Typical and extreme entropies of long-lived isolated quantum systems

Anthony Aguirre; Dana Faiez; Dominik \v{S}afr\'anek; J. M. Deutsch

REVIEW 3 major objections 5 minor 71 references

Typical and extreme entropies of long-lived isolated quantum systems

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Observational entropy behaves like classical Boltzmann entropy even at its extreme values, while bipartite entanglement entropy does not.

desk verdict Solid numerical case that observational entropy, unlike entanglement entropy, behaves Boltzmann-like at its extremes, though the quantitative lower bound leans on a same-group analytic result that is only extrapolated to the thermodynamic limit. read the letter →

arxiv 1908.07083 v2 pith:K7M5P2NK submitted 2019-08-19 quant-ph cond-mat.quant-gascond-mat.stat-mech

Dana Faiez , Dominik Šafránek , J. M. Deutsch , Anthony Aguirre This is my paper · ORCID

classification quant-phcond-mat.quant-gascond-mat.stat-mech

keywords observationalentropyentanglementrandompurethermalstatequantumthermalizationfluctuationslocalizationprobabilityisolatedsystemfermioniclattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An isolated quantum system that starts in a random pure thermal state spends its long unitary evolution near many different states, and this paper asks how two entropy definitions behave at the extremes of that evolution. The central claim is that observational entropy $S_{xE}$—the entropy an observer assigns using only coarse-grained position and energy measurements—behaves like classical Boltzmann entropy even in its extreme values: its maximum is close to thermal equilibrium with particles spread uniformly, and its minimum, although achieved by packing particles into one spatial region, cannot fall below roughly half of that maximum. Bipartite entanglement entropy, in contrast, can drop almost to zero by emptying the observed subsystem into the rest of the lattice, and reaches its maximum for a non-typical, non-uniform configuration. This distinction matters because it determines which entropy can meaningfully describe isolated, far-from-equilibrium, long-lived quantum systems.

What carries the argument

The load-bearing object is the maximum localization probability $P_{\mathrm{max}}$: the largest probability, reachable by evolving a random pure thermal state unitarily, that all particles are found in a chosen spatial region of size $\Delta x$. A prior analytical result [47] bounds $P_{\mathrm{max}}\le 1/2$ for real initial states and $P_{\mathrm{max}}\le \pi^2/16$ for complex ones, provided the localized subspace is much smaller than the full Hilbert space, the region is larger than the thermal wavelength, and the interactions are local; the paper checks these conditions numerically on its fermionic lattice. Through the identity $S_{xE}(\mathrm{loc}) \ge (1-P_{\mathrm{max}})S_{\mathrm{th}}$, this bound turns the physical impossibility of concentrating all particles into a small region into a universal floor on observational entropy. The contrasting object for entanglement entropy is the upper bound in Eq. (5), derived in [59], which depends only on particle number and partition sizes and explains why the maximal entanglement entropy stays constant as the bath grows.

What would settle it

Directly maximize $P_{\mathrm{max}}$ over unitary evolution for many complex random pure thermal states in the same fermionic lattice with $N_p=3$, $L=30$ or larger, and a localized region satisfying the three stated conditions; if the mean maximum localization probability clearly exceeds $\pi^2/16$, the localization bound that produces the floor in Eq. (9) is false. Alternatively, prepare a cold-atom lattice in a thermal pure state, reconstruct $S_{xE}$ by coarse-grained counting, and look for a measured value below $(1-P_{\mathrm{max}})S_{\mathrm{th}}$.

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Extended reading notes

Core claim

Using exact diagonalization of a one-dimensional fermionic lattice with $N_p=2$ or $3$ particles, the paper shows that over infinite-time unitary evolution starting from a random pure thermal state, the maximum and minimum of $S_{xE}$ are both close to its typical value and to the thermodynamic entropy $S_{\mathrm{th}}$, while the minimum of the bipartite entanglement entropy $S_{\mathrm{ent}}$ approaches zero and its maximum approaches a constant, $\ln 6$ for the parameters studied, that is independent of the bath size and much larger than the average. The quantitative anchor is the maximum localization probability $P_{\mathrm{max}}$: for initial real random pure thermal states $P_{\mathrm{max}}\le 1/2$, and for complex ones $P_{\mathrm{max}}\le \pi^2/16$, so a localized state obeys $S_{xE}(\mathrm{loc})\ge (1-P_{\mathrm{max}})S_{\mathrm{th}}$ and the minimum observational entropy stays at a fixed fraction of the thermodynamic entropy. The paper verifies the localization bound numerically for its interacting lattice, identifies the localized state as the one minimizing $S_{xE}$ (but not $S_{\mathrm{ent}}$), and concludes that observational entropy tracks Boltzmann entropy through the correspondence between macrostate size $\Omega$ and entropy, whereas entanglement entropy does not.

