{"id":"8136bac6-22fd-4e94-ad54-30da82a502ec","arxiv_id":"2501.01823","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In many-body localized systems with local conserved charges, the projected ensemble of a subsystem converges to the Scrooge ensemble for generic measurement bases, unlike systems with global charges.","lead":"This paper studies what happens to the distribution of quantum states in a small region when the rest of a disordered, many-body-localized system is measured. It finds that for most measurement choices this distribution becomes a known universal ensemble, the Scrooge ensemble, independent of the measurement basis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Analytic proof of Scrooge convergence depends on unproved independent Porter-Thomas ansatz for conditional bitstring probabilities; this is precisely what fails near α=0 and is untested for k>3.","rationale":"The paper is numerically careful and transparent: Figs. 1 and 2 show exponential-in-L decay of trace distances for k=2,3 at α=π/8 and π/2 for both the Floquet MBL model and the ℓ-bit model, and Appendix B supports similarity of the temporal ensemble to the random-phase ensemble at α=π/8. The authors also explicitly flag the Porter-Thomas/decorrelation result as unproved. The reader's conditional verdict is therefore proportionate: the physical claim may be true, but its only analytic support is conditional on a plausible but open probabilistic conjecture. I identify the independence of p_t(b|z_A) across z_A as the single most load-bearing step because it is the point where the derivation would fail even for the k=2,3 moments the numerics test; without it Eq. 40 cannot be replaced by Eq. 41, and comparison with Eq. 45 does not follow. The α=0 limit and the small-α data of Fig. 4 show that this assumption is not harmless, and the paper gives no theorem or rigorous bound delimiting the α→0 crossover. The proposed test directly measures the relevant joint distribution in the temporal ensemble, so it would settle whether the assumption holds in the large-L limit for fixed nonzero α. The reader's request for a proof or controlled error bound on the ansatz, plus extended checks to higher moments, remains exactly the right prescription; no change to the conditional verdict is needed.","tokens_in":19949,"tokens_out":23351,"duration_ms":233996,"concrete_test":"For the ℓ-bit model of Eq. 17, fix two measurement angles, a generic one α=π/8 and a small one α=π/32. Pick a fixed bitstring b and distinct subsystem bitstrings z_A≠z'_A. Using the temporal ensemble Etemp of Eq. 21 (late-time sampling), estimate the mutual information I(p_t(b|z_A); p_t(b|z'_A)) between the two conditional probabilities, normalizing the variables by the mean μ(b) of Eq. 39. Average this mutual information over b and over many random pairs (z_A,z'_A), and plot it versus L=12,14,16,18,20. Separately compute the KL divergence of the marginal distribution of p_t(b|z_A) from the exponential with mean μ(b). If the averaged mutual information does not trend to zero as L grows at fixed α, or the marginal KL divergence does not vanish, then the independence/Porter-Thomas assumption underlying Eq. 40 fails, and the derivation of Eq.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the semi-analytic derivation in Sec. VI, specifically the step from Eq. 38 to Eq. 41. That step assumes two empirical inputs: (i) each conditional probability p_t(b|z_A), defined via φ_t in Eq. 33, is exponentially (Porter-Thomas) distributed with mean μ(b) from Eq. 39, and (ii) these random variables are statistically independent across distinct z_A. The authors explicitly state in Sec. V that they have not been able to prove these results. This is not a cosmetic gap: Eq. 40 replaces the temporal average of the ratio in Eq. 38 by an integral over a product of independent Porter-Thomas densities, and the factorization step is exactly where the independence assumption is used. Without independence, the denominator [Σ z_A p(z_A) p_t(b|z_A)]^{k−1} couples all z_A, and the Y-variable change leading to Eq. 41 does not go through, so the equality with the Scrooge expression Eq. 45 is not established. The failure mode is concrete: at α=0 the p_t(b|z_A) are perfectly correlated (independent of z_A), and the Fig. 4 data at α=0.05π do not show the clean exponential L-decay seen for α≳0.1π. The paper provides no bound on how close to α=0 the convergence can fail, or how large L must be at fixed small α before the independent-Porter-Thomas regime sets in. Thus the proof establishes the claimed late-time, large-system equivalence only conditionally on an unproved decorrelation ansatz, leaving finite-size numerics as the sole evidence in the small-α and higher-moment (k>3) sectors.