Projected ensemble in a system with conserved charges with local support

G. J. Sreejith; Sandipan Manna; Sthitadhi Roy

REVIEW 3 major objections 4 minor 1 cited by

Projected ensemble in a system with conserved charges with local support

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Deep thermalization survives many-body localization: the projected ensemble of a subsystem converges to the maximum-entropy Scrooge ensemble for almost any measurement basis, unless the basis aligns with the locally conserved charges.

desk verdict New and worth refereeing: first projected-ensemble analysis for systems with extensively many local conserved charges, with solid numerics for k=2,3 and an openly conditional analytic derivation whose unproved input is the real soft spot. read the letter →

arxiv 2501.01823 v1 pith:UXBB4M7M submitted 2025-01-03 cond-mat.stat-mech cond-mat.dis-nnquant-ph

Sandipan Manna , Sthitadhi Roy , G. J. Sreejith This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.dis-nnquant-ph

keywords projectedensembledeepthermalizationScroogemany-bodylocalizationℓ-bitmodelPorter-Thomasdistributionlocalconservedchargesmeasurementbasis

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

The paper asks whether a quantum system whose only conserved quantities are many local charges—the defining feature of many-body-localized systems—still exhibits deep thermalization, meaning that the full distribution of pure states of a small subsystem becomes a universal ensemble at late times, not merely the subsystem's average density matrix. The answer it argues for is yes: starting from random product states, the projected ensemble built by projective measurements on the complement converges at late times and in the large-system limit to the Scrooge ensemble, the maximum-entropy distribution of pure states consistent with the subsystem's reduced density matrix, for essentially any single-site measurement basis. This contrasts with systems carrying one global conserved charge, where the limiting ensemble varies continuously with the measurement basis; with locally supported charges, the basis matters only in the singular case where the measurement operator is nearly aligned with a conserved charge. The supporting argument is semi-analytical: when the conditional bitstring probabilities are uncorrelated and exponentially (Porter-Thomas) distributed in the temporal ensemble—a fact verified numerically in Section V—the late-time matrix elements of the $k$-th projected-ensemble moment (Eq. 41) are exactly the Scrooge-ensemble moment matrix elements (Eq. 45).

What carries the argument

The load-bearing identity is the equality between two integrals: the $k$-th moment of the projected ensemble after temporal averaging (Eq. 41) and the $k$-th moment of the Scrooge ensemble computed by Haar averaging (Eq. 45). The Scrooge ensemble is the named central object: the unique ensemble of pure states realizing a given reduced density matrix $\rho$ while minimizing accessible information, formally the distortion of the Haar ensemble $E_{\mathrm{Scrooge}}(\rho)=\{D\langle\psi|\rho|\psi\rangle,\ \sqrt{\rho}|\psi\rangle/\sqrt{\langle\psi|\rho|\psi\rangle}\}$. The projected-ensemble side of the equality is carried by the empirical Porter-Thomas structure of the temporal ensemble—each conditional bitstring probability $p_t(b|z_A)$ is independently exponentially distributed with mean $\mu(b)$ and uncorrelated across subsystem bitstrings $z_A$—and it is precisely this structure that, after a change of variables, turns the projected-ensemble integral into the Scrooge integral. The Scrooge side is evaluated by standard Haar integration with a Laplace-transform trick, so the whole proof reduces the physics of deep thermalization to a single statistical property of measurement-outcome probabilities.

What would settle it

Compute the joint distribution of the conditional probabilities $p_t(b|z_A)$ and $p_t(b|z'_A)$ for distinct subsystem bitstrings in the late-time temporal ensemble of the $\ell$-bit model: if their correlation does not vanish as $L\to\infty$ at any fixed $\alpha>0$, the projected-ensemble integral in Eq. 41 is not the correct late-time matrix element and the Scrooge identity fails. At exact alignment the paper supplies a direct check of the framework: at $\alpha=0$ the projected-ensemble moments are exactly the product in Eq. C5, which differs from the Scrooge value in Eq. C6, so a numerical comparison of these two expressions over accessible system sizes settles the matter. An experiment-ready version is to measure the empirical distribution of bitstring probabilities on a many-body-localized quantum simulator: at $\alpha=\pi/2$ it must be exponential with mean $1/2^L$, and any other shape contradicts the Porter-Thomas assumption the proof rests on.

