arxiv: 2605.03829 · v1 · submitted 2026-05-05 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

A Berry-Esseen Bound for Quantum Lattice Systems

Marcus Cramer , Fernando G.S.L. Brand\~ao , M\u{a}d\u{a}lin Gu\c{t}\u{a} , \'Alvaro M. Alhambra , Matteo Scandi

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:09 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords Berry-Esseen boundquantum lattice systemscentral limit theoremcorrelation lengthlocal observablesquantum fluctuationsnormal distribution

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The pith

Quantum states with finite correlation length obey a Berry-Esseen bound so that local observable measurements converge to normal distribution with error O(N^{-1/2} polylog N).

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

The paper proves a quantitative version of the central limit theorem for quantum lattice systems. For a local Hamiltonian on N particles and a state with finite correlation length, it shows that the distribution of measuring local observables such as energy is close to Gaussian. The bound on the distance to the normal distribution scales as O(N^{-1/2} polylog N) and is optimal up to the logarithmic factors. This matters because it replaces the usual asymptotic claim with an explicit rate that works for large but finite systems, allowing precise control over approximation errors in statistical mechanics and quantum information.

Core claim

Given a local quantum Hamiltonian on N particles and a quantum state with a finite correlation length, the measurement of local observables such as the energy follows a normal distribution, up to an error scaling as O(N^{-1/2} polylog(N)), which is optimal up to logarithmic factors.

What carries the argument

The Berry-Esseen bound for quantum lattice systems, which supplies a rigorous rate of convergence to the normal distribution for local measurements on states with finite correlation length.

If this is right

Explicit quantitative error estimates become available when approximating fluctuations of local observables in finite-size quantum many-body systems.
The central limit theorem applies with a concrete rate to any local observable in lattice models whose states have exponentially decaying correlations.
The same machinery yields bounds for other extensive quantities beyond energy, provided they are sums of local terms.
The near-optimality of the rate implies that further improvement would require stronger assumptions on the state or the observable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bound could be used to derive finite-size corrections in thermodynamic calculations for gapped quantum matter.
Extensions to critical states without finite correlation length would require separate analysis, possibly via renormalization-group methods.
The result may allow direct comparison with classical Berry-Esseen bounds through suitable quantum-to-classical reductions.

Load-bearing premise

The quantum state must have a finite correlation length.

What would settle it

Numerical computation or exact solution for a concrete finite-correlation-length model, such as a gapped 1D spin chain, showing that the distance from the local observable distribution to the nearest Gaussian exceeds C N^{-1/2} polylog(N) for some constant C and sufficiently large N.

Figures

Figures reproduced from arXiv: 2605.03829 by \'Alvaro M. Alhambra, Fernando G.S.L. Brand\~ao, Marcus Cramer, Matteo Scandi, M\u{a}d\u{a}lin Gu\c{t}\u{a}.

**Figure 1.** Figure 1: Representation of the operators defined in Eq. (62). Each view at source ↗

read the original abstract

It is expected that the statistical fluctuations of local observables in large quantum systems obey the central limit theorem, and approximate a normal distribution as their size grows. Here, we prove a version of the Berry-Esseen theorem for quantum lattice systems, which strengthens that central limit theorem by providing a rigorous convergence estimate towards the normal distribution for large but finite system size. Given a local quantum Hamiltonian on $N$ particles and a quantum state with a finite correlation length, the result states that the measurement of local observables such as the energy follows a normal distribution, up to an error scaling as $\mathcal{O}\left(N^{-\frac{1}{2}} \text{polylog}(N)\right)$, which is optimal up to logarithmic factors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves a Berry-Esseen bound giving O(N^{-1/2} polylog N) error for local observables in quantum lattices with finite correlation length.

read the letter

This paper establishes a quantitative version of the central limit theorem for local observables in quantum lattice systems. Given a local Hamiltonian on N sites and a state with finite correlation length, the distribution of extensive quantities like the energy is within O(N^{-1/2} polylog N) of a Gaussian in Kolmogorov distance. That rate is the main new piece: it matches the classical Berry-Esseen scaling up to logs and applies directly to quantum many-body settings where only qualitative CLTs were available before. The abstract states the assumptions cleanly and claims the bound is optimal up to the logarithmic factors, which is a reasonable bar for this type of result. The work is useful because finite-size corrections matter in numerical studies and in understanding when the thermodynamic limit is a good approximation. The finite-correlation-length condition is stated up front and is necessary for the rate; without it the bound would not hold in general, so the limitation is explicit rather than hidden. The main soft spot is that the full proof is not visible here, so one cannot yet check the technical steps or whether the polylog arises from standard Lieb-Robinson or clustering estimates. If the derivation is clean and avoids hidden constants that grow with system size, the result is solid. This is aimed at people in quantum statistical mechanics and many-body theory who already work with correlation lengths and want explicit error control for fluctuations. It is the sort of mathematical statement that deserves a serious referee: the claim is precise, the setting is standard, and the potential applications to finite-size scaling are clear. I would send it to review rather than desk-reject.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a Berry-Esseen bound for quantum lattice systems. For a local Hamiltonian on N particles and a quantum state with finite correlation length (independent of N), it establishes that the distribution of measurement outcomes for extensive local observables such as the energy converges to a normal distribution with total variation error O(N^{-1/2} polylog(N)). The bound is stated to be optimal up to logarithmic factors and strengthens the central limit theorem with an explicit finite-N rate.

