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Equilibration and the eigenstate thermalization hypothesis as limits to observing macroscopic quantum superpositions

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Under generic many-body unitary evolution, a perfectly isolated GHZ state becomes operationally indistinguishable from its classical mixture, because thermalization suppresses the coherence signal for all observables whose operator norm gro

desk verdict A clean ETH-based argument that unitary thermalization hides GHZ coherence; the main risk is an unverified extension of ETH to a fully nonlocal observable. read the letter →

arxiv 2512.11522 v2 pith:2FRE7B6A submitted 2025-12-12 quant-ph

Gabriel Dias Carvalho , Pedro S. Correia , Thiago R. de Oliveira This is my paper
classification quant-ph
keywords macroscopicquantumsuperpositioneigenstatethermalizationhypothesisequilibrationGHZstateunitarydynamicsdecoherencequantumness
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

This paper argues that macroscopic quantum superpositions cannot be observed even in perfectly isolated systems if the many-body dynamics are generic. Using the GHZ state as a test case, it shows that unitary evolution under an eigenstate-thermalizing Hamiltonian makes the superposition and its classical mixture operationally indistinguishable for almost all times: the time-averaged difference in any observable expectation value falls exponentially with particle number. The same dynamics also suppresses the standard measures of macroscopic quantumness. If correct, this identifies intrinsic unitary thermalization—not environmental decoherence—as a fundamental limit on the emergence of macroscopic quantum effects.

What carries the argument

The key machinery is the eigenstate thermalization hypothesis (ETH) applied to both eigenstate overlaps and off-diagonal matrix elements, combined with the long-time average of the squared distinguishability. For a generic Hamiltonian with non-degenerate gaps, the time average of |Δ⟨A(t)⟩|² picks out only stationary terms, which are then estimated via ETH: overlaps scale as O(2^{-N/2}) and off-diagonal matrix elements scale as O(‖A‖ 2^{-N/2}) for distinct eigenstates. Together these give (Δ⟨A⟩)² ∼ O(‖A‖² 2^{-N}). The paper also uses a double-commutator coherence measure (the trace norm of the double commutator with a local observable, maximized over local observables) to quantify macroscopic

What would settle it

Compute the off-diagonal matrix elements of σ_x^{⊗N} between eigenstates of a chaotic spin chain (e.g., the XYZ model used in the paper) for N=10–14 and check whether they decay as 2^{-N/2}; if they decay slower, or if the time-averaged (Δ⟨σ_x^{⊗N}(t)⟩)² fails to approach O(2^{-N}), the central claim collapses. Alternatively, directly measure the time-dependent contrast in a system with N≈10 qubits and see whether it equilibrates to the predicted exponential bound.

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Extended reading notes

Core claim

The central claim is that the dephasing mechanism behind equilibration erases the coherence signature of a macroscopic superposition. For a GHZ state, the off-diagonal coherence elements are suppressed by generic many-body evolution: using eigenstate thermalization, the paper derives that the time-averaged squared distinguishability (Δ⟨A(t)⟩)² scales as O(‖A‖² 2^{-N}) for any observable A whose operator norm grows at most polynomially with system size N. Consequently, the GHZ state and its classical mixture become operationally indistinguishable for any such observable in the thermodynamic limit, and the macroscopic-quantumness index q drops from 2 toward the mixture's value. The paper suppo

Load-bearing premise

The paper assumes that the eigenstate thermalization hypothesis—established mainly for local observables—also gives exponentially suppressed off-diagonal matrix elements for a fixed fully correlated bounded operator like σ_x^{⊗N}; if that scaling fails, the loss of distinguishability for global measurements is not established.

Editorial extensions

If this is right

  • Even with perfect isolation, generic many-body dynamics destroy the operational signature of a macroscopic superposition; environmental decoherence is not required.
  • No observable with operator norm growing at most polynomially in N can distinguish the GHZ state from its mixture after equilibration, so distinguishing them requires exponentially large measurement resources.
  • The measures of macroscopic quantumness also decay: the index q goes from 2 to values close to the mixture's, indicating the dynamics destroy macroscopic coherence rather than merely hide it.
  • Preserving macroscopic coherence in a thermalizing system requires non-generic spectra (e.g., many-body localization), active error correction, or observables with super-polynomial norm.
  • Coarse-grained versions of the GHZ state, which are more realistic experimentally, should exhibit the same loss of distinguishability because the mechanism depends only on effective dimension and coherent dephasing.