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Extended reading notes

Core claim

On the paper's own terms, the central result is that deep thermalization reaches the Scrooge ensemble in a system with an extensive number of locally supported conserved charges. Using a strongly disordered Floquet spin chain with a many-body-localized regime and the phenomenological $\ell$-bit model, whose conserved charges are the strictly 1-local operators $\{\sigma_i^z\}$, the authors find numerically for the second and third moments that the trace distance between the projected ensemble and the Scrooge ensemble decays as a power law in time and then exponentially in system size, for measurement angles $\alpha \gtrsim 0.1\pi$; at $\alpha=0$, where the measurement is along the conserved charges, the projected ensemble instead reaches a different, exactly computed ensemble and the trace distance shows no decay with system size. For the $\ell$-bit model this numerical statement is elevated to a semi-analytical proof: under the $k$-th no-resonance condition, the late-time temporal average of the projected-ensemble moment matrix elements is exactly the integral in Eq. 41 whenever the conditional probabilities $p_t(b|z_A)$ are independent and exponentially distributed with means $\mu(b)=\sum_{z_B}|M_{z_B}|^2|\langle b|z_B\rangle|^2$, and that integral is term-by-term identical to the Haar-averaged Scrooge moment in Eq. 45.

Load-bearing premise

The proof assumes, on the basis of numerical evidence rather than derivation, that at late times the conditional bitstring probabilities fluctuate independently in time and follow the same exponential (Porter-Thomas) distribution with bitstring-dependent means, so that averaging the projected-ensemble moments over time reproduces the Scrooge moments; this fails when the measurement is aligned with a conserved charge, and if the residual correlations seen at small alignment angles survive for large systems, the claimed equality collapses.

Editorial extensions

If this is right

Deep thermalization is not blocked by non-thermalizing dynamics: a many-body-localized system, which never thermalizes in the usual sense, still generates a universal maximum-entropy distribution of subsystem pure states at late times.
When conserved charges are local, the limiting ensemble is essentially independent of the measurement basis, in contrast to the global-charge case where it varies continuously with the basis; this makes the local-versus-global structure of charges the controlling factor for the projected ensemble.
The projected ensemble of the $\ell$-bit model forms an approximate Scrooge $k$-design (verified numerically for $k=2,3$), so repeated single-shot measurements on an MBL system at late times present a local observer with Scrooge-distributed pure states.
Convergence is controlled and predictable in practice: trace distances decay as power laws in time with an angle-dependent exponent and then exponentially in system size at a rate $\lambda\approx0.23$ for $\alpha\gtrsim \pi/8$, the same rate in both models, so modest system sizes should already show near-Scrooge ensembles.
The exactly solvable aligned case ($\alpha=0$) provides a controlled counterpoint: there the projected-ensemble moments differ from the Scrooge moments in a computable way (Eq. C5 versus Eq. C6), giving a sharp diagnostic of when measurement and conserved charges align.

Reading between the lines

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

A generalization the paper leaves implicit: any dephasing dynamics with a complete set of local integrals of motion and a non-resonant spectrum—including non-chaotic and non-MBL integrable chains—should show the same Scrooge-level deep thermalization, since the proof uses only the Porter-Thomas independence of conditional probabilities, not chaos.
A testable experimental signature follows: on a current-generation many-body-localized quantum simulator, the histogram of bitstring measurement probabilities at late times should be exponential with bitstring-dependent mean $\mu(b)$, which would certify Scrooge-level deep thermalization from repeated single-shot readouts alone, without state tomography.
The measurement-angle dependence could be turned into a probe of the localized regime: because the residual deviation from Scrooge at $\alpha\gtrsim0.1\pi$ decays exponentially with system size while the $\alpha\to0$ deviation is controlled by correlations of conditional probabilities, measuring the second moment as a function of $\alpha$ could estimate the localization length or detect the onset
If the Porter-Thomas independence follows from the no-resonance condition rather than from chaos, then integrable systems with resonant spectra (such as clean free-particle chains) should deviate from the Scrooge ensemble in a calculable way, providing a sharp test of the paper's mechanism on platforms that are already available.

