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Filter width sets thermalization rate at square-root scaling

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load-bearing objection O(√δ) thermalization bound for filtered states, derived from Floquet ETH via a block-diagonal RDM structure the 3 major comments →

arxiv 2607.06847 v1 pith:HBUR6BTQ submitted 2026-07-07 quant-ph

How thermal is a filtered state?

Yilun Yang , J. Ignacio Cirac , Mari Carmen Ba\~nuls This is my paper
classification quant-ph
keywords statesthermalfloquetenergyfilterfilteredwidthallows
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks a precise quantitative question: if you start with a simple pure state and apply an energy filter that narrows its energy spread to width δ, how close do you get to a genuine thermal state? The authors answer by working in the Floquet (discrete-time) regime, where the filter becomes a weighted sum of time-evolved states. They show that the reduced density matrix of a Floquet-filtered state is effectively a direct sum of the density matrices at each time step, because Loschmidt echoes between different times decay exponentially with system size. This block-diagonal structure lets them reproduce known scaling laws for Rényi entropies — logarithmic for α>1, linear for α=1, and superlinear for α<1 — confirming that polynomially narrow filters do not produce fully thermal entanglement. The central result is that, under the Floquet eigenstate thermalization hypothesis (ETH), the deviation of any local observable from its thermal value is bounded by the square root of the filter width, O(√δ). This bound is derived by showing that the filter suppresses off-diagonal matrix elements in the eigenbasis as a Gaussian, and then bounding the spectral radius of the resulting random-band-like matrix using moment methods. The authors extend the result to the standard Hamiltonian setting by grouping consecutive time-evolution terms into blocks of duration Θ(1), effectively constructing an equivalent Floquet problem, and show the same O(√δ) convergence holds.

Core claim

The trace distance between a filtered state and its thermal counterpart scales as the square root of the filter width δ, not as δ itself. This means the convergence to thermal behavior is slower than one might naively expect from the energy variance, but it is still a concrete, controllable rate that follows directly from ETH without additional independence assumptions on the off-diagonal matrix elements.

What carries the argument

The argument rests on three pieces. First, the Loschmidt echo decay makes the reduced density matrix of a Floquet-filtered state approximately block-diagonal, with each block corresponding to a single time step. Second, the normalized filter coefficients form a Gaussian in the time index, so the filter acts as a Gaussian suppression of off-diagonal elements in the Floquet eigenbasis. Third, the matrix whose spectral radius must be bounded — call it Q(δ) — has entries that are ETH off-diagonal elements multiplied by this Gaussian suppression. Its moments are bounded using a generalized ETH ansatz for higher-order correlations, yielding tr[Q(δ)^{2l}] = O(D δ^l), which forces the eigenvalues to

Load-bearing premise

The bound relies on a generalized ETH ansatz for higher-order correlations among off-diagonal matrix elements, specifically that products of n such elements along a closed loop scale as D^{1−n/2}. This is not derived from standard ETH but is motivated by consistency requirements for bounded observables. If these correlations have a different structure, the spectral radius bound on Q(δ) — and thus the O(√δ) convergence rate — could weaken.

What would settle it

A Floquet or Hamiltonian system satisfying standard ETH but violating the generalized higher-order correlation ansatz (Eq. 40), such that the moments tr[Q(δ)^{2l}] grow faster than O(D δ^l), would break the O(√δ) bound and potentially slow the convergence rate.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The O(√δ) rate quantifies how many filter applications are needed to reach a target thermal precision, directly informing the cost of quantum algorithms that approximate thermal states via filtering.
  • The equivalence between Floquet-filtered and Hamiltonian-filtered states means discrete-time Floquet filters can replace continuous-time Hamiltonian filters, reducing the number of distinct evolution steps by a factor of order √N for product initial states.
  • The logarithmic Rényi entropy scaling for α>1 shows that filtered states with polynomially narrow width are not thermal pure quantum states in the entanglement sense, even though local observables may already be close to thermal.
  • The moment-based bound on Q(δ) suggests that if higher-order ETH correlations deviate from the assumed scaling form, the convergence rate could change, providing a diagnostic for beyond-ETH physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This manuscript addresses how narrow an energy filter must be for a filtered pure state to approximate a thermal state. The authors work primarily in the Floquet regime, where the filtered state decomposes into a weighted sum of stroboscopically evolved states. They show that the reduced density matrix is approximately block-diagonal (with blocks corresponding to different time steps), which allows them to compute Rényi-α entanglement entropies and reproduce the distinct scalings found by Morettini et al. [PRL 133, 240401 (2024)]. The central new result is that, under Floquet ETH, the trace distance between filtered and thermal states is bounded by O(√δ_F) (Eq. 42), and this extends to the Hamiltonian regime as Õ(O(√δ_H)) (Eq. 46). The argument proceeds by mapping the observable deviation to the spectral radius of a matrix Q(δ_F) (Eq. 38), whose eigenvalues are bounded via the moment method using a generalized ETH correlation ansatz (Eq. 40).

