REVIEW 3 major objections 4 minor 7 references
Entropy of small subsystems in thermalizing systems
T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read For thermalizing quantum systems, the equilibrium von Neumann entropy of a small subsystem is $\ln d_A$ minus a computable correction fixed by the Hamiltonian and the initial state.
desk verdict A clean, honestly presented corollary: the entropy formula is new and parameter-free, but it inherits its central premise from the author's companion paper, so Referee and reader both need to check Eq. (4) before trusting Eq. (14). read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the ETH ansatz for the equilibrated reduced density matrix, Eq. (4): for a subsystem of $O(1)$ size, $\psi_\infty^A = I_A/d_A + c X_A + O(1/N^2)$ in trace norm. The scalar $c=(\tilde v-v)/(2N\tilde v^2)$ measures how far the initial state's energy variance lies from the infinite-temperature value, and the operator $X_A$ is assembled from traces of $H$, $H^2$, and $H^3$ over the complement. Expanding $-(1/d_A+\mu)\ln(1/d_A+\mu)$ and using $\sum_j \mu_j=0$ turns this ansatz into the entropy formula.
What would settle it
For the same spin chain at a different value of the transverse fields, compute the time-averaged reduced density matrix by exact diagonalization for $N=20$ to $34$, fit the entropy to $\ln 8 - \alpha((v-\tilde v)/(2N))^2$, and check whether $\alpha$ tends to $d_A\operatorname{tr}(X_A^2)/(2\tilde v^4)$. If the fitted prefactor moves away from the predicted value as $N$ increases, the $O(1/N^2)$ ansatz is wrong.
Extended reading notes
Core claim
The paper establishes that, under the eigenstate thermalization hypothesis, the von Neumann entropy of a fixed-size subsystem after equilibration is $\ln d_A - \frac{c^2 d_A}{2}\operatorname{tr}(X_A^2) + O(1/N^3)$, where $c$ encodes the difference between the initial state's energy variance and the infinite-temperature variance, and $X_A$ is an explicit operator built from $\operatorname{tr}(H^3)$, $\operatorname{tr}_{\bar A}(H)$, $\operatorname{tr}(H^2)$, and $\operatorname{tr}_{\bar A}(H^2)$. This is the leading finite-size correction to the maximally mixed entropy, and it contains no fitting parameters.
Load-bearing premise
The derivation depends on the assumed form of the equilibrated reduced density matrix of a small subsystem, namely the maximally mixed state plus a specific operator correction with an error that scales as $1/N^2$; if the true correction contains additional operators at that order, the entropy formula fails.
Editorial extensions
If this is right
- The equilibrium entropy of a small subsystem depends on the initial state's energy variance through $c$, not just on the Hamiltonian alone.
- For any local traceless Hamiltonian, the leading correction to $\ln d_A$ is negative, so equilibration slightly suppresses subsystem entropy below its maximal value.
- The formula provides an analytic explanation of the finite-size scaling seen in exact-diagonalization studies of thermalizing spin chains.
- The prediction applies to non-translation-invariant local Hamiltonians and to any initial state with exponential correlation decay and zero total energy.
Reading between the lines
- A natural extension, not pursued in the paper, is that the same expansion should hold for R\'enyi entropies with an $\alpha$-dependent prefactor replacing $1/2$.
- The formula could serve as a diagnostic for numerical thermalization simulations: the fitted prefactor should settle at the predicted value as $N$ grows.
- A direct test of the underlying ansatz would be to diagonalize the time-averaged reduced density matrix at finite $N$ and check that its leading correction lies in the operator subspace spanned by $X_A$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the equilibrium von Neumann entropy of an O(1) subsystem in a local, translation-non-invariant Hamiltonian on N sites, assuming the eigenstate thermalization hypothesis (ETH). For an initial state with exponential correlation decay and zero total energy, it derives S(psi_inf_A) = ln dA - (c^2 dA tr(X_A^2))/2 + O(1/N^3), where c and X_A are expressed through Hamiltonian moments and the initial energy variance (Eqs. (5), (6), (8)). The derivation expands the entropy around the infinite-temperature state, relying on Eq. (4), which states that the time-averaged reduced density matrix is I_A/d_A + c X_A up to O(1/N^2) in trace norm. The paper applies this result to a spin-1/2 chain and compares the predicted quadratic coefficient with exact diagonalization data from Maceira and Läuchli, reporting semi-quantitative agreement.
Significance. If the central premise Eq. (4) is correct, the paper provides a parameter-free, analytical prediction for the leading finite-size correction to subsystem entropy in thermalizing systems, expressed directly in terms of the Hamiltonian and initial state. The derivation from Eq. (4) to Eq. (8) is transparent and internally consistent, and the explicit example calculation is valuable as a concrete test. The numerical comparison, though semi-quantitative, addresses a recent exact diagonalization study and the prediction in Eq. (14) is falsifiable. However, the main result is entirely conditional on an imported ETH expansion from Ref. [7], and the numerical support is not yet conclusive. The paper is concise and mostly clear, but its central claim requires either a self-contained derivation of Eq. (4) or a much stronger numerical verification.
major comments (3)
- [Section 2, Eq. (4)] The entire derivation of Eq. (8) rests on Eq. (4), which is imported from Eq. (15) of Ref. [7] without derivation or a precise statement of the assumptions under which the O(1/N^2) trace-norm bound holds. Since Ref. [7] is a companion paper by the same author, the present manuscript should make the central premise self-contained: either provide a derivation of Eq. (4) in an appendix, state it as an explicit assumption, or quote the precise theorem from Ref. [7] with its hypotheses. As written, the main result is a conditional statement whose key hypothesis is not established in this paper.