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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

The genuinely new result here is that the projected ensemble for MBL-like systems with extensively many local conserved charges converges to a measurement-basis-independent Scrooge ensemble, in sharp contrast to the global-charge case where the limiting ensemble varies continuously with the basis. I think that claim holds for generic measurement bases, and the paper earns its keep on the numerics: exponential decay of trace distance in L for k=2 and k=3 is shown in two distinct models, and the measurement-angle dependence is clearly demonstrated. The analysis of the alpha=0 pathology and the contrast with the global-charge case are also well laid out.

The soft spot is real and it is exactly where the stress-test note lands. The analytic derivation in Section VI from Eq. 38 to Eq. 41 requires two inputs: Porter-Thomas distributed conditional bitstring probabilities p_t(b|z_A), and statistical independence across distinct z_A. The authors say plainly they could not prove these. The independence assumption is load-bearing: it is what lets the temporal average factor into a product of single-variable integrals, and without it the Y-variable change leading to Eq. 41 does not go through. The failure at alpha=0 is concrete, and the numerics for small alpha show slower, non-exponential decay. So the proof is conditional, not a full derivation, and the paper is honest about that. I do not think this is fatal: the central claim is still supported by the finite-size numerics and the analytic argument works conditional on an empirically well-tested ansatz for alpha not too close to zero. But the paper should be asked to bound or prove the independence/Porter-Thomas input, or at minimum to provide a detailed numerical study of how the crossover alpha scale and required L behave as alpha shrinks. It would also help to see k>3 moments; the current claim of a Scrooge ensemble is only tested as a state 3-design.

Minor point: the measurement-basis dependence is studied only for single-site Pauli measurement axes parametrized by alpha. That is a reasonable first step, but the phrase "generic measurement basis" is a bit stronger than the evidence supports.

Who gets value from this: anyone working on deep thermalization, projected ensembles, MBL phenomenology, and the structure of emergent conserved charges. It deserves a serious referee and, likely, a conditional accept after the requested additions. I would send it out.

Referee Report

3 major / 4 minor

Summary. The paper studies the projected ensemble (PE) in a strongly disordered spin chain exhibiting many-body localization and in the phenomenological l-bit model with 1-local conserved charges. For initial product states, the authors find numerically that for k=2 and k=3 the late-time PE converges to the Scrooge ensemble built from the infinite-time reduced density matrix, with trace distances decaying exponentially in subsystem size L for measurement angles alpha not too close to zero. The main analytic contribution is a derivation, in Sec. VI, showing that if the conditional bitstring probabilities p_t(b|z_A) are independently Porter-Thomas distributed with mean mu(b), then the matrix elements of the k-th PE moment equal those of the Scrooge ensemble. The paper explicitly leaves the Porter-Thomas and independence assumptions as empirical inputs; the small-alpha regime (alpha close to 0), where the measurements overlap the conserved charges, is found to deviate from Scrooge behaviour in the accessible system sizes.

Significance. If the claimed equivalence holds, it identifies a new scenario for deep thermalization: a system with an extensive set of locally supported conserved charges still produces a basis-independent Scrooge projected ensemble at late times and large sizes, in contrast to systems with global conserved charges where a generalized Scrooge ensemble arises. The numerical results are solid within their stated range: trace-distance decay in L is seen for both the disordered Floquet chain and the l-bit model, for k=2 and k=3, and the conditional analytic step is a clean reduction of PE moments to Scrooge moments. The main weakness is that the load-bearing statistical input—independent Porter-Thomas behaviour of conditional bitstring probabilities—is numerically motivated but not proved, and the paper's own data show that the claimed regime 'not close to the conserved charges' is not sharply delineated. The manuscript is honest about these limitations and would be strengthened by either proving the ansatz in the random-phase ensemble or making the conditional nature of the claim explicit in the abstract and conclusion.

major comments (3)