Significance. The question of how filter width controls thermality is timely and practically important for quantum simulation. The O(√δ) bound is a concrete, falsifiable prediction that refines the canonical universality hypothesis of Dymarsky and Liu [14] and is consistent with existing numerical evidence. The block-diagonal RDM structure and its consequences for entropy scaling constitute a useful conceptual contribution. The reduction from Hamiltonian to Floquet filters (Section V) has practical implications for quantum algorithms, potentially reducing measurement overhead by O(√N). The two exactly solvable models in Appendix F (the Floquet toy model and the global random unitary model) provide valuable independent verification of the entropy scaling claims.

major comments (3)
  1. Appendix D, Eq. (41): The central O(√δ_F) bound on the spectral radius of Q(δ_F) is established via the moment method, where the leading contribution to each 2l-th moment comes from 'conjugated pair' contractions (R_{m,n}R_{n,m}), while 'longer loop' terms involving all-distinct indices are claimed to contribute only higher-order corrections in δ_F. This claim is explicitly verified only for l=1 (Eq. D1) and l=2 (Eqs. D4–D6). The generalization to arbitrary l is stated as analogous but not proven. Since the spectral radius bound λ_p = O(√δ_F) (Eq. 39) requires the moment bound tr[Q(δ_F)^{2l}] = O(D δ_F^l) to hold for all l (or at least for l growing with D), the unproven generalization is load-bearing for the central claim. The authors should either provide the general inductive argument or restrict the claim to a statement about low-order moments with a corresponding weakening of the O(
  2. Section IV, Eqs. (37)–(42): The generalized ETH correlation ansatz R_{k1,k2}R_{k2,k3}···R_{kn,k1} = F_n/D^{n/2−1} (Eq. 40), drawn from [47, 48], is acknowledged as not implied by standard ETH. The paper correctly notes that the independence assumption used in [14] is stronger than what is needed here. However, the precise conditions on F_n under which the moment bound holds are not stated. If F_n grows with n (e.g., F_n ~ n^c for some c > 0), the longer-loop contributions in Appendix D could scale differently. The authors should state explicitly what growth rate of F_n is compatible with the O(√δ_F) bound, and whether this condition is physically motivated by existing results in the literature. This would clarify the logical status of the result: is it 'O(√δ_F) under standard Floquet ETH plus a mild condition on F_n,' or does it require a stronger assumption?
  3. Section V, Eq. (45) and Appendix E: The trace distance between the Hamiltonian-filtered state |ψ_δH⟩ and its Floquet approximation |ẽψ_δH⟩ is bounded as Õ(O(√δ_H)). The derivation in Appendix E bounds the Euclidean norm of the difference |q⟩ by O(δ_H √log N) (Eq. E3), which then gives d = O(√(∥|q⟩∥)) = Õ(O(√δ_H)). However, the step from ∥|q⟩∥ = O(δ_H √log N) to d = O(√δ_H) involves a square root of the norm, not the norm itself. The trace distance is d = √(1 − |⟨ψ_δH|ẽψ_δH⟩|²/ẽN²), and the bound d = O(√(∥|q⟩∥)) appears to use ∥|q⟩∥ as a proxy for 1 − |⟨ψ_δH|ẽψ_δH⟩|²/ẽN². The authors should verify that the relationship between ∥|q⟩∥² and the overlap is tight enough to support the claimed Õ(O(√δ_H)) scaling, particularly regarding the log N factor. Also, the Loschmidt echo bound in Eq. (E4) uses e^{−O(log² N)}, which is sub-exponential; the statement that these terms have 'sub-exponentally
minor comments (6)
  1. Table I: The entry for the truncated Floquet filter with α < 1 reads O(x/δ_F) + log(1/δ_F), while the text in Eq. (31) gives S_{α<1}(ρ^{trunc}_{A,δ_F}) ~ x·k_α T/δ_F + O(log(1/δ_F)). The table omits the k_α T factor. Consider adding it for consistency.
  2. Section III.B, Eq. (27): The coefficient of the linear term is k_1 T/(√(2π) δ_F), where k_1 is the entanglement growth rate. The text states this is '√(2π) times slower' than direct time evolution, but the factor should be √(2π) in the denominator (i.e., the growth rate is divided by √(2π)), which is indeed slower. The statement is correct but could be stated more precisely.
  3. Fig. 5: The system size is N=15, which is small. While the authors acknowledge this is exact diagonalization, the O(√δ) scaling is observed over roughly one decade in 1/δ. A comment on the expected finite-size effects and whether larger systems (even N=18–20) could be accessible would strengthen the numerical evidence.
  4. Eq. (33): The constants α_{δ_F} and β_{δ_F} depend on δ_F, but their dependence is not specified. Since β_{δ_F} = γ_min/(1 + 3/(2μ)) and γ_min = min_{|n|≤xM} γ_n, β_{δ_F} could in principle decrease as δ_F decreases (if smaller filter widths probe times where γ_n is smaller). A brief comment on whether β_{δ_F} remains O(1) for the relevant parameter range would clarify the bound for small δ_F.
  5. Section II.B: The distinction between 'cosine' and 'truncated' Floquet filters is introduced but the notation P_{δ_F} is used without superscript when the distinction is 'irrelevant.' Given that the entropy scalings differ between the two (Table I), the authors should be more careful about which filter is being used in each subsequent equation, particularly in Section IV where the thermality bound is derived.
  6. The reference to 'canonical universality' in [14] is discussed in Section IV, but the relationship between the O(√δ) bound and the η ≈ 2 found numerically in [14] could be made more precise. Specifically, η = 2 corresponds to O(δ^{1/2}), which matches the bound; this connection is noted but could be highlighted as direct confirmation of those numerical findings.