- [Section 3, Fig. 1 and Eq. (13)] The numerical validation is only a single-parameter, semi-quantitative test. The dots in Fig. 1 do not clearly converge to the dashed theoretical line by N=30, and the paper explicitly admits that 'one can not conclude whether the dots approach the dashed line as N → ∞'. This does not support the abstract's claim of providing a theoretical explanation for Ref. [6]'s numerical findings. A stronger test would involve multiple parameter sets, a finite-size scaling analysis with estimated O(1/N^3) corrections, or a direct check of Eq. (4) itself rather than only the final entropy coefficient.
- [Section 3, Eq. (13) versus Eq. (12)] The fitted quantity α in Eq. (13) is extracted from finite-N exact diagonalization data, but the theoretical prediction in Eq. (14) is the coefficient in the thermodynamic limit. The paper does not discuss how the O(1/N^3) error term in Eq. (12) (or the finite-size corrections hidden in 'O(1/N^3)') affects the extracted α at N=20–30. Without such a discussion, the comparison between finite-N fits and the infinite-N prediction is uncontrolled.
minor comments (4)
- [Section 2, Eq. (6) and text after Eq. (7)] The expansion ln(1/d_A + μ_j) = ln(1/d_A) + d_A μ_j - (d_A^2 μ_j^2)/2 + ... is valid only if |d_A μ_j| < 1. For finite N this could fail for small systems; the paper should state explicitly that N is taken large enough so that this holds uniformly for all j.
- [Section 3, Eq. (10) and Eq. (11)] The expression for d_A tr(X_A^2) in Eq. (11) is lengthy and provided without derivation. Including a short derivation or a supplementary note would improve verifiability.
- [Section 3, Eq. (9)] The Hamiltonian in Eq. (9) uses periodic boundary conditions implicitly (σ_i^z σ_{i+1}^z for i=N should wrap around), but this is not stated. Please clarify the boundary condition.
- [Section 3, Fig. 1] The figure would be clearer with error bars or a discussion of the uncertainty in the fitted α values, and with a statement of the initial state used in the exact diagonalization data of Ref. [6].
Circularity Check
No significant circularity: Eq. (8) follows from the imported ETH result Eq. (4) by a parameter-free Taylor expansion, and the predicted prefactor is compared with independent ED data rather than fitted.
full rationale
The derivation chain is: assume ETH, import Eq. (4) from Ref. [7], expand the eigenvalues of the reduced density matrix, compute the von Neumann entropy by a Taylor expansion, and substitute the definition of X_A to obtain Eq. (8). The only imported input is Eq. (4), which is a self-citation, but it is a parameter-free statement about the reduced density matrix derived from ETH in prior work, with assumptions (local traceless Hamiltonian, initial state with exponential correlation decay, zero mean energy) that do not include the target entropy formula. It is not fitted to the exact-diagonalization data and is not defined in terms of the entropy. The later steps are algebraic identities: the expansion of x ln x up to second order, the use of tr(X_A) = 0, and substitution of the definitions of c and X_A. The numerical comparison in Eq. (14) is a genuine prediction with no adjustable parameters: the ED data are used to extract a fitted alpha per system size, but the theoretical value is computed ab initio from the Hamiltonian and the initial-state energy variance. Therefore no step reduces to its own inputs by construction. The self-citation is load-bearing in that the paper does not re-derive Eq. (4), but under the stated criteria this is independent support rather than circularity.
Assumptions & free parameters
assumptions (3)
- domain assumption Eigenstate thermalization hypothesis holds for the considered local Hamiltonians.
- domain assumption The equilibrated reduced density matrix obeys psi_inf_A = I_A/d_A + c X_A + O(1/N^2), from Eq. (4).
- domain assumption Initial state has exponential decay of correlations and zero total energy; H is extensive with bounded local terms.
Cite this review
Pith. "Pith review of Entropy of small subsystems in thermalizing systems." pith.science (2026). https://pith.science/paper/VO4FBYPC
@misc{pith2026250118611,
author = {Pith},
title = {Pith review of: Entropy of small subsystems in thermalizing systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/VO4FBYPC}},
note = {Machine review of arXiv:2501.18611}
}
read the original abstract
We study the entropy of small subsystems in thermalizing quantum many-body systems governed by local Hamiltonians. Assuming the eigenstate thermalization hypothesis, we derive an analytical formula for the von Neumann entropy of equilibrated subsystems. This formula reveals how subsystem entropy depends on the microscopic parameters of the Hamiltonian and the macroscopic properties of the initial state. Furthermore, our results provide a theoretical explanation for recent numerical findings by Maceira and L\"auchli, obtained via exact diagonalization.
Figures
Reference graph
Works this paper leans on
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[7]
Y. Huang. High-precision simulation of finite-size thermalizing syst ems at long times. arXiv:2406.05399. 5
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[6]
I. A. Maceira and A. M. L¨ auchli. Thermalization dynamics in closed q uantum many body systems: a precision large scale exact diagonalization study. a rXiv:2409.18863. 4
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L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol. From quantum c haos and eigen- state thermalization to statistical mechanics and thermodynamics . Advances in Physics , 65(3):239–362, 2016
work page 2016
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J. M. Deutsch. Eigenstate thermalization hypothesis. Reports on Progress in Physics , 81(8):082001, 2018
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Reviewed August 10, 2026 · model on record in the stance chip above.
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