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REVIEW 3 major objections 5 minor 69 references

Non-classicality at equilibrium and efficient predictions under non-commuting charges

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For systems with non-commuting charges, the equilibrium distribution of a coarse observable is a generalized Gibbs law written in terms of weak values of energy and charge.

desk verdict Solid extension of observable statistical mechanics to non-commuting charges, but the headline 'efficient prediction' currently rests on an in-sample consistency check, not the advertised linear ansatz end-to-end. read the letter →

arxiv 2507.22882 v1 pith:745MVUTM submitted 2025-07-30 quant-ph cond-mat.stat-mech

Lodovico Scarpa , Nishan Ranabhat , Amit Te'eni , Abdulla Alhajri , Vlatko Vedral , Fabio Anza , Luis Pedro García-Pintos This is my paper
classification quant-phcond-mat.stat-mech
keywords non-commutingchargesobservablestatisticalmechanicsweakvaluesKirkwood-Diracquasiprobabilitiesmaximumentropyprinciplenon-AbelianthermalstateequilibrationdegenerateHamiltonians
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 tries to show that even when a quantum system conserves charges that do not commute with one another, the equilibrium distribution of a sufficiently coarse observable can be predicted without diagonalizing the Hamiltonian. The prediction is a generalized Gibbs form, Eq. (14), in which the effective single-outcome energy and charge values are the real parts of weak values of H and of the charges conditioned on that outcome. The authors verify the formula numerically on a non-integrable SU(2)-symmetric spin chain, reaching total variation distances below 1% for one- and two-site observables and beating the usual non-Abelian thermal state for two-body observables. They also prove that Hamiltonian degeneracy is equivalent to the existence of non-commuting charges, and that the imaginary parts of the charges' weak values can be nonzero at equilibrium, which signals non-classicality that would be absent if the charges commuted.

What carries the argument

The machinery is the generalized Gibbs-type equilibrium formula of Eq. (14), built from weak values defined by $O_w(\rho,A_j) = \mathrm{Tr}(\rho O A_j)/\mathrm{Tr}(\rho A_j)$. Its real parts, $\varepsilon_j = \mathrm{Re}[E_w(\rho,A_j)]$ and $q^a_j = \mathrm{Re}[Q^a_w(\rho,A_j)]$, act as effective outcome-resolved energy and charge entering a maximum-entropy weight; the imaginary parts must satisfy the first Equilibrium Equation via the chemical potentials. The derivation relies on a degeneracy theorem (Appendix A3) that makes the fine-grained outcome probabilities uniform within each highly degenerate eigenspace, on a small-mutual-information argument that justifies using only first moments of $H$ and $Q^a$ as constraints, and on Theorem I.1 linking degeneracy to non-Abelian symmetry.

What would settle it

Compute the classical mutual information $I_{\mathrm{eq}}(A,H)$ and $I_{\mathrm{eq}}(A,Q^a)$ for a specific coarse observable in an SU(2)-symmetric spin chain, pick an initial state or observable where these are not much smaller than the corresponding Shannon entropies, and check whether the total variation distance between Eq. (14) and the time-averaged distribution stays below 1%; a clear violation would refute the generality of the prediction.

Watch

Extended reading notes

Core claim

The central claim is that observable-level equilibrium in the presence of non-commuting charges is governed by the constrained maximum of the observable's Shannon entropy, and that the resulting distribution is characterized by weak values: $p^{\mathrm{est}}_j = d_j e^{-\beta_A \mathrm{Re}[E_w(\rho,A_j)] - \sum_a \mu_a \mathrm{Re}[Q^a_w(\rho,A_j)]}/Z_A$, where $\mathrm{Re}[E_w]$ and $\mathrm{Re}[Q^a_w]$ are the real parts of weak values of the Hamiltonian and of each conserved charge postselected on outcome $j$. The paper derives this from the first and second Equilibrium Equations, uses a degeneracy theorem to justify reducing the fine-grained distribution to the coarse one, and numerically confirms the formula for $N = 12, 14, 16$ with errors under 1% in total variation distance. It also proves a structural equivalence: a Hamiltonian is degenerate if and only if its symmetry group is non-Abelian, which links degeneracy directly to non-commuting charges. Finally, because the charges do not commute with the equilibrium state, the imaginary parts of their weak values can be nonzero at equilibrium; the paper shows this happens in about 20% of the explored SU(2)-breaking initial-state cases, witnessing Kirkwood-Dirac non-classicality without violating the Equilibrium Equations.