Significance. If the proof is correct, the result supplies a near-optimal quantitative rate for the quantum central limit theorem under a standard locality assumption. This is significant for mathematical physics and quantum information, as it yields rigorous error estimates for fluctuations of local observables in finite but large systems, with direct relevance to statistical mechanics and sampling algorithms. The explicit polylog factor and optimality claim (relative to the classical 1/sqrt(N) rate) are notable strengths.

minor comments (2)

The abstract and introduction should explicitly state the precise class of observables to which the bound applies (e.g., whether it covers all local operators or only the Hamiltonian itself) and clarify the dependence of the correlation length on system parameters.
Notation for the finite-correlation-length assumption and the error term could be standardized across the statement of the main theorem and its corollaries to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures the main contribution: a Berry-Esseen bound establishing convergence to normality with rate O(N^{-1/2} polylog N) for local observables in quantum lattice systems with finite correlation length, under the stated assumptions on the Hamiltonian and state.

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained mathematical proof

full rationale

The paper establishes a conditional Berry-Esseen bound as a theorem for quantum lattice systems, explicitly requiring a quantum state with finite correlation length (independent of N) as a prerequisite. The error bound O(N^{-1/2} polylog(N)) for local observables is derived via standard concentration and approximation techniques in probability theory applied to quantum systems, without any reduction of the central claim to fitted parameters, self-definitions, or load-bearing self-citations that presuppose the result. The finite-correlation-length assumption is stated upfront and does not depend on the bound being proved. No steps in the derivation chain collapse by construction to the inputs; the proof is externally verifiable against classical Berry-Esseen results and quantum information tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of finite correlation length in the quantum state and locality of the Hamiltonian; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The quantum state has a finite correlation length
Explicitly required in the abstract for the bound to apply to local observables.

pith-pipeline@v0.9.0 · 5447 in / 1081 out tokens · 49042 ms · 2026-05-07T17:09:25.958539+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

91 extracted references · 16 canonical work pages

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For allN >max{e 2,4ξ (logN) 2 log 2 }we have: ∆≤f 1(N) (logN) 2D √ N ,(17) for some explicit functionf1(N)(see Eq.(31)) which can be upper-bounded by a constant

letα(ℓ) :=L 0e−ℓ/ξ, andξ >0. For allN >max{e 2,4ξ (logN) 2 log 2 }we have: ∆≤f 1(N) (logN) 2D √ N ,(17) for some explicit functionf1(N)(see Eq.(31)) which can be upper-bounded by a constant. 5
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letα(ℓ) :=L 0ℓ−(D+β), andβ > D+ 1. For anyε∈(0, β−D 2(β+3D−1) )andN > max{⌊2β+3D−1 ⌋, 4RN 1 β+3D−1 + ε 2D (logN) log 2 }we have: ∆≤f 2(N) 1 N 1 2 (β−D)/(β+3D−1)−ε ,(18) for some explicit functionf 2(N)(see Eq.(35)) which can be upper-bounded by a constant. Moreover, in the limitε→0we obtain: ∆≤f 2(N) logN N 1 2 (β−D)/(β+3D−1) .(19)
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letα(ℓ) :=L 0ℓ−(D+β), andβ > D. Assume the decay rate to satisfy the condition stronger than Eq.(3): ⟨AB⟩ρ − ⟨A⟩ρ ⟨B⟩ρ ≤ ∥A∥ ∥B∥α(d(A,B)).(20) Then, for anyε∈(0, β−D 2(β+3D) )andN >max{⌊2 β+3D ⌋, 4RN 1 β+3D + ε 2D (logN) log 2 }we have: ∆≤ ˜f2(N) 1 N 1 2 (β−D)/(β+3D)−ε ,(21) for some explicit function ˜f2(N)(see Eq.(38)) which can be upper-bounded by a co...
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Then, applying this inequality to the first term in Eq. (42), we obtain: e− ω2 2 Z ω 0 dˆω e R ˆω 0 d˜ω η(˜ω)η(˜ω) ≤e − ω2 2 e ω2 3 Z ω 0 d˜ω|η(˜ω)| ≤e− ω2 6 ω2 c1 2 + c2 3 ω (44) where in the last step we used the fact thatω∈[0,1−c1 2c2 ]. The second term in Eq. (42) needs additional steps, due to the exponential inside of the integral. First, let use Eq...
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It remains to bound the linear term. First, one can give the trivial bound: c3 e−(1−c1) ω2 4 Z ω 0 dω1 e(1−c1) ω2 1 4 ≤c 3 Z ω 0 dω1 =c 3 ω ,(49) where we used Hölder’s inequality. Additionally, forω≥ p 4/(1−c 1), it also hold that: c3 e−(1−c1) ω2 4 Z ω 0 dω1 e(1−c1) ω2 1 4 ≤ c3 4 (1−c 1)ω ≤ c3 8 ω ,(50) This can be shown as follows. First notice that for...
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for ally, it holds that|G ′(y)| ≤A
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(12) sinceFbH(y)is non-decreasing (being a cumulative function),G(y)de- fined in Eq