Circularity Check

0 steps flagged

No significant circularity: the central O(√δ) bound derives from external ETH assumptions, not from self-citation or definitional equivalence.

full rationale

The paper's central result—the O(√δ_F) trace distance bound between Floquet-filtered states and thermal states (Eq. 42)—is derived from Floquet ETH (Eq. 36, cited from [20] by D'Alessio and Rigol) plus a generalized correlation ansatz for higher-order ETH matrix element correlations (Eq. 40, cited from [47, 48] by Foini and Kurchan). Neither of these load-bearing assumptions is authored by the present paper's authors, so there is no self-citation chain. The derivation proceeds through the moment method (Appendix D): the spectral radius of Q(δ_F) is bounded by computing tr[Q(δ_F)^{2l}] = O(D δ_F^l) (Eq. 41), where the leading contributions come from conjugated pair contractions and longer-loop terms are shown to be higher-order in δ_F under the ansatz. This is a genuine mathematical reduction, not a definitional equivalence. The entropy scaling results (Section III B) reproduce the findings of Morettini et al. [1] (external authors) using an independent block-diagonal RDM argument. Self-citations [4–8, 15] provide the filter construction framework but the O(√δ) thermality bound does not reduce to these prior works. The Hamiltonian regime extension (Section V) maps to an effective Floquet problem via a grouping argument (Eq. 43–44) with a trace distance bound (Eq. 45) that is independently derived. The assumptions may be unproven (as the paper itself acknowledges regarding Eq. 40), but unproven assumptions are a correctness risk, not circularity. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities or free parameters. All axioms are domain assumptions from the quantum thermalization literature or standard mathematical results. The most fragile axiom is the generalized ETH correlation ansatz (Eq. 40), which is not a consequence of standard ETH and is load-bearing for the central result.

axioms (5)
  • domain assumption Floquet ETH: observable matrix elements satisfy O_{k,k'} = Ō δ_{k,k'} + √(O²/D) R_{k,k'} with R_{k,k'} Gaussian-distributed (Eq. 36, Section IV)
    Standard Floquet ETH ansatz from [20], used to decompose observable matrix elements into diagonal and off-diagonal parts.
  • domain assumption Generalized ETH correlation ansatz: R_{k1,k2}R_{k2,k3}···R_{kn,k1} = F_n/D^{n/2−1} for distinct indices (Eq. 40, Section IV)
    Higher-order correlation structure from [47, 48], not implied by standard ETH. Load-bearing for the spectral radius bound on Q(δ_F) and thus the O(√δ) trace distance result.
  • domain assumption Exponential decay of Loschmidt echo: γ_n := inf_{|ψ⟩} f(nT) > 0 for n ≠ 0 (Eq. 15, Section III.A)
    Assumes product-state Loschmidt echoes decay exponentially at all stroboscopic times. Underlies the block-diagonal RDM structure. Expected generically but can fail for scar states [26-30].
  • domain assumption Linear entanglement growth: S_α(ρ_{A,mm}) = k_α |m|T grows linearly in m for any α (Section III.B)
    Standard expectation for generic local Hamiltonians [32, 37-41], used to compute Rényi entropy scalings.
  • standard math Lieb-Robinson bound for quasi-locality of V(t) (Eq. B7, Appendix B)
    Standard Lieb-Robinson bound from [36], used to factorize the evolution operator and bound off-diagonal RDM blocks.

pith-pipeline@v1.1.0-glm · 33312 in / 2539 out tokens · 325463 ms · 2026-07-10T00:19:22.888391+00:00 · methodology

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read the original abstract

Quantum many-body states with sufficiently low energy variance can serve as approximations to thermal states, and they may be prepared by energy filtering simple pure states. In this work, we examine how narrow the filter width must be to guarantee thermal behavior. To this end, we analyze the problem in the Floquet regime, where filtered states are found to be equivalent to time averages. This equivalence allows us to reproduce the distinct R\'enyi-$\alpha$ entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.

Figures

Figures reproduced from arXiv: 2607.06847 by J. Ignacio Cirac, Mari Carmen Ba\~nuls, Yilun Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. Action of the cosine filtered state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The normalized filter coefficients [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagram for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Overlap between diagonal blocks [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The difference between filtered state and diagonal [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scaling of half-chain von Neumann entanglement en [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Factorization of the system [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Evolution of the Floquet toy model. The blue points [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The largest eigenvalue of bipartite RDM of the filtered [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Density histograms of eigenvalues of the bipartite [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Von Neumann and R´enyi-2 entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

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Reference graph

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