Load-bearing premise

The load-bearing premise is that a coarse observable shares so little mutual information with the energy and charges that maximizing entropy with only the first moments of H and Qa yields the right equilibrium distribution; if that smallness fails for some observable, Eq. (14) would need extra constraints and the prediction would break down.

Editorial extensions

If this is right

  • For any sufficiently degenerate observable, the equilibrium distribution is computable from weak values of H and the charges, without energy eigenvectors or eigenvalues.
  • Because degeneracy and non-commuting charges are equivalent, relaxation effects blamed on degeneracy cannot be separated from non-Abelian symmetry effects.
  • Nonzero imaginary parts of charge weak values at equilibrium certify that the equilibrium description cannot be captured by a non-contextual classical model, even while the first Equilibrium Equation holds.
  • For the spin model studied, the method improves on the non-Abelian thermal state precisely where coupling is not weak, which is also where the NATS derivation is not justified.
  • The numerical agreement improves with system size from N = 12 to N = 16, suggesting the estimate becomes increasingly accurate in the thermodynamic limit.

Reading between the lines

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

  • One could test Eq. (14) on a model with an SU(3) or larger non-Abelian symmetry: the paper predicts anomalies in imaginary weak values should appear for appropriate symmetry-breaking initial states, while the real parts may eventually become anomalous too.
  • If the small-mutual-information condition is the true validity boundary, then deliberately constructing an observable that is finely correlated with an energy window or a charge sector should make Eq. (14) fail by an amount proportional to that mutual information; this would give a quantitative falsifier.
  • The linear ansatz connecting $\varepsilon_j$ and $q^a_j$ to first moments of $H$ and $Q^a$ is empirical; an analytical derivation would turn the method into a parameter-free predictive scheme, but until then the method needs a few calibration points, so its practical status is interpolation rather than prediction from first principles.
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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

3 major / 5 minor

Summary. The paper generalizes Observable Statistical Mechanics to Hamiltonians with non-commuting conserved charges. It derives first and second equilibrium equations from a maximum-entropy principle, expresses the solution as a Gibbs-like distribution (Eq. 14) whose effective 'energy levels' and 'charge levels' are the real parts of weak values of H and the charges, and shows that these weak values can be anomalous (non-classical) at equilibrium. It also proves an equivalence between Hamiltonian degeneracy and the presence of a non-Abelian symmetry group (Theorem I.1). Numerical simulations on a non-integrable SU(2)-symmetric spin chain are used to verify the equilibrium equations and to compare the resulting distribution with the non-Abelian thermal state.

Significance. If the framework is correct, it provides a novel structural characterization of equilibrium distributions of coarse observables under non-commuting charges, connects equilibrium physics to weak values and Kirkwood-Dirac quasiprobabilities, and offers a potential route to predictions that avoid full diagonalization of the Hamiltonian. The derivation in Appendix A4 is careful and explicit, Theorem I.1 is clean and correctly proved for both finite and infinite dimensions, and the numerical checks of the First and Second Equilibrium Equations are extensive and supportive. The main weakness is that the 'efficient predictions' advertised in the title and abstract are not tested end-to-end: the numerical validation of Eq. (14) uses quantities extracted from the very equilibrium data being predicted, and the empirical linear ansatz (Eqs. 29-30) that is supposed to make the scheme predictive is validated only for the weak-value real parts, not for the resulting probability distributions.