[Sec. VI, Eqs. (38)-(41)] The analytic derivation of the PE moments is conditional on two unproved statistical inputs: (i) each p_t(b|z_A) is exponentially (Porter-Thomas) distributed with mean mu(b) from Eq. (39), and (ii) these variables are statistically independent across distinct z_A. The independence assumption is used precisely in Eq. (40), where the temporal average of the ratio in Eq. (38) is replaced by a product of independent Porter-Thomas densities, and again in the change of variables leading to Eq. (41). Without independence, the denominator [sum_{z_A} p(z_A) p_t(b|z_A)]^{k-1} couples all z_A and the identification with the Scrooge expression (45) does not follow. The authors themselves state in Sec. V and Sec. VII that these empirical results have not been proved. The central claim therefore holds conditionally, and the text should say so explicitly; ideally the random-phase ensemble of Appendix B, which already provides a cleaner stationary ensemble, could be used to test or prove the ansatz.
[Sec. IV B and Fig. 4] The abstract and conclusion claim convergence to the Scrooge ensemble 'except when the measurement operator is close to the conserved charges', but no quantitative characterization of 'close' is given. The numerical evidence shows clean exponential decay for alpha ≳ 0.1 pi, a slower non-exponential trend for alpha = 0.05 pi, and a qualitatively different limit at alpha = 0 (Appendix C). Without a bound on how close to alpha = 0 the convergence can fail, or a statement that for any fixed alpha > 0 convergence holds for sufficiently large L, the claimed regime is not precisely defined and the central claim is not fully falsifiable. The authors should either specify the threshold and its L-dependence or explicitly restrict the claim to alpha bounded away from 0.
[Sec. VI A, no-resonance condition] The derivation also relies on the k-th no-resonance condition and on temporal self-averaging of the matrix elements (Fig. 5). These are reasonable for the l-bit model, but the k-dependence of the condition and the assumed decay of temporal fluctuations with L are not discussed beyond the numerical evidence. Since the proof is already conditional on the Porter-Thomas ansatz, this additional assumption should at least be stated as an assumption, and preferably checked for k>3 where no numerical moments are reported.

minor comments (4)

[Throughout] There are several typos that should be fixed: 'efffective' in Sec. I, 'with has' in Sec. VII, 'satisfies satisfies' in Sec. VI A, and 'p(b|zAA)' in Eq. (40).
[Eq. (40) and Sec. V] The notation PoPb and PoP' is used with inconsistent spacing and font; using a single notation such as PoP_b and PoP would improve clarity.
[Fig. 3 caption] The caption says the initial state has theta_1 and theta_2 specified, but the text in Sec. V and Eq. (11) samples these angles uniformly; it would be helpful to clarify whether the figure uses a fixed representative initial state or an average over states.
[Sec. VII] The conclusion says numerical tests on the 'first three moments' confirm the k=3 design; since the paper reports k=2 and k=3 moments, this phrasing should be adjusted to avoid implying a separate k=1 check or a moment beyond the third.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Scrooge-convergence claim is supported by direct numerical comparison and a transparently conditional analytic derivation whose Porter-Thomas input is an acknowledged empirical hypothesis, not a fitted or definitional surrogate for the target result.

full rationale

The paper's central derivation in Sec. VI is a genuine conditional implication rather than a reduction by construction. Starting from the exact expression for the projected-ensemble moment (Eq. 38), the authors introduce two statistical assumptions about the temporal ensemble: that each p_t(b|z_A) is Porter-Thomas distributed with mean mu(b) given by Eq. 39, and that these variables are independent across distinct z_A. Under these assumptions, the temporal average in Eq. 40 evaluates to the integral in Eq. 41, which is then shown to be identical to the independently computed Scrooge-ensemble moment in Eq. 45. This is a nontrivial mathematical identity: the input is a statement about bitstring probability distributions, while the output is an equality between two many-body ensemble moments. The Porter-Thomas ansatz is not fitted to the Scrooge moments, nor is it defined in terms of them; it is an empirical property of the same system, verified numerically in Sec. V (Figs. 6-9) and explicitly acknowledged as unproved in the conclusion. An unproved hypothesis is a rigor gap, not circularity. Moreover, the paper provides direct numerical evidence for Scrooge convergence that is independent of the Porter-Thomas ansatz: Sec. III and Sec. IV compute the trace distance (Eq. 16) between the projected-ensemble moments and the Scrooge moments over time and system size (Figs. 1-4), showing exponential decay with L for alpha >~ 0.1 pi. This external benchmark means the central claim does not rest solely on the conditional derivation. There are also no load-bearing self-citations: the Scrooge ensemble construction is cited from prior work, but the equality with the projected ensemble is derived here rather than imported, and no uniqueness or ansatz is smuggled in through a citation to the authors' own earlier results. The paper is unusually explicit about its limitation, stating in Sec. VII that 'we have not been able to prove these empirical results,' which further confirms that the Porter-Thomas input is an openly stated assumption rather than a hidden circular step. Accordingly, no specific circular reduction can be exhibited, and the appropriate finding is no significant circularity.

