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Scrooge k-designs are introduced and proven to emerge from late-time chaotic dynamics, from projected measurements of global Scrooge 2k-designs, and from scrambled-basis measurements on arbitrary entangled states.

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2026-08-03 13:05 UTC pith:VSXI7KBA

load-bearing objection A serious framework with two solid transfer theorems, but Theorem 1's advertised error bound needs an extra assumption to mean what the abstract says. the 3 major comments →

arxiv 2601.00266 v3 pith:VSXI7KBA submitted 2026-01-01 quant-ph cond-mat.stat-mechmath-phmath.MP

Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems

Wai-Keong Mok , Tobias Haug , Wen Wei Ho , John Preskill This is my paper
classification quant-ph cond-mat.stat-mechmath-phmath.MP
keywords scroogeensemblesquantumdesignglobalmany-bodyapproximatebehavior
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

When a quantum system is in an entangled state, measuring part of it leaves the rest in a randomly selected pure state. The collection of such states, the projected ensemble, can reveal statistics beyond ordinary thermal averages, a phenomenon called deep thermalization. In idealized infinite-temperature settings this ensemble becomes the uniform Haar distribution. But with real constraints, such as finite temperature or conserved charges, the maximal-randomness answer is different: the Scrooge ensemble, the distribution that hides the least classical information while still reproducing the observed average state. This paper's first contribution is a precision tool: Scrooge k-designs, finite collections of states that match the first k moments of the Scrooge ensemble. Its second contribution is three rigorous theorems. Theorem 1: a chaotic Hamiltonian run for long times produces a global Scrooge design from dynamics alone, no measurement needed, provided the spectrum has no resonances and the equilibrium state has low purity. Theorem 2: if the global state is drawn from a Scrooge 2k-design, then any fixed-basis measurement of the environment projects the system onto a mixture of local Scrooge k-designs. Theorem 3: for any entangled state whatsoever, measuring the environment in a basis scrambled by an efficiently implementable Haar 2k-design produces a local Scrooge k-design. These come with explicit error bounds that shrink as the bath grows. Numerical experiments then argue that three resources, coherence, magic (non-stabilizerness), and non-local information scrambling, are each necessary for this behavior, and that ground states of integrable models can also produce Scrooge designs when measured in the right basis.

Core claim

That approximate Scrooge k-designs emerge rigorously in three physical settings: (1) the temporal ensemble of a chaotic Hamiltonian obeying the kth no-resonance condition forms, in the low-purity regime k^2||sigma_diag||_2 << 1, a Scrooge(sigma_diag) k-design with additive error epsilon = O((D||sigma_diag||_infty)^k k^2/D + k||sigma_diag||_2) (Theorem 1, Eq. 16); (2) measuring B in any fixed basis, a generator drawn from a Scrooge 2k-design with relative error epsilon yields a projected ensemble that is a mixture of local Scrooge k-designs with average trace error O(sqrt(D_A^k (epsilon + ||sigma||_2))) (Theorem 2, Eq. 17); (3) an arbitrary bipartite entangled state, measured after scrambling B with a Haar 2k-design unitary, forms a local Scrooge k-design with the error bound of Eq. (20) (Theorem 3). If correct, deep thermalization universality extends beyond infinite temperature to finite-temperature and conserved-charge settings, with quantitative finite-size and finite-time errors.