Load-bearing premise

The argument assumes that the localization bounds on $P_{\mathrm{max}}$, proved for random-vector and weakly-interacting-gas eigenstates, transfer to the physical non-random eigenstates of the interacting fermionic lattice and to the thermodynamic limit, with the constants $1/2$ and $\pi^2/16$ approached quickly enough.

Editorial extensions

If this is right

If $S_{xE}$ is Boltzmann-like, then a single coarse-grained measurement protocol gives a practical measure of how far an isolated system is from thermal equilibrium, including during rare large fluctuations.
Starting from a random pure thermal state, observational entropy cannot fluctuate below about half of the thermodynamic entropy, so spontaneous concentration of particles into a small region is limited in entropy cost; entanglement entropy has no such floor and can vanish.
The maximal entanglement entropy is not a symptom of a large macrostate: it is achieved by non-typical configurations with balanced particle numbers in the two partitions, and it stays constant as the bath size grows.
The gap between maximal and typical entanglement entropy grows with system size, so any claim that uses extreme entanglement values as a thermodynamic entropy proxy becomes worse for larger systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the localization bound is the true origin of the $S_{xE}$ floor, then the same floor should appear for other spatial coarse-grainings finer than the thermal wavelength; varying $\Delta x$ and comparing measured minima with Eq. (9) would test this directly.
The dip in $S_{xE}(\mathrm{min})$ near $\beta\approx 0.5$ suggests a temperature-controlled crossover in the ability to localize particles at the coarse-graining scale, which might be observable in cold-atom experiments as a change in the rarity of low-entropy configurations.
A closely related question not settled here is whether the sum of local von Neumann entropies, a multipartite entanglement measure, also obeys a $(1-P_{\mathrm{max}})$-type floor; this is a direct extension of the paper's closing observation.
For cosmological questions such as the arrow of time, the relevant entropy should be one that is bounded below even under extreme fluctuations and remains defined for a closed system; observational entropy is a concrete candidate if its localization floor survives in a gravitational state space.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: this paper is a careful numerical comparison of two entropy definitions in a small fermionic lattice, and it makes a real point — observational entropy tracks Boltzmann entropy even at its extreme values, while bipartite entanglement entropy doesn't. The new content is the first detailed mapping of extreme and typical entropy values versus system size and temperature for this model, and the clean physical picture that emerges: SxE is minimized by localizing particles, Sent by emptying a subsystem.

The paper does several things well. It uses random pure thermal states, computes histograms of fluctuations, and finds extrema with a simplex search, with error bars over 6 initial states. The figures are convincing that SxE(min) stays around half of Sth while Sent(min) drops to near zero as L grows. The contrast with Sent(max) being L-independent is explained well via an upper bound from Ref. [59]. The paper is also honest about numerical limitations.

The main soft spot is exactly what the stress-test flags: Eq. (9) inherits the Pmax ≤ pi^2/16 bound from Ref. [47] (three shared authors), and the transfer to this interacting lattice is only tested numerically for Np=3 and L up to 30, where the ratio N/M^2 is not huge. The paper itself notes remaining fluctuations. So the thermodynamic-limit claim that SxE(min) >= (1-pi^2/16) Sth is an extrapolation. If physical eigenstates allowed larger localization, the lower bound would erode. That doesn't break the qualitative picture for the sizes studied, but it does mean the central quantitative anchor is not fully established. A second, minor issue: the simplex search is local, not global, so the 'extreme' values may be local extrema; for small systems this could be checked by enumeration. Also, no code or data is provided, which is a reproducibility gap for a numerical paper.

Overall, this is a solid, readable paper that deserves a serious referee. I'd send it to peer review, with requests for the numerical data/code, a more careful discussion of the Pmax transfer and its finite-size scaling, and a check of the extremum search on small cases. It's a useful contribution for anyone working on entropy in isolated systems, quantum thermodynamics, or foundations of statistical mechanics.

Referee Report

3 major / 5 minor

Summary. This paper compares two entropy measures for isolated quantum systems out of equilibrium: bipartite entanglement entropy and observational entropy with position-and-energy coarse graining. Using exact diagonalization of a one-dimensional lattice of interacting spinless fermions (N_p = 2 or 3, L up to 30, starting from random pure thermal states), the authors characterize the typical, maximal, and minimal values reached under unitary evolution. They report that observational entropy behaves qualitatively like classical Boltzmann entropy even at its extremes: the maximum is close to the uniform thermal state, and the minimum is bounded below at roughly half the thermodynamic entropy, achieved by localizing particles as much as possible. In contrast, entanglement entropy is minimized by emptying the subsystem and maximized by splitting particles between the two partitions, and neither extreme is typical. The quantitative anchor is Eq. (9), S_xE(loc) ≥ (1 - P_max) S_th, where P_max ≤ 1/2 (real pure thermal states) or π²/16 (complex pure thermal states) comes from the authors' earlier work [47].