major comments (3)
  1. [Sec. IV.D and Fig. 3] The numerical verification of Eq. (14) is in-sample. The text states that to compute pest_j the authors 'first obtain εj ≈ Rj(t)/pj(t) and qa_j ≈ Ra_j(t)/pj(t)' from the time-averaged evolution, and then fix β_A and μ_a by fitting or optimization on the same data. Thus Fig. 3 shows that the equilibrium distribution is consistent with a Gibbs form whose level parameters are read off from that very distribution; it does not demonstrate prediction of equilibrium without knowledge of the energy eigenvalues and eigenvectors. The genuinely predictive route, Eqs. (29)-(30), is never propagated to the probability distributions, and no total-variation distance is reported for the end-to-end scheme. I request an out-of-sample test: estimate εj and qa_j via Eqs. (29)-(30) for held-out initial states or observables, compute pest_j, and report the TVD against the time-averaged pj(t).
  2. [Sec. IV.C, Eqs. (29)-(30), and Fig. 2] The linear ansatz for εj and qa_j is explicitly admitted to be empirical and to lack an analytical derivation. Fig. 2 validates the ansatz only for εj and qa_j, using two or three data points to fix the coefficients, and it does not quantify how errors in the ansatz propagate into the final probability distribution. Since the 'efficient prediction' claim rests on this ansatz, the paper should either supply a derivation or provide a more extensive out-of-sample validation that includes the resulting distributions, not just the intermediate weak-value estimates.
  3. [Sec. II.D and Appendix A2] The validity of Eq. (14) is conditional on small mutual information between the coarse observable and the energy and charges. The paper gives heuristic arguments for this condition, but it never directly verifies the condition in the numerical experiments. Given that this small-mutual-information assumption is load-bearing for the maximum-entropy step, I suggest computing Ieq(A,H) and Ieq(A,Qa) from the simulated data and reporting them alongside the equilibrium-equation checks. This would confirm that the studied observables actually lie in the regime where the few-constraint maximum-entropy treatment is justified.
minor comments (5)
  1. [Eqs. (9), (10), (16), (18)] The notation '!' before '=' is used nonstandardly, apparently to mean 'is set to zero' or 'should equal'. This is confusing, especially combined with the surrounding equations; please define the notation explicitly or replace it with standard equality/stationarity notation.
  2. [Sec. IV.D] The procedure for determining β_A and μ_a^A is described only vaguely as 'either analytically or via numerical optimization'. Please specify for each observable which method was used, and report the fitted parameter values or the optimization objective, so that the results are reproducible.
  3. [Fig. 2 inset] The inset legend states that the grey line 'in this case coinciding with the green and orange lines' is the average energy E(θ). This makes the plot hard to read; using distinct markers or colors for overlapping curves would improve clarity.
  4. [Sec. IV.A vs. Appendix A5] The charges are defined with a 1/N factor in the main text (Eq. (24)) and in parts of Appendix A5, but the NATS construction in Appendix A5 uses Qa = Σ_i σ_i^a without the 1/N factor. Please flag this rescaling explicitly when moving between the two conventions, as it affects the magnitudes of qa and Δqa and could confuse readers.
  5. [Abstract and Sec. V] The abstract claims the framework can 'accurately estimate the equilibrium distribution of coarse observables without access to the energy eigenvalues and eigenvectors'. As argued in Major Comment 1, the numerical support for this specific claim is currently missing. Please either temper the claim to what is demonstrated or add the missing end-to-end test.

Circularity Check

1 steps flagged · score 6.0 of 10

Main numerical validation of Eq. (14) is in-sample: εj and qa_j are extracted from the same equilibrium distribution being predicted, and β, μ are fitted; the parameter-light linear-ansatz route is never tested end-to-end.