Load-bearing premise

The low-purity condition k^2||sigma||_2 << 1 (and its local analogue k^2||sigma_hat_{A|z}||_2 << 1) underpins Lemma 1 (Appendix B), the technical backbone of all three theorems. Lemma 1 shows the normalized Scrooge ensemble can be replaced by the analytically tractable unnormalized ensemble {sqrt(D sigma)|phi>} with error O(k||sigma||_2); if the density matrix has significant purity, e.g., a finite-size system at low temperature, the error bound grows and the conclusions of Theorems 1-3 no longer follow. The paper's own finite-temperature application (Sec. III C 1) confines Theorem 3 to beta < beta_c for precisely this reason. Theorem 1 additionally depends on the kth no-resonance condition on the spectrum (Sec. II B 2), and the relative-error proofs also inherit Lemma 5 of Ref. [61], which bounds negative moments only for q < ||sigma||_infty^{-1}.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces the notion of approximate Scrooge k-designs and proves three main theorems: (1) the temporal ensemble of a chaotic Hamiltonian satisfying the kth no-resonance condition forms an approximate Scrooge(σ_diag) k-design with additive error given by Eq. (16) when k^2||σ_diag||_2 << 1; (2) measuring a subsystem B of a global state drawn from a Scrooge(σ) 2k-design with relative error ε yields a projected ensemble close to a mixture of local Scrooge k-designs, with the bound of Eq. (17); (3) measuring an arbitrary bipartite entangled state after applying a Haar 2k-design unitary to B yields a local Scrooge k-design with the bound of Eq. (20). The paper also presents numerical simulations that identify coherence, magic, and information scrambling as resources for emergent Scrooge behavior. The appendices contain detailed proofs with a clear lemma structure, including the key unnormalized-to-normalized Scrooge approximation (Lemma 1) and an honest discussion of the relative-versus-additive error gap between Theorems 1 and 2.

Significance. If the three theorems are valid as stated, the work substantially extends the theory of deep thermalization beyond infinite-temperature Haar randomness, providing quantitative finite-size and finite-time error bounds for Scrooge universality in settings with energy conservation and other constraints. The paper's strengths include a systematic proof structure in the appendices, the nontrivial approximation Lemma 1, explicit error bounds that track dependence on system parameters, and numerical support for resource requirements. The stated resource-efficiency claims (e.g., local random circuits of logarithmic depth for Theorem 3) are also of practical interest. However, the correctness of the central claims is currently compromised by a missing condition in Theorem 1 and by scaling issues in Theorem 3, as detailed below; the overall framework is promising but the theorems need correction before the conclusions can be relied upon.

major comments (3)
  1. [§III.A, Theorem 1 / Eq. (16), Appendix C] The low-purity condition k^2||σ_diag||_2 << 1 does not control the first term (D||σ_diag||_∞)^k k^2/D in Eq. (16). For a microcanonical initial state supported on M energy eigenstates, ||σ_diag||_∞=1/M and ||σ_diag||_2=1/√M; the condition only requires M >> k^4, whereas the first error term equals k^2 D^{k-1}/M^k. For D=2^20, M=10^3, k=3 this term is ≈10^4 while k^2||σ_diag||_2≈0.28. Thus the theorem's stated assumptions do not ensure the additive error is small, and the conclusion that a Scrooge k-design is formed is vacuous in finite-temperature or low-entropy regimes. The proof in Appendix C, Eq. (C4), shows the origin: the Haar-approximation error k^2/D is amplified by (D||σ_diag||_∞)^k. The theorem needs an explicit condition such as (D||σ_diag||_∞)^k k^2/D << 1, or the claim must be restricted to the limit ||σ_diag||_∞ = O(1/D).
  2. [§III.B, Proposition 1 (formal version: Theorem 9, Appendix D.2)] The proof of Proposition 1 bounds the error of the normalized projected ensemble by O(Δβ^{1/2}), where Δβ is only 'argued to be exponentially small' in typical many-body systems; no rigorous bound is provided. Since Proposition 1 is used to close the additive/relative gap between Theorems 1 and 2 and to justify the claim that dynamically generated global states produce local Scrooge behavior, this missing quantitative control is load-bearing. The manuscript should either prove a useful bound on Δβ or explicitly label the result as conditional on an unproven assumption.
  3. [§III.C, Theorem 3 / Eq. (20)] The error bound in Eq. (20) appears inconsistent with the claimed infinite-temperature limit. When σ_A=I/D_A, one has D_{A,eff}=D_A and D_A||σ_A||_2^2=1, so the first error term becomes O(√(ε+k^2D_A)), which does not vanish as D_B→∞ for fixed k and large D_A. This contradicts the assertion that Theorem 3 generalizes the Wilming–Roth result, which gives a vanishing error in that limit. The scaling with D_{A,eff} (in the numerator) may be a typo for k^2/D_{A,eff}, but as stated the theorem does not guarantee a Scrooge k-design under its conditions. Please revisit the derivation of Eq. (20).
minor comments (4)
  1. [§III.A after Theorem 1] The sentence 'Applying √Dσ_diag to these random phase states yields Eq. (8)' is confusing: Eq. (8) defines the random-phase ensemble, not the result of the application. Consider rewording.
  2. [General notation, Definition 2 vs Theorems 1–3] The theorems state trace-distance bounds without the factor 1/2 that appears in Definition 2 (Eq. (13)). Please clarify whether the 1/2 factor is included in the theorem statements or if the definitions are inconsistent.
  3. [Fig. 3c inset] The inset shows ν=2 but the definition of this exponent and the fitting procedure are not stated; a brief caption explanation would help.
  4. [Appendix D, Lemma 3] In the chain after Eq. (A42), '⪯' is used for scalar inequalities; this should be '≤'.