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20211,"tokens_out":2899,"duration_ms":34492,"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":[{"comment":"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.","section":"Sec. VI, Eqs. (38)-(41)"},{"comment":"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.","section":"Sec. IV B and Fig. 4"},{"comment":"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.","section":"Sec. VI A, no-resonance condition"}],"minor_comments":[{"comment":"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).","section":"Throughout"},{"comment":"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.","section":"Eq. (40) and Sec. V"},{"comment":"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.","section":"Fig. 3 caption"},{"comment":"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.","section":"Sec. VII"}],"recommendation":"major_revision","confidential_remarks":"The paper's main claim is presented as a convergence result but its analytic proof is conditional on an unproved independent Porter-Thomas ansatz; the abstract and conclusion should be tempered to reflect this. The small-alpha regime, which the authors themselves identify as exceptional, is exactly where the ansatz fails, so a precise statement of the regime of validity is needed. A major revision that either proves the ansatz in the random-phase ensemble or makes the conditional status explicit, and that sharpens the alpha-dependence statement, would make the contribution suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nMinor 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.\n\nWho 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.","headline":"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.","tokens_in":20798,"tokens_out":1106,"would_cite":true,"duration_ms":13793,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["projected ensemble","deep thermalization","Scrooge ensemble","many-body localization","ℓ-bit model","Porter-Thomas distribution","local conserved charges","measurement basis"],"falsifier":"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.","tokens_in":19700,"feed_emoji":"⚛️","tokens_out":35946,"duration_ms":265692,"temperature":0.7,"pith_summary":"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).","feed_headline":"Local conserved charges don't block the Scrooge ensemble","feed_subtitle":"The full state distribution becomes the maximum-entropy Scrooge ensemble even in nonthermalizing chains.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the projected ensemble and frames deep thermalization as the approach of this ensemble to a universal late-time limit, the question this paper pursues.","marker":"[14]"},{"why":"Supplies the maximum-entropy principle for deep thermalization, the random-phase ensemble and no-resonance averaging, and the Scrooge-moment computation that Section VI and Appendix A of the paper follow.","marker":"[19]"},{"why":"Defines the Scrooge ensemble as the unique ensemble with a prescribed density matrix that minimizes accessible information, making it the paper's target limiting ensemble.","marker":"[20]"},{"why":"Supports the identification of the Scrooge ensemble as the universal late-time distribution of subsystem wavefunctions in equilibrium, which the paper extends to localized dynamics.","marker":"[21]"},{"why":"Establishes the generalized Scrooge ensemble for dynamics with a single global conserved charge, the measurement-basis-dependent contrast case against which the paper's basis-independence result is set.","marker":"[22]"},{"why":"Provides the local integrals of motion (l-bits) in many-body-localized systems, the locally supported conserved charges that define the setting of the paper.","marker":"[32]"},{"why":"Introduces the phenomenological l-bit Hamiltonian, Eq. 17, that carries the paper's analytical derivation and its main numerical computations.","marker":"[33]"},{"why":"Supplies the disordered Floquet model, Eq. 10, whose many-body-localized regime provides the paper's first numerical demonstration of convergence to the Scrooge ensemble.","marker":"[39]"},{"why":"Shows dephasing-driven approach to equilibrium in the many-body-localized phase, the mechanism that lets the projected ensemble reach a steady late-time limit.","marker":"[36]"}],"fun_headline_variants":["Scrooge ensemble emerges despite many local conserved charges","Local charges don't prevent deep thermalization to Scrooge","Even with many local charges, projected ensemble is Scrooge","Scrooge ensemble forms in MBL with local conserved charges","Deep thermalization attains Scrooge ensemble with local charges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Scrooge ensemble emerges despite many local conserved charges","Local charges don't prevent deep thermalization to Scrooge","Even with many local charges, projected ensemble is Scrooge","Scrooge ensemble forms in MBL with local conserved charges","Deep thermalization attains Scrooge ensemble with local charges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000885,"raw_usage":{"total_tokens":3901,"prompt_tokens":1105,"completion_tokens":2796,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":2712}},"tokens_in":721,"tokens_out":2796,"duration_ms":20981,"temperature":1.0,"reasoning_tokens":2712,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:19:25.408310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the projected ensemble and frames deep thermalization as the approach of this ensemble to a universal late-time limit, the question this paper pursues."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-entropy principle for deep thermalization, the random-phase ensemble and no-resonance averaging, and the Scrooge-moment computation that Section VI and Appendix A of the paper follow."},{"cited_title":"Jozsa, D","cited_arxiv_id":null,"evidence_quote":"Defines the Scrooge ensemble as the unique ensemble with a prescribed density matrix that minimizes accessible information, making it the paper's target limiting ensemble."},{"cited_title":"Goldstein, J","cited_arxiv_id":null,"evidence_quote":"Supports the identification of the Scrooge ensemble as the universal late-time distribution of subsystem wavefunctions in equilibrium, which the paper extends to localized dynamics."}],"review_version":1}