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

The central claim rests on standard quantum many-body dynamics, the l-bit model structure, and two empirical assumptions: the no-resonance condition and the Porter-Thomas distribution with uncorrelated conditional probabilities. No free parameters are fitted to the target result; all model parameters are taken from prior literature. No invented entities are introduced.

assumptions (4)

domain assumption The l-bit Hamiltonian satisfies the k-th no-resonance condition.
Invoked in Section VI after Eq. 36 to argue that off-diagonal matrix elements of the projected ensemble vanish at late times. The paper states it 'expects' this due to random interactions, but does not prove it.
domain assumption In the late-time temporal ensemble, the conditional bitstring probabilities p_t(b|z_A) are independent Porter-Thomas (exponential) random variables with mean mu(b) given by Eq. 39, and are uncorrelated across z_A.
This is the central input to the analytic derivation in Section VI, Eqs. 39-41. It is verified numerically in Section V but not derived.
domain assumption The temporal ensemble average captures the late-time steady-state values of projected ensemble moments, with fluctuations vanishing as system size increases.
Used in Sections V and VI to replace time-dependent moments by their temporal averages; supported numerically by Fig. 5.
standard math Standard techniques for Haar integration and the Scrooge moment formula from Ref. [19].
Used in Appendix A to evaluate the Scrooge moments, following the derivation in Ref. [19].

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Pith. "Pith review of Projected ensemble in a system with conserved charges with local support." pith.science (2026). https://pith.science/paper/UXBB4M7M

@misc{pith2026250101823,
  author       = {Pith},
  title        = {Pith review of: Projected ensemble in a system with conserved charges with local support},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXBB4M7M}},
  note         = {Machine review of arXiv:2501.01823}
}

abstract

The investigation of ergodicity or lack thereof in isolated quantum many-body systems has conventionally focused on the description of the reduced density matrices of local subsystems in the contexts of thermalization, integrability, and localization. Recent experimental capabilities to measure the full distribution of quantum states in Hilbert space and the emergence of specific state ensembles have extended this to questions of {\textit{deep thermalization}}, by introducing the notion of the {\textit{projected ensemble}} -- ensembles of pure states of a subsystem obtained by projective measurements on its complement. While previous work examined chaotic unitary circuits, Hamiltonian evolution, and systems with global conserved charges, we study the projected ensemble in systems where there are an extensive number of conserved charges all of which have (quasi)local support. We employ a strongly disordered quantum spin chain which shows many-body localized dynamics over long timescales as well as the $\ell$-bit model, a phenomenological archetype of a many-body localized system, with the charges being $1$-local in the latter. In particular, we discuss the dependence of the projected ensemble on the measurement basis. Starting with random direct product states, we find that the projected ensemble constructed from time-evolved states converges to a Scrooge ensemble at late times and in the large system limit except when the measurement operator is close to the conserved charges. This is in contrast to systems with global conserved charges where the ensemble varies continuously with the measurement basis. We relate these observations to the emergence of Porter-Thomas distribution in the probability distribution of bitstring measurement probabilities.

Figures

Figures reproduced from arXiv: 2501.01823 by the authors.

**Figure 2.** FIG. 2. Distance between the second and third moments of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Asymptotic trace distance between Scrooge ensemble [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 6.** Figure 6: FIG. 6. Distribution of probabilities of measurement bit [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 5.** Figure 5: FIG. 5. The matrix elements in second moments of PE [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7. The distribution of KL divergence between the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. The PoP [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. The random phase ensemble (dotted lines) is compared with the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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