Circularity Check

0 steps flagged

No significant circularity: the derivation chain is self-contained; the flagged Eq. (16) bound concern is a vacuity/correctness issue, not a circular reduction.

full rationale

The central claims reduce to independent ingredients: (i) the random-phase ensemble is a Haar k-design [52,110]; (ii) Lemma 1 approximates the normalized Scrooge kth moment by the unnormalized moment (Dσ)^{⊗k}ρ_Haar with error controlled by k||σ||₂; and (iii) Weingarten/design-hierarchy calculations for projected ensembles. None of these steps defines the target quantity in terms of itself. Lemma 1 is an internal approximation result proved in Appendix B, not an assumption of its own conclusion. The kth no-resonance condition and the low-purity condition k²||σ||₂≪1 are stated assumptions; the temporal-ensemble-to-random-phase reduction is cited to Mark et al. [31], whose authors do not overlap with the present paper. The negative-moment bound is taken from McGinley–Schuster [61], also external. The Scrooge ensemble definition and its construction Eq. (11) come from the external Ref. [48]. The paper's self-citations (e.g., Refs. [24,33,35,36]) are contextual or used for numerical interpretation and are not load-bearing in the proofs of Theorems 1–3. The skeptically flagged growth of (D||σ_diag||_∞)^k k²/D for microcanonical initial states is a potential vacuousness/validity problem for the stated assumptions, but it does not exhibit a quantity that is equivalent to its own input by construction; accordingly it is a correctness risk, not circularity. I therefore find no circular step.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The analytic claims contain zero fitted parameters: every error bound is proven from stated assumptions (no-resonance, low purity, design inputs). The numerics introduce only illustrative fit exponents (nu=2, alpha values, nu=1 collapse). The load-bearing premises are domain assumptions about physical genericity plus two imported technical results ([61]'s negative-moment bound; [31]'s no-resonance-to-random-phase equivalence and Delta_beta estimate). No new physical entities are postulated; Scrooge k-designs are a mathematical definition with a precise falsifiable (moment-matching) meaning.