Significance. If established, the paper's central distinction is significant: observational entropy would be a Boltzmann-like entropy that remains physically meaningful far from equilibrium and at extreme fluctuations, whereas bipartite entanglement entropy would not. This matters for discussions of thermalization, entropy production, and arrow-of-time questions in closed systems. The manuscript is transparent about its numerical setup, uses exact diagonalization with explicitly stated parameters, reports error bars from six initial states, and connects its results to analytic bounds. It does not, however, ship code or data, and its quantitative claims rely on a transfer of results from a companion paper [47] that is tested here only on very small systems.

major comments (3)

[Sec. VI, Eqs. (8)-(9)] The lower bound S_xE(loc) ≥ (1 - P_max) S_th is load-bearing for the paper's central claim that observational entropy cannot fall below about half its maximum, but the constants 1/2 and π²/16 are proved in Ref. [47] for random energy eigenvectors and a weakly interacting gas, not for the interacting fermionic lattice used here. The numerical transfer test in Fig. 8 uses N_p = 3 and L ≤ 30, where N/M² ≈ 40.6 at L = 30 (N = C(30,3) = 4060, M = C(5,3) = 10); this is a modest ratio, and the paper itself notes that 'the presence of some fluctuations is expected since our model is a real system with non-random energy eigenvectors.' Since a larger P_max in the thermodynamic limit would weaken or erase the lower bound, please add a finite-size scaling analysis of P_max as a function of N/M² at fixed β and Δx, and investigate the sensitivity of the limiting P_max to the interaction strengths V, V′ and to N_p. Without this, the claim that S_xE(min) cannot fall below roughly half of S_th is an extrapolation rather than an established result.
[Sec. III (simplex search algorithm)] The reported global minima and maxima of Sent and SxE are obtained with a local simplex search over phases φ_E = Et, with no demonstration that the search is globally convergent. For nonconvex entropy landscapes this is a real risk: the values labeled 'Sent(min)', 'Sent(max)', and 'SxE(min)' could be local extrema, and the quantitative statements such as 'Sent(min) ≈ 0.05' or 'SxE(min) ≈ 63% of the maximum' could be artifacts of the starting point. Because the paper's central claims concern extreme values, please establish robustness of the extrema, for example by repeated optimization from many random phase initializations, or by an exhaustive phase grid for small L where the Hilbert-space dimension is small enough, and report the spread of the resulting extrema.
[Sec. VII, Eq. (8)] Equation (8) is stated without derivation and is attributed to Ref. [47], which shares three authors with this paper. Since Eq. (9) is the quantitative anchor for the main conclusion, the manuscript should either reproduce the derivation in an appendix or state precisely which conditions from Ref. [47] guarantee the formula, including the large-L assumption and the treatment of finite-size corrections. Note also that Eq. (8) is then plotted in Fig. 9 for system sizes as small as L = 8, even though it is introduced as a large-L expression; the expected finite-size error at these small L should be quantified or the plot restricted to the regime of validity.

minor comments (5)

[Sec. I and Sec. III] The word 'compliment' appears where 'complement' is meant (e.g., 'between a subsystem and its compliment').
[Sec. VI] The condition on the Hilbert-space dimensions appears to be typeset as 'N ≥ M²'; it should be 'N ≫ M²' to match the condition required by the argument in Ref. [47].
[Sec. II] The paper takes t = t′ = 1.9 and V = V′ = 0.5 for all simulations, but does not discuss whether this parameter set is in the non-integrable regime or how the qualitative conclusions depend on the choice of couplings; a brief justification or a test with different couplings would make the numerics more convincing.
[Sec. III] The description of the histogram construction says the system is evolved 'for a long time' and sampled 'at each small fixed time step,' but the actual time step and total duration are not stated; these details matter for assessing whether the sampled distribution is representative of the long-time behavior.
[Fig. 10 and associated text] The presentation of macrostate sizes and the percentages in Fig. 10 is hard to follow; please clarify the definitions of Ω and the percentages in the caption or in the main text so that the claimed correspondence between macrostate size and observational entropy can be checked directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eqs. (8)/(9) rest on an independent parameter-free bound (Ref. [47]) with numerical transfer checks in Sec. VI, and no fitted input is relabeled as a prediction.