  1. fitted input called prediction [Section IV.D (Predicting the equilibrium distribution), around Eq. (14) and Fig. 3; see also Eqs. (15) and (20).]
    "To compute our estimate pest_j ∝ e^{−βA εj − Σ_a μ_a qa_j}, we first obtain εj ≈ Rj(t)/pj(t) and qa_j ≈ Ra_j(t)/pj(t). Then, we calculate βA and the {μa_A}a either analytically or via numerical optimization. We quantify the closeness with pj(t) using the total variation distance."

    By Eqs. (15) and (20), εj and qa_j are defined as Rj/pj and Ra_j/pj for the equilibrium state. The numerical recipe computes them from exactly the same time-averaged pj(t) that the estimate is later compared with, and βA, μa are fitted on the same data. The reported TVD < 1% therefore checks internal consistency of the max-entropy equations, not an out-of-sample prediction. The genuinely parameter-light route, Eqs. (29)-(30), is validated only for εj and qa_j individually (Fig. 2), and the end-to-end TVD obtained by inserting those linear-ansatz estimates into Eq. (14) is never reported. Thus the headline 'efficient predictions without spectral information' rests on an in-sample refit plus an untested empirical ansatz.

full rationale

The core theoretical derivation of Eq. (14) is not circular: it follows from a Jaynes-type constrained entropy maximization (Sec. II.A-II.B) with the Lagrange-multiplier solution, and the connection to weak values (Eq. (20)) is a mathematical identity. Theorem I.1 is proved in the paper, and the Anza-Gogolin-Huber-type theorem used to coarse-grain pjs to pj is restated and proved in Appendix A3, so those are independent mathematical inputs. The self-citations to Ref. [25] for the heuristic 'only first moments suffice' and for the linear ansatz Eqs. (29)-(30) are explicitly described as empirical and lacking a proper analytical derivation, so they are not disguised as external theorems; this lowers the circularity weight. The significant circularity is in the numerical validation of the central predictive claim: Sec. IV.D computes εj and qa_j from the same equilibrium data it then predicts, and fits β and μ on that data. Consequently Fig. 3 demonstrates that the equilibrium distribution can be parametrized in the Gibbs-like form of Eq. (14) with coefficients read off from the distribution itself, not that the scheme predicts the distribution from readily available information. The out-of-sample linear fits of Eqs. (29)-(30) cover only εj and qa_j, and the paper does not report the total variation distance of the resulting distributions. This leaves the end-to-end 'efficient prediction' claim partially supported by construction rather than by an independent test, warranting a score of 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The core derivation relies on the maximum-entropy principle and the heuristic small-mutual-information condition, both domain assumptions. The practical predictions introduce several fitted parameters (linear ansatz coefficients, effective temperature and chemical potentials, and NATS baseline parameters). No new entities are postulated.

free parameters (5)
  • γj, ηj, χj (linear fit coefficients for εj) = not stated (fit to 2-3 data points)
    Eq. (29) is an empirical linear relation εj ≈ γj E + ηj ΔE + χj; the authors state no analytical derivation exists. The coefficients are fixed using data points in Fig. 2.
  • γa_j, χa_j (linear fit coefficients for qa_j) = not stated
    Eq. (30) is the analogous empirical fit for the real parts of charge weak values.
  • βA (effective inverse temperature) = obtained analytically or via numerical optimization
    The effective inverse temperature in Eq. (14) is set by numerical optimization in Sec. IV.D.
  • µa_A (effective chemical potentials) = obtained analytically or via numerical optimization
    The chemical potentials in Eq. (14) are set by numerical optimization.
  • β, µa (NATS inverse temperature and chemical potentials) = fit to data in Appendix A5
    The baseline NATS predictions require fitting the inverse temperature and chemical potentials to the simulated data.
assumptions (6)
  • domain assumption Maximum entropy principle: the equilibrium distribution of a coarse observable is the one maximizing Shannon entropy under normalization, mean energy, and mean charge constraints.
    Invoked in Sec. II.A; the paper provides heuristic support (small mutual information) but no rigorous derivation from unitary dynamics.
  • domain assumption The classical mutual information between a coarse observable and the energy/charges is much smaller than the Shannon entropy bounds, so only first moments are needed as constraints.
    Assumed in Sec. II.A and argued heuristically in Appendix A2; if false, Eq. (14) would require additional constraints.
  • domain assumption The initial state has an approximate microcanonical subspace, with charge standard deviations Δqa scaling at most as √N.
    Assumed in Sec. I.C following Ref. [6]; required for the existence of a microcanonical subspace.
  • domain assumption The observables are coarse enough to satisfy dj(dj-1) ≥ nE+1, so the generalized Anza-Gogolin-Huber theorem applies.
    Condition stated in Sec. II.B and proved in Appendix A3; needed to relate pjs to pj.
  • domain assumption The observable equilibrates in the time-average sense.
    Standard assumption in equilibration theory, stated in Sec. II.D.
  • standard math Spectral theorem and direct integral decomposition for the infinite-dimensional version of Theorem I.1.
    Used in Appendix A1 to define degeneracy for non-pure-point spectra.