free parameters (3)
  • coherence-transition scaling exponent nu (Fig. 3c inset) = nu = 2
    Finite-size scaling collapse of Delta^(2) vs theta*N_B^(1/nu) in Sec. V A; fitted to numerics, illustrative of the predicted transition, not used in the theorems.
  • Scrooge-distance decay rates (Fig. 5b) = alpha_Clifford ~ 0.23, alpha_Haar ~ 0.5
    Exponents in Delta^(2) ~ 2^(-alpha N_B) fitted in Sec. V C; interpretive estimates of scrambling power, not load-bearing for the theorems.
  • Ising critical exponent nu (Fig. 5c) = nu = 1 (taken from the Ising universality class)
    Used to collapse Delta^(2) data near h_c = 1 as evidence of universal critical behavior; assumed from the literature rather than measured, reported without uncertainty.
axioms (5)
  • domain assumption kth no-resonance condition on the Hamiltonian spectrum
    Theorem 1 equates the temporal ensemble with the random phase ensemble only under this condition (Sec. II B 2, following Mark et al. [31]); standard genericity assumption for chaotic spectra, fails for integrable or symmetric spectra.
  • domain assumption Low purity: k^2 ||sigma||_2 << 1 and k^2 ||sigma_hat_{A|z}||_2 << 1
    Lemma 1's approximation of the normalized Scrooge ensemble by the unnormalized ensemble requires it; all three theorems inherit it (Appendix B). Breaks at low temperature or for small systems.
  • standard math Negative-moment bound on Haar overlaps (Lemma 5 of Ref. [61])
    E[<phi|D sigma|phi>^{-q}] <= exp(q^2/(2(m-q))) for q < m = floor(||sigma||_infty^{-1}); used in the relative-error proofs of Lemma 1 and the cTPQ theorem (Appendices B, C). Adopted from concurrent work; not re-proven here.
  • domain assumption Gaussian bulk spectral density of local Hamiltonians
    Eq. (22) in Sec. III C 1 approximates ||sigma_A||_p for Gibbs states; restricts Theorem 3's finite-temperature application to beta below a model-dependent threshold beta_c.
  • standard math Existence of efficient approximate unitary 2k-designs
    Theorem 3's practical implementability and the abstract's resource claim rely on logarithmic-depth unitary design constructions (Refs. [60, 63]).

pith-pipeline@v1.3.0-alltime-deepseek · 84315 in / 19375 out tokens · 184197 ms · 2026-08-03T13:05:57.468858+00:00 · methodology

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read the original abstract

Recent advances in quantum simulators allow direct experimental access to ensembles of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics governed by maximum-entropy principles, known as deep thermalization. At infinite temperature this universality is characterized by Haar-random ensembles. More generally, physical constraints such as finite temperature or conservation laws lead to Scrooge ensembles, which are maximally entropic distributions of pure states consistent with these constraints. Here we introduce Scrooge $k$-designs, which approximate Scrooge ensembles, and use this framework to sharpen the conditions under which Scrooge-like behavior emerges. We first show that global Scrooge designs arise from long-time chaotic unitary dynamics alone, without measurements. Second, we show that measuring a complementary subsystem of a scrambled global state drawn from a global Scrooge $2k$-design induces a local Scrooge $k$-design. Third, we show that a local Scrooge $k$-design arises from an arbitrary entangled state when the complementary system is measured in a scrambled basis induced by a unitary drawn from a Haar $2k$-design. These results show that the resources required to generate approximate Scrooge ensembles scale only with the desired degree of approximation, enabling efficient implementations. Complementing our analytical results, numerical simulations identify coherence, entanglement, non-stabilizerness, and information scrambling as essential ingredients for the emergence of Scrooge-like behavior. Together, our findings advance theoretical explanations for maximally entropic, information-stingy randomness in quantum many-body systems.

Figures

Figures reproduced from arXiv: 2601.00266 by John Preskill, Tobias Haug, Wai-Keong Mok, Wen Wei Ho.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: c that ∆(2) → 0 as NB → ∞. This indicates that deep thermalization is obstructed for θ = 0, but occurs for any nonzero rotation angle. This observation aligns with the discussion in Ref. [36]. Those authors argued that a combination of the coher￾ence of the initial state and the coherence of the mea￾surement basis determines whether the projected ensem￾ble on A is deeply thermalized or not. Now, random pha… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

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Reference graph

Works this paper leans on

16 extracted references · cited by 9 Pith papers

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    Norms of operators and random variables For an arbitrary operatorA, we denote its Schatten-pnorm by∥A∥ p, where ∥A∥p = [Tr(|A|p)]1/p, p∈[1,∞),(A1) and∥A∥ ∞ is the operator norm. Schatten-pnorms satisfy monotonicity: for 1≤p≤q≤ ∞, ∥A∥p ≥ ∥A∥q .(A2) For an arbitrary random variableX, we denote itsL p-norm by∥X∥ Lp , where ∥X∥ Lp = [E(|X| p)]1/p, p∈[1,∞),(A3...