full rationale

The central quantitative claim, SxE(loc) >= (1 - Pmax) Sth, is not produced by fitting the present data. It is taken from Ref. [47], a separate parameter-free derivation for random energy eigenvectors and a weakly interacting gas, and the paper then independently tests the relevant bound Pmax <= 1/2 (real) or pi^2/16 (complex) on the interacting fermionic lattice in Sec. VI (Fig. 8). Section VII compares the resulting Eq. (8) with numerically minimized SxE, so the comparison is evidence, not an input. The qualitative statement that SxE is Boltzmann-like is indeed partly built into the definition of observational entropy as a coarse-grained Shannon entropy over macrostate volumes (Appendix A): this is a design feature of the quantity rather than a hidden derivation step, and the paper's nontrivial content lies in the extreme-value characterization and the contrast with entanglement entropy, which are computed independently. The remaining concerns about finite-size extrapolation from L <= 30 are correctness risks, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or forces. The quantitative claims rest on hand-chosen Hamiltonian and state parameters, on the transfer of a maximum-localization bound from a same-group companion paper, and on the assumption that local optimization finds global entropy extrema. The theoretical lower bound for SxE(min) is imported rather than derived here.

free parameters (4)

Hamiltonian couplings t=t'=1.9, V=V'=0.5
Chosen by hand to make the system non-integrable; quantitative entropy values and the approach of Pmax to constants may depend on these couplings, though the qualitative comparison likely does not.
Inverse temperature beta=0.01 for most runs
Chosen to approximate the high-temperature limit; Figs. 6 and 7 show strong beta dependence, so this representative value is a modeling choice.
Subsystem and partition size Delta x=4, or 5 in Sec. VI
Fixed partition size determines which localization region is used; the bound on SxE(min) and the constant Sent(max) depend on this size.
Number of particles Np=2 or 3
Exact diagonalization restricts the study to a few particles; the thermodynamic-limit scaling claims are extrapolated from these small particle numbers.

assumptions (5)

domain assumption The energy eigenstates of the fermionic lattice are sufficiently ETH-like that the Pmax bounds of Ref. [47] apply.
Section VI requires N >> M^2, Delta x >> lambda_T, and weak correlations; the paper checks finite-size behavior but cannot prove the asymptotic limit for physical eigenstates.
domain assumption The Nelder-Mead simplex search over phases phi_E attains the global entropy extrema.
Section III uses a local optimization method; no proof or multi-start verification is given, so the reported minima and maxima may be local rather than global.
standard math Maximizing over phases phi_E = E t is equivalent to maximizing over all times because energy spacings are effectively irrational.
Stated in Sec. III; this is valid for non-degenerate spectra with rationally independent energy gaps.
domain assumption Finite-size results with Np <= 3 and L <= 30 represent the thermodynamic limit trends.
Claims such as 'Sent(min) approaches zero in the limit of large L' and 'SxE(min) is about half of its maximum' are extrapolations from small lattices.
domain assumption A random pure thermal state is a valid stand-in for the canonical thermal state at the chosen beta.
RPTS emulates thermal equilibrium for local observables, following Refs. [55-57]; the extreme-value statistics are computed for a small number of random states.

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Cite this review

Pith. "Pith review of Typical and extreme entropies of long-lived isolated quantum systems." pith.science (2026). https://pith.science/paper/K7M5P2NK

@misc{pith2026190807083,
  author       = {Pith},
  title        = {Pith review of: Typical and extreme entropies of long-lived isolated quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7M5P2NK}},
  note         = {Machine review of arXiv:1908.07083}
}

read the original abstract

In this paper, we investigate and compare two well-developed definitions of entropy relevant for describing the dynamics of isolated quantum systems: bipartite entanglement entropy and observational entropy. In a model system of interacting particles in a one-dimensional lattice, we numerically solve for the full quantum behavior of the system. We characterize the fluctuations, and find the maximal, minimal, and typical entropy of each type that the system can eventually attain through its evolution. While both entropies are low for some "special" configurations and high for more "generic" ones, there are several fundamental differences in their behavior. Observational entropy behaves in accord with classical Boltzmann entropy (e.g. equilibrium is a condition of near-maximal entropy and uniformly distributed particles, and minimal entropy is a very compact configuration). Entanglement entropy is rather different: minimal entropy "empties out" one partition while maximal entropy apportions the particles between the partitions, and neither is typical. Beyond these qualitative results, we characterize both entropies and their fluctuations in some detail as they depend on temperature, particle number, and box size.

Figures

Figures reproduced from arXiv: 1908.07083 by the authors.

**Figure 2.** FIG. 2. Semi-log probability histogram of entanglement [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. The minimum (blue dots), maximum (orange cir [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: FIG. 5. The minimum (blue dots), maximum (orange [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. The minimum (lower blue stars), maximum (orange [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. The minimum (blue lower stars), maximum (or [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) Maximum probability [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. This illustration shows entropies for various types of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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