Reading between the lines

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

  • The argument suggests a sharper statement than the paper makes: if ETH applies to any fixed bounded operator, then essentially all pure states—not just GHZ—will become indistinguishable from their time-averaged mixtures with respect to polynomial-norm observables, implying a generic classical limit for large systems.
  • The result may bear on the quantum measurement problem: if unitary dynamics alone can make macroscopic superpositions unobservable, the point at which 'collapse' appears could be governed by equilibration rather than by environment interaction.
  • The timescale question remains open; one can test whether the equilibration time stays short for increasing N or grows, which would determine whether the effect is practically relevant for mesoscopic systems.
  • One could test the ETH assumption for the fully correlated observable by exact-diagonalization computation of the off-diagonal matrix elements of σ_x^{⊗N} in larger chaotic spin chains; a slower decay would invalidate the central bound.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper argues that even in a perfectly isolated system, generic unitary many-body evolution under the eigenstate thermalization hypothesis (ETH) can suppress the operational distinguishability between a macroscopic GHZ superposition and its classical mixture. Using the time average of the squared difference Δ⟨A(t)⟩, the authors claim a scaling O(||A||² 2^{-N}) for observables with polynomially bounded norm, including the fully nonlocal operator σ_x^⊗N. They also study the Shimizu–Morimae macroscopicity indices p and q, reporting that the infinite-time averaged states have q<2, indicating loss of macroscopic quantumness. Numerical simulations on a chaotic spin-1/2 XYZ chain with next-nearest-neighbor interactions and a local defect, for N≤12 and 25 parameter sets, are presented as support.

Significance. If the central scaling result holds, the paper identifies unitary thermalization—independent of environmental decoherence—as a fundamental mechanism limiting the observation of macroscopic quantum superpositions, thereby connecting the ETH to the quantum–classical boundary. The paper is self-contained, does not fit free parameters to the central 2^{-N} scaling, uses established macroscopicity measures, and provides nontrivial numerical support. However, the central derivation has a technical omission in the time-average step, and the key extension of ETH to the fully nonlocal observable σ_x^⊗N is asserted rather than demonstrated. The conclusion is therefore plausible but not yet fully established.

major comments (2)
  1. [§IV, Eqs. (12)–(14)] The time averaging of |f(t)|² and f(t)² omits the zero-gap stationary blocks. Under the non-degenerate-gap condition stated in §III, the condition E_n−E_m = E_q−E_p also admits m=n, p=q; similarly f(t)² admits m=n, p=q. These contribute an additional nonnegative term |Σ_m A~_{m,m}|² to Eq. (14). The estimate (15) for |A~_{m,n}|² is also not valid at m=n: there |A~_{m,m}|² ~ O(||A||² 2^{-2N}), not O(||A||² 2^{-3N}). A separate ETH estimate for Σ_m A~_{m,m} = Σ_m ⟨E_m|A|E_m⟩⟨0|E_m⟩⟨E_m|1⟩, using random-phase overlaps ⟨0|E_m⟩⟨E_m|1⟩ ~ 2^{-N}, gives O(||A||² 2^{-N})—the same order as the claimed result. The derivation should be corrected to include these terms explicitly and their scaling shown.
  2. [§IV after Eq. (16); §V] The claim for the fully correlated observable A_NL = σ_x^⊗N rests entirely on the assertion that 'the same ETH scaling applies because the argument does not rely on locality but only on the observable being a fixed bounded operator.' This is not established. ETH is firmly verified for local few-body observables; global operators can have qualitatively different eigenbasis matrix elements (e.g., symmetry-sector selection rules or O(1) off-diagonal elements). The numerics in §V only show time-averaged values of Δ⟨A_NL(t)⟩ for N≤12 and do not directly extract the eigenbasis matrix elements of A_NL. Since Eq. (17) is the paper's headline nonlocal result, the authors should provide direct numerical evidence, e.g., plots of |⟨E_m|σ_x^⊗N|E_n⟩| versus energy gap for a fixed N showing the variance scaling ~2^{-N}, or a rigorous random-matrix argument for a fixed operator in a Haar-random eigenbas
minor comments (4)
  1. [§VI, Eq. (20)] The notation in Eq. (20) is confusing: the left side should be something like \max\{N, \max_A\|[A,[A,ρ]]\|_1\}, but the printed form is ambiguous. Please clarify the definition of the outer maximization and the role of the factor N.
  2. [Captions of Figs. 5 and 6] The captions refer to 'the second bold column in Table 4', but there is no Table 4 in the manuscript. These references should be corrected to point to the parameter sets described in Fig. 4.
  3. [Various] There are several typos and incomplete sentences: 'drive by' in the Introduction; 'a established measures' in §VI; 'and andefrom' in §VI; the footnote to [13] contains an unfinished sentence ('Later, it was found that von Neumann already obtained similar results. A not-so-recent but also important and usually forgotten reference is [29].').
  4. [§VI] The paragraph introducing the Shimizu–Morimae measure says 'we employ a established measures' and the index q is defined through a double commutator. Please ensure the normalization conventions for the trace norm and the local observable A are stated unambiguously, since q is extracted from power-law fits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(2^{-N}) distinguishability decay is a conditional consequence of the externally assumed ETH scaling, not a re-labeled fit or self-citation chain.