  2. [2]

    To each permutationπ∈S k (where Sk is the symmetric group of orderk), we can associate a corresponding permutation operator ˆπ∈ L H⊗k that permutes between thekcopies

    Symmetric subspace, permutation operators, and moments of the Haar ensemble Given thek-fold Hilbert spaceH ⊗k, the symmetric subspace ofH ⊗k, denotedH (k) sym, is the vector space spanned by all states that are invariant under an arbitrary permutation of thekreplicas. To each permutationπ∈S k (where Sk is the symmetric group of orderk), we can associate a...

  3. [3]

    Here, we will briefly introduce the results relevant for this work, and establish the notation used throughout the manuscript

    W eingarten calculus for the unitary group The Weingarten calculus provides a very useful tool for evaluating polynomial functions of Haar random states and unitaries. Here, we will briefly introduce the results relevant for this work, and establish the notation used throughout the manuscript. A detailed treatment can be found in Refs. [71–73]. For permut...

  4. [4]

    Starting from a bipartite quantum state|Ψ⟩ AB, which we refer to as thegenerator state, we measure subsystemBin a complete orthonormal basis{|z⟩} DB z=1

    Projected ensemble Here, we give a concise review of the projected ensemble, and establish the notation used in this work. Starting from a bipartite quantum state|Ψ⟩ AB, which we refer to as thegenerator state, we measure subsystemBin a complete orthonormal basis{|z⟩} DB z=1. By default, we will choose the measurement basis to be the computational basis, ...

  5. [5]

    D⟨ϕ|σ|ϕ⟩ (√σ|ϕ⟩⟨ϕ| √σ)⊗k ⟨ϕ|σ|ϕ⟩k # =DE ϕ∼Haar(D)

    Scrooge ensemble The Scrooge ensemble is defined in Ref. [48] as the state ensembleEthat attains the minimum accessible information Iacc(E), among all ensembles that realize a given density matrixσ. Here, we briefly review the concept of accessible 24 information [67, 68], which can be understood by the following scenario. Bob samples a state drawn from t...

  6. [6]

    We can generalize this definition to Scrooge(σ)k-designs, which match the firstk moments of Scrooge(σ)

    Scroogek-designs Statek-designs provide a low-order approximation of the Haar ensemble, by matching only the firstkstatistical moments of the Haar ensemble. We can generalize this definition to Scrooge(σ)k-designs, which match the firstk moments of Scrooge(σ). The moments do not need to match exactly; for practical purposes it suffice for the moments to b...

  7. [7]

    Approximation of thekth moment To aid the analysis in the paper, we introduce the following technical lemma, which provides an approximation to thekth moment of an ensemble. Lemma 2.[kth moment approximation] Consider the ensemble of pure states E= pj =⟨ψ ′ j|ψ′ j⟩,|ψ j⟩= |ψ′ j⟩ √pj j=1,...,|E| (A31) withkth moment ρ(k) E = |E|X j=1 (|ψ′ j⟩⟨ψ′ j|)⊗k pk−1 ...

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    Thus, we have the lower bound X π∈Sk Tr σ⊗k ˆπ ≥1 + k 2 ∥σ∥2 2 .(B6) To prove the upper bound, let us examine the contribution from all permutations with a fixed Cayley distancel≥1

    All the remaining terms omitted are positive. Thus, we have the lower bound X π∈Sk Tr σ⊗k ˆπ ≥1 + k 2 ∥σ∥2 2 .(B6) To prove the upper bound, let us examine the contribution from all permutations with a fixed Cayley distancel≥1. There are at most k 2 l such permutations, and the contribution of each permutation can be upper bounded by Tr σ⊗k ˆπ ≤Tr σl+1 . ...