the following integral is bounded: ε:= Z Ω 0 dω | ˆf(ω)−ˆg(ω)| |ω| .(A1) 18 Then, for everyk >1there exists a finite constantc(k)(which only depends onk) such that: ∆ = sup y |F(y)−G(y)| ≤ c(k)A Ω + k k−1 ε π .(A2) We can apply this Theorem to prove Eq. (12) sinceFbH(y)is non-decreasing (being a cumulative function),G(y)de- fined in Eq. (10) is of bounded...
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ˆh(ω)has support on[−1,1]and it is bounded by1, i.e., ˆh(ω)≤1. We present here a choice of such a function, but the argument is independent of the specific choice. Defineˆk(ω)as: ˆk(ω) := ( 1− |ω|ifω∈[−1,1] 0otherwise ,(A6) and letK(y)be its Fourier transform: K(y) = 1 2π Z ∞ −∞ dω e −iωyˆk(ω) = 1 2π sin(y/2) (y/2) 2 .(A7) Then, we define: H(y) := 2 3π K(...
[77]

Boundingη 1,j(ω, K) Let us begin by rewriting the term inside of the parenthesis as: ξ1 j (ω) = eiωbH ℓ j(1)RM 1,j(ω)−I = Z ω 0 dω1 eiω1bH ℓ j(1) i bH ℓ j(1)RM 1,j(ω1) + (∂˜ωRM 1,j(˜ω)) ˜ω=ω1 =(B1) =iω bH ℓ j(1) + Z ω 0 dω1 Z ω1 0 dω2 ∂˜ω eiˆωbH ℓ j(1) i bH ℓ j(1)RM 1,j(ˆω) + (∂˜ωRM 1,j(˜ω)) ˜ω=ˆω ˆω=ω2 ,(B2) where we implicitly used the fact thatRM 1,j(0...
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Moreover, using the fact that⟨bhj⟩ρ = 0and the decay of correlations (see Eq

Boundingη 2,j(ω, K) The supports ofbhi andbhj are at least at a distanced(i, j)−2R. Moreover, using the fact that⟨bhj⟩ρ = 0and the decay of correlations (see Eq. (3)), it follows that: |η2,j(ω, K)|=ω D bhjzℓ j(1) E ρ ≤ ω σH X i∈X d(i,j)>2Rℓ D bhjbhi E ρ ≤(B9) ≤ ω σH X i∈X d(i,j)>2Rℓ R α(d(i, j)−2R)E 2 ≤ cD E2R σH ∞X r=2Rℓ+1 α(r−2R)r D−1 ! ω ,(B10) where w...
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ξ m j (ω) E ρ

Boundingη 3,j(ω, K) In order to boundη3,j(ω, K), let us first focus on averages of the type D bhj ξ1 j (ω). . . ξ m j (ω) E ρ . First, repeating the step of taking a derivative and integrating again, we obtain: ξk j (ω) = eiωbH ℓ j(k)RM k,j(ω)−I = Z ω 0 dω1 eiω1bH ℓ j(k) i bH ℓ j(k)RM k,j(ω1) + (∂˜ωRM k,j(˜ω)) ˜ω=ω1 .(B11) Thanks to this expression, in th...
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B1, B2 and B3

Bound onη(ω, K) In this section, we put together the results from Sec. B1, B2 and B3. First, it should be noticed that multiplying the results of Sec. B1 and B3 byN/σ H gives an estimate over the whole space. On the other hand, the term in Sec. B2 needs a bit more care. In this case by summing overjwe obtain: |η2(ω, K)| ≤ 1 σH X j∈X |η2,j(ω, K)| ≤ cDN E 2...
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Boundingν 1,j(ω, K) We can boundν1,j(ω, K)through a direct application of Eq. (B15). Indeed, we have: |ν1,j(ω, K)|= D bhj ξ0 j (ω). . . ξ K j (ω)eiωz ℓ j(K) E ρ ≤E(K!) D−1 B2 ℓD σH K (Eω) K (B29) where we implicitly used the fact that∥eiωz ℓ j(K) ∥= 1

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