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Entanglement governs early-time growth of randomness in projected ensembles

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Thuwaragesh Jayachandran, Wai-Keong Mok

Quantum Physics · Submitted 23 Aug 2026 · 70 references

Pith's verdict: A clean, genuinely new short-time result for projected ensembles; the ensemble-average O(t^4) concern raised in triage is resolved by the paper's own basis-choice argument.

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arXiv ID2608.22158 (version 1)
Pith IDULU3E4FQ, Pith's permanent record of this paper and its review
CitationThuwaragesh Jayachandran, Wai-Keong Mok. "Entanglement governs early-time growth of randomness in projected ensembles." arXiv:2608.22158 (2026). https://pith.science/paper/ULU3E4FQ
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The paper

The paper claims

For initially unentangled states, up to cubic order in time every moment of a projected ensemble's randomness is fixed by the second Rényi entropy of the unmeasured subsystem: early randomness growth is entanglement growth.

The reading

The paper asks what drives the initial growth of quantum randomness in projected ensembles — the distribution of pure states on a subsystem obtained by measuring its complement. Its central claim is that, for initially unentangled states whose bath part has non-zero overlap with every measurement outcome, all statistical moments of this distribution are controlled up to third order in time by a single quantity: the purity of the unmeasured subsystem, i.e. the second Rényi entanglement entropy. Early randomness growth is therefore nothing but entanglement growth, occurring on the entanglement timescale and independent of the measurement basis, whose influence is delayed to quartic order. For unitarily invariant Hamiltonian ensembles the same early-time growth is tied to the $4k$-point spectral form factor of the global dynamics, giving a direct quantitative bridge between local and global randomness.

What carries the argument

The load-bearing object is the short-time expansion of each normalized projected state around the initial subsystem state, $|\psi_z(t)\rangle = f_z(t)|\phi_A\rangle + t|\eta_z\rangle + t^2|\eta'_z\rangle + O(t^3)$, with $\langle\phi_A|\eta_z\rangle = 0$. It makes the pairwise overlap deficit $\Delta_{z_1,z_2}(t) \equiv 1 - |\langle\psi_{z_1}|\psi_{z_2}\rangle|^2$ equal to $\|\eta_{z_1}-\eta_{z_2}\|^2 t^2 + O(t^3)$, so the $l$th moment $M_l \equiv \mathbb{E}_P[\Delta^l]$ scales as $t^{2l}$. Writing the frame potential as a binomial series $F^{(k)} = \sum_{l=0}^{k} \binom{k}{l}(-1)^l M_l$ shows that below fourth order only $M_0$ and $M_1$ contribute, and $M_1 = 1 - F^{(1)}(t)$ is exactly the subsystem purity, yielding $F^{(k)} = (F^{(1)})^k$ up to $O(t^4)$. For the random-matrix results, the machinery is the factorization of a unitarily invariant ensemble into Haar-random eigenvectors and an independent eigenvalue distribution, evaluated with Weingarten calculus, together with the expansion $|Z(t)|^2 = D^2(1 - \Sigma_H^2 t^2/D^2 + O(t^4))$ of the analytically continued partition function feeding the spectral form factor.

Extended reading notes

Core claim

The paper establishes a quantitative relation between the randomness of a projected ensemble and bipartite entanglement. Its main result is that the $k$th frame potential $F^{(k)}(t)$, the standard measure of how far the ensemble's $k$th moment sits from Haar-random, satisfies $F^{(k)}(t) = 1 - k(1-F^{(1)}(t)) + O(t^4) = \bigl(F^{(1)}(t)\bigr)^k + O(t^4)$ for every moment order $k$ and any Hamiltonian, provided the initial bath state has full support in the measurement basis. Since $F^{(1)}(t)$ is the purity of subsystem $A$, all moments share one early-time driver: the entanglement generated between $A$ and the measured bath $B$. Substituting the known short-time expansion of purity yields $F^{(k)}(t) = 1 - 2k\,(t/T_{\mathrm{ent}})^2 + O(t^3)$, with $T_{\mathrm{ent}} = \| \Pi_0 H_I |\Psi_0\rangle \|^{-1}$ the entanglement timescale, so the onset of randomness is set by the same timescale as entanglement itself. For unitarily invariant ensembles, the ensemble-averaged frame potential is shown to match $R_{4k}(t)/D^{4k}$ up to corrections suppressed in large dimension — the $4k$-point spectral form factor, equivalently the $2k$th frame potential of the global unitary ensemble, a '$2k\to k$' correspondence between local and global randomness.