full rationale

The paper's core derivation is conditional on ETH, cited to the external review [18]; Eqs. (11)-(15) translate ETH overlap and off-diagonal scalings into an O(||A||^2 2^{-N}) bound on the time-averaged squared distinguishability by counting 2^{2N} terms each of size O(||A||^2 2^{-3N}). No constant is fitted to the target decay in this chain; the Sec. VI q-index fits are post hoc diagnostics and do not enter Eqs. (16)-(17). The self-citations ([6], [17], [23]) are contextual motivation, a peripheral extension for time-dependent Hamiltonians, and a coarse-graining remark in the conclusions—none is used to force the central claim or to forbid alternatives, so they are not load-bearing. Legitimate caveats exist, but they are not circularity: applying ETH to the nonlocal observable A_NL rests on the explicit assertion 'the same ETH scaling applies because the argument does not rely on locality but only on the observable being a fixed bounded operator' (Sec. IV), and the stationary-phase reduction in Eq. (14) does not display the zero-gap diagonal terms of f(t)^2. If those matrix elements are not exponentially suppressed, the bound may fail; however, that would be a failure of the assumed ETH scaling or a calculational omission, not a reduction of the result to its own inputs. Therefore no circular step is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central scaling argument rests on ETH rather than on fitted parameters; the only hand-chosen numbers are Hamiltonian parameters used to generate numerical examples. No new entities are postulated.

free parameters (1)
  • Hamiltonian parameters (J1,J2,d,hx,hz,e) = varied; e.g. J1=1.0, J2=1.35, d=0.5, hx=0.2, hz in [0.1,0.6], e in [0.1,0.5]
    Chosen by hand for the numerical spin chain; 25 parameter sets scanned to demonstrate robustness. They do not enter the analytical ETH scaling and are not fitted to the target result.
assumptions (4)
  • domain assumption Eigenstate thermalization hypothesis (ETH): off-diagonal matrix elements of a bounded observable decay as ⟨E_m|A|E_n⟩ ∼ O(||A|| 2^{-N/2}) and eigenstate overlaps with product states are ∼ 2^{-N/2}.
    Invoked in Sec. IV before Eq. (15) to estimate |A~mn|²; for the nonlocal operator A_NL the paper asserts the same scaling without proof.
  • domain assumption Non-degenerate energy gaps (except trivial zero gaps) for the Hamiltonian.
    Used in Secs. III-IV to keep only stationary terms in the time-averaged signal; believed generic for chaotic many-body systems.
  • domain assumption The extended Heisenberg-XYZ Hamiltonian (18) is chaotic/ETH-thermalizing for the chosen parameters.
    Sec. V states the model is 'known to exhibit chaotic dynamics for generic parameter choices, ensuring ETH-type behavior.'
  • domain assumption Observables considered have operator norm growing at most polynomially with N.
    Part of the stated scope in Sec. IV; excludes measurements with super-polynomial norm that could distinguish the states.

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Cite this review

Pith. "Pith review of Equilibration and the eigenstate thermalization hypothesis as limits to observing macroscopic quantum superpositions." pith.science (2026). https://pith.science/paper/2FRE7B6A

@misc{pith2026251211522,
  author       = {Pith},
  title        = {Pith review of: Equilibration and the eigenstate thermalization hypothesis as limits to observing macroscopic quantum superpositions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2FRE7B6A}},
  note         = {Machine review of arXiv:2512.11522}
}
read the original abstract

Macroscopic quantum superpositions are widely believed to be unobservable because large systems cannot be perfectly isolated from their environments. Here, we show that even under perfect isolation, intrinsic unitary dynamics in generic many-body systems, as analyzed through the eigenstate thermalization hypothesis and random matrix theory, can suppress the observable signatures of macroscopic coherence. Using the Greenberger-Horne-Zeilinger (GHZ) state as a representative example, we demonstrate that while fully correlated measurements can initially distinguish a macroscopic superposition from its corresponding classical mixture, generic many-body evolution renders them operationally indistinguishable for most times. By analyzing both distinguishability measures and established quantifiers of macroscopic quantumness, we find that equilibration not only hides coherence from accessible observables but also suppresses the corresponding signatures of macroscopic quantumness, in particular within the additive-local framework considered here. These results identify unitary thermalization, independent of environmental decoherence, as a fundamental mechanism that limits the observation of macroscopic quantum effects.

Figures

Figures reproduced from arXiv: 2512.11522 by the authors.

Figure 1
Figure 1. FIG. 1. Purity (inverse of the effective dimension) for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Maximization over [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Maximization over [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Indices [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Obtaining the values of index [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Equilibration of generalized subsystems: a quantum-channel approach

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