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Cite this review

Pith. "Pith review of Non-classicality at equilibrium and efficient predictions under non-commuting charges." pith.science (2026). https://pith.science/paper/745MVUTM

@misc{pith2026250722882,
  author       = {Pith},
  title        = {Pith review of: Non-classicality at equilibrium and efficient predictions under non-commuting charges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/745MVUTM}},
  note         = {Machine review of arXiv:2507.22882}
}
read the original abstract

A quantum thermodynamic system can conserve non-commuting observables, but the consequences of this phenomenon on relaxation are still not fully understood. We investigate this problem by leveraging an observable-dependent approach to equilibration and thermalization in isolated quantum systems. We extend such approach to scenarios with non-commuting charges, and show that it can accurately estimate the equilibrium distribution of coarse observables without access to the energy eigenvalues and eigenvectors. Our predictions do not require weak coupling and are not restricted to local observables, thus providing an advantage over the non-Abelian thermal state. Within this approach, weak values and quasiprobability distributions emerge naturally and play a crucial role in characterizing the equilibrium distributions of observables. We show and numerically confirm that, due to charges' non-commutativity, these weak values can be anomalous even at equilibrium, which has been proven to be a proxy for non-classicality. Our work thus uncovers a novel connection between the relaxation of observables under non-commuting charges, weak values, and Kirkwood-Dirac quasiprobability distributions.

Figures

Figures reproduced from arXiv: 2507.22882 by the authors.

Figure 1
Figure 1. The imaginary part Imh Qz w  ρ∞(θ),Ay,N/2 00 i ||Qz||op of the weak value of the charge Q z at equilibrium, normalized by ||Q z ||op = 1, for different initial states parametrized by the angle θ, for N = 16. Such quantity is computed with re￾spect to the eigenprojector A y,N/2 00 , which projects onto the eigensubspace corresponding to outcome 00 of the observable σ y N 2 σ y N 2 +1 . The non-zero values of Im h Q… view at source ↗
Figure 2
Figure 2. Comparison between our predictions for the real [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The total variation distance D(pj (t), pest j ) between the true probability distribution at equilibrium pj (t) and our estimate for it, p est j , plotted against the initial state parametrized by an angle θ. The figure includes all ob￾servables considered, for N = 16. In the inset, we plot the same quantity but for the estimate based on the NATS p NATS j := Tr(ρNATSAj ), i.e., D(pj (t), pNATS j ). Note for sim￾plic… view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Time evolution of the Shannon entropy of the probability distributions [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Here we show all anomalous imaginary parts of weak values found in the numerical simulations described in Section [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: We confirm the Equilibrium Equations are respected for all initial states, observables and (independent) eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: In blue: histogram of the total variation distance [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: The difference (absolute improvement) in total variation distance (TVD) ∆ [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: The cumulative distribution function of the total variation distance [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 11
Figure 11. Figure 11: We show that even according to the third definition of weak coupling [Eq. (A34)] the interaction is not sufficiently [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

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