Load-bearing premise

For the headline formula, the bath half of the initial state must have non-zero overlap with every measurement outcome; for the quartic-order accuracy of the ensemble-averaged results, the averaged frame potential must be even in time, a property the proof establishes only for initial states that are real in the measurement basis.

Editorial extensions

If this is right

  • Measurement-basis dependence, which decides whether late-time deep thermalization forms state designs, cannot affect the projected ensemble before fourth order in time.
  • The onset timescale of local randomness is the entanglement timescale $T_{\mathrm{ent}}$, independent of subsystem sizes and of the measurement basis.
  • All higher moments inherit their leading early-time behaviour from the purity: $F^{(k)}(t) = (F^{(1)}(t))^k + O(t^4)$, so measuring subsystem purity determines how random the ensemble is at early times.
  • For unitarily invariant ensembles, early-time randomness growth depends only on the spectral variance and so does not by itself diagnose quantum chaos.
  • The '$2k\to k$' correspondence ties local projected-ensemble randomness at order $k$ to global unitary randomness at order $2k$ through the $4k$-point spectral form factor.

Reading between the lines

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

  • Computing the quartic term would expose the first basis-dependent coefficient; a natural test is to measure $F^{(2)}(t) - (F^{(1)}(t))^2$ at fixed small $t$ for two measurement bases related by a rotation on the bath and compare the $t^4$ coefficients, which the paper leaves unquantified.
  • The full-support condition is the operative constraint for experiments: states that formally satisfy it but have exponentially small overlap with some outcomes show a proportionally shorter window of validity, as the paper's own interpolation numerics indicate, so near-single-basis baths should deviate early.
  • The connection between $F^{(k)}$ and the spectral form factor suggests a possible route to estimating $4k$-point spectral correlations of a chaotic system from local measurements on a small subsystem at early times, without accessing the global state.
  • If the hierarchy persists to higher orders, the first basis-dependent terms may carry signatures that distinguish chaotic from integrable dynamics, since at quadratic order the growth depends only on the spectral variance and cannot.
Original abstract

