations that the perturbative ansatz follows."},{"cited_title":"Facchi, M","cited_arxiv_id":null,"evidence_quote":"supplies the general theory of effective Hamiltonians behind the three requirements the constructed generator must satisfy."}],"review_version":1}