  9. [9]

    Random phase ensemble Here, we prove Theorem 1 in the main text. Since the temporal ensemble obtained by late-time Hamiltonian dynamics is described by the random phase ensembleE Random Phase, assuming the HamiltonianHsatisfies thekth no-resonance condition [31], it suffices to show that the random phase ensemble indeed forms a Scroogek-design. Theorem 4(...

  10. [10]

    Theorem 5(cTPQ states form Scroogek-designs).LetE cTPQ be the ensemble of canonical pure thermal quantum states defined in Eq

    Canonical thermal pure quantum (cTPQ) states Next, we show that the cTPQ ensemble forms a Scroogek-design in relative error, with respect to the density matrix σβ = e−βH Tr (e−βH ) (C7) for an arbitrary HamiltonianH. Theorem 5(cTPQ states form Scroogek-designs).LetE cTPQ be the ensemble of canonical pure thermal quantum states defined in Eq. (C2). Fork 2 ...

  11. [11]

    (B1), the projected ensemble is locally close to a generalized Scroogek-design (Lemma 7)

    First, we show that for the unnormalized generator state sampled from ˜Scrooge(σ) in Eq. (B1), the projected ensemble is locally close to a generalized Scroogek-design (Lemma 7)

  12. [12]

    Next, with the help of Lemma 7, we show that the projected ensemble generated by a global state drawn from an exact Scrooge 2k-design forms a generalized Scroogek-design (Theorem 6)

  13. [13]

    Using the results of Theorem 6, we arrive at Theorem 2 in the main text, which we reproduce here (Theorem 7)

    Finally, we relax the assumption that the generator state is drawn from an exact Scrooge(σ) 2k-design, and instead consider the case where the generator state is drawn from an approximate Scrooge(σ) 2k-design. Using the results of Theorem 6, we arrive at Theorem 2 in the main text, which we reproduce here (Theorem 7). We then specialize our results to the...

  14. [14]

    In this limit, it turns out that the error bound can be improved compared to Theorem 2, for technical reasons

    Projected ensemble generated by a state drawn from a Haar2k-design In the special case whereσ=I/Dis maximally mixed, the Scrooge ensemble reduces to the Haar ensemble, giving Corollary 1. In this limit, it turns out that the error bound can be improved compared to Theorem 2, for technical reasons. The proof follows similarly as Theorem 7 above. Theorem 8(...

  15. [15]

    (DA ∥σA∥2 2)kO k2 ∥σA∥4 4 ∥σA∥4 2 +ε ! +O Dk Akk+2 DB #1/2 =

    Late-time chaotic Hamiltonian dynamics Consider a generator state|Ψ⟩ AB obtained by evolving the initial state|Ψ 0⟩AB under an ergodic HamiltonianH, for a late-timet. Generically, it is reasonable to expect|Ψ⟩ AB to be modeled by the random phase ensemble ERandom Phase =    dDφ (2π)D , DX j=1 | ⟨Ej|Ψ0⟩ |eiφj |Ej⟩    ,(D75) with the diagonal density ...

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    As stated in the main text, we can define the effective dimension ofσ A via DA,eff = ∥σA∥2 ∥σA∥4 4 ,(E33) Ifσ A =I A/DA is the maximally mixed state, thenD A,eff =D A

    Application: Local Hamiltonian at finite temperatures Theorem 3 requires the conditionk≪ ∥σ A∥2 /∥σ A∥4, for the projected ensemble to converge to Scrooge(σ A). As stated in the main text, we can define the effective dimension ofσ A via DA,eff = ∥σA∥2 ∥σA∥4 4 ,(E33) Ifσ A =I A/DA is the maximally mixed state, thenD A,eff =D A. Then, the above condition re...