Deep thermalization concerns the emergence of universal pure-state statistics in projected ensembles at late times, yet the mechanism governing the initial growth of randomness remains unclear. Here, we study the short-time dynamics of projected ensembles generated from initially unentangled states, quantifying their randomness using frame potentials. For arbitrary Hamiltonians, provided the initial bath state has full support in the measurement basis, we show that the frame potentials to cubic order in time are determined entirely by the subsystem purity, and hence by the bipartite entanglement generated between the unmeasured subsystem and its complement. The entanglement timescale therefore sets the initial timescale for the growth of local randomness, independently of the bath measurement basis. For unitarily invariant Hamiltonian ensembles, we further relate the projected-ensemble frame potential at order $k$ to the $4k$-point spectral form factor, or equivalently to the $2k$th frame potential of the global unitary dynamics, establishing a direct connection between local and global randomness. Our results identify entanglement as the mechanism governing the onset of randomness in projected ensembles and clarify how local randomness emerges from global quantum dynamics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,140 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.54 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Deep thermalization concerns the emergence of universal pure-state statistics in projected ensembles at late times, yet the mechanism governing the initial growth of randomness remains unclear. Here, we study the short-time dynamics of projected ensembles generated from initially unentangled states, quantifying their randomness using frame potentials. For arbitrary Hamiltonians, provided the initial bath state has full support in the measurement basis, we show that the frame potentials to cubic order in time are determined entirely by the subsystem purity, and hence by the bipartite entanglement generated between the unmeasured subsystem and its complement. The entanglement timescale therefore sets the initial timescale for the growth of local randomness, independently of the bath measurement basis. For unitarily invariant Hamiltonian ensembles, we further relate the projected-ensemble frame potential at order $k$ to the $4k$-point spectral form factor, or equivalently to the $2k$th frame potential of the global unitary dynamics, establishing a direct connection between local and global randomness. Our results identify entanglement as the mechanism governing the onset of randomness in projected ensembles and clarify how local randomness emerges from global quantum dynamics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Deep thermalization concerns the emergence of universal pure-state statistics in projected ensembles at late times, yet the mechanism governing the initial growth of randomness remains unclear. gap 0.87 0.04 0.04 asserted 0.87
s1 Here, we study the short-time dynamics of projected ensembles generated from initially unentangled states, quantifying their randomness using frame potentials. contribution 0.81 0.95 0.03 asserted 0.90
s2 For arbitrary Hamiltonians, provided the initial bath state has full support in the measurement basis, we show that the frame potentials to cubic order in time are determined entirely by the subsystem purity, and hence by the bipartite entanglement generated between the unmeasured subsystem and its complement. result 1.00 0.94 0.51 asserted 1.00
s3 The entanglement timescale therefore sets the initial timescale for the growth of local randomness, independently of the bath measurement basis. interpretation 0.82 0.88 0.47 asserted 1.00
s4 For unitarily invariant Hamiltonian ensembles, we further relate the projected-ensemble frame potential at order $k$ to the $4k$-point spectral form factor, or equivalently to the $2k$th frame potential of the global unitary dynamics, establishing a direct connection between local and global randomness. result 0.98 0.91 0.09 asserted 1.00
s5 Our results identify entanglement as the mechanism governing the onset of randomness in projected ensembles and clarify how local randomness emerges from global quantum dynamics. interpretation 0.96 0.94 0.95 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Numerical
Figure 2. shows that the finite-time discrepancy grows with γ, consistent with the shrinking pmin and the re￾sulting reduction of the useful short-time window. The full-support assumption holds for every finite γ and fails only in the limit γ → ∞. IV. UNITARILY INVARIANT ENSEMBLES AND SPECTRAL VARIANCE The preceding results hold for arbitrary Hamiltonians. We now specialize to a unitarily invariant ensemble EU, whose distribution is invariant under H 7→ UHU† for any unitary U. We assume that E H∼EU ∥H∥ 4 < ∞, for the results below to hold. A prominent example is the Gaus￾sian Unitary Ensem
Figure 3. compares E H∼EGUE F (k) with D−4kR4k for k = 1, 2, 3. We fix DB = 45 and vary DA ∈ {2, 10, 45}, find￾ing better agreement as DA increases. This agreement requires DA, DB ≫ 1, but not DB ≫ DA; the latter con￾dition is needed only for the formation of approximate projected k-designs. As discussed in Sec. III, the analytical results assume full support in the measurement basis [Eq. (20)]
Figure 4. FIG. 4. Numerical validation of the theory for initial states
Figure 5. FIG. 5. Early-time approximations to
Figure 6. FIG. 6. Early-time approximations from Fig.
Figure 7. FIG. 7. Frame potential
Figure 8. FIG. 8. Approximation time

Figures reproduced from arXiv:2608.22158 by the authors.

References
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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 2 sentences without a confident label

Compares with 1 paper
Cites as background 9 papers
  • A general framework describing the nonequilibrium dynamics of projected ensembles was recently developed by Anza and Hahn [ref] .

    Non-equilibrium theory of projected ensembles · Introduction · 0.94

  • Previous work showed that projected ensembles generated by GUE dynamics form $k$-designs at certain $1$ and late times [ref] ; see Ghosh et al. [ref] for details, including a generalization to $$.

    Deep thermalization with and without quantum chaos · Unitarily invariant ensembles and spectral variance · 0.81

  • Previous work has largely focused on the late-time equilibrium behavior of projected ensembles [ref] , as well as how projected ensembles dynamically approach $k$-designs in various models of quantum dynamics [ref] .

    Emergence of the Scrooge Ensemble in the Sachdev-Ye-Kitaev Model · Introduction · 0.96

Discussion

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