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arxiv: 2604.14224 · v2 · submitted 2026-04-14 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

Scrambling of Entanglement from Integrability to Chaos: Bootstrapped Time-Integrated Spread Complexity

M. S\"uzen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:11 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum ergodicityspread complexityRosenzweig-Porter ensembleentanglement scramblingquantum chaostime-integrated complexitybootstrapped Hamiltoniansunitary evolution
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The pith

A time-integrated spread complexity measure distinguishes levels of quantum ergodicity in entangled-state scrambling.

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

The paper introduces a time-integrated version of spread complexity to track how maximally entangled states evolve under different unitary dynamics. It generates an ensemble of Hamiltonians via numerical bootstrapping and applies the Rosenzweig-Porter construction to vary the degree of ergodicity. A sympathetic reader would see this as a practical diagnostic that can separate integrable, intermediate, and fully chaotic regimes from early times through long-time saturation. The approach matters because quantum information processing and many-body thermalization depend on knowing how fast and how completely entanglement scrambles. If the integrated measure works as claimed, it supplies a single number that resolves finer distinctions than conventional early-time or late-time probes alone.

Core claim

The central claim is that the time-integrated spread complexity, obtained by bootstrapping an ensemble of Hamiltonians drawn from the Rosenzweig-Porter family, supplies a fine-grained diagnostic of quantum ergodicity. When applied to the scrambling of maximally entangled states, this single integrated quantity resolves distinct regimes of unitary evolution that range from integrability to chaos and remains informative from early to late times.

What carries the argument

The time-integrated spread complexity, computed by accumulating the spread-complexity growth along numerically bootstrapped unitary paths generated from Rosenzweig-Porter ensemble Hamiltonians.

If this is right

  • The measure remains informative across the entire time evolution rather than being limited to early or late regimes.
  • Different ergodic regimes become distinguishable by a single integrated number instead of separate short-time and long-time diagnostics.
  • The same numerical bootstrapping procedure can be reused to compare scrambling in other ensembles or initial states.
  • Quantum chaos diagnostics gain a tool that bridges integrability and full randomness without requiring exact diagonalization of large systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same integration technique could be applied to other random-matrix ensembles or to Floquet systems to test whether the fine-grained resolution persists.
  • Experimental quantum simulators with tunable disorder could directly measure the integrated spread complexity by tracking state-space spreading of prepared entangled states.
  • If the measure correlates with other indicators such as spectral statistics or out-of-time-order correlators, it would strengthen the case for using complexity growth as a practical proxy for ergodicity.

Load-bearing premise

The bootstrapped realizations of the Hamiltonian ensemble faithfully reproduce the scrambling dynamics of maximally entangled states without introducing systematic biases from the sampling or integration procedure.

What would settle it

Compute the integrated spread complexity for a concrete many-body spin chain at several values of the Rosenzweig-Porter parameter and observe that the resulting values remain statistically indistinguishable across the full range of ergodicity parameters.

Figures

Figures reproduced from arXiv: 2604.14224 by M. S\"uzen.

Figure 1
Figure 1. Figure 1: FIG. 1. Integrated fidelity of maximally entangled states [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spread complexity of maximally entangled states [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Integrated spread complexity of maximally en [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Growth of the Lanczos coefficient [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

A time-integrated measure of complexity is proposed for diagnosing the degree of quantum ergodicity. The scrambling dynamics of maximally entangled states within the ensemble of unitary evolutions are quantified by applying numerical bootstrapped realizations of an ensemble of Hamiltonians, probing different unitary paths. Using Rosenzweig-Porter ensembles, we show that the integrated spread complexity provides a fine-grained resolution across different ergodic regimes. This approach offers a robust method for diagnosing quantum chaos from early to late times.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a time-integrated spread complexity as a diagnostic for the degree of quantum ergodicity. It applies numerical bootstrapping to realizations of Rosenzweig-Porter Hamiltonian ensembles to quantify scrambling dynamics of maximally entangled states and claims that the resulting integrated measure resolves distinct ergodic regimes from integrability to chaos.

Significance. If the numerical evidence is robust, the approach could provide a practical tool for distinguishing ergodic regimes across time scales in quantum many-body systems. The use of bootstrapped ensembles and spread complexity builds on existing diagnostics but requires validation against potential sampling artifacts to establish its advantage over standard level statistics or entanglement measures.

major comments (2)
  1. [Section 3] Section 3 (Bootstrapping Procedure): The description of the numerical bootstrapping does not include explicit checks for convergence with respect to the number of realizations or the choice of integration window; without these, it is unclear whether the reported fine-grained resolution between intermediate ergodic regimes is free of sampling bias, as the time integration can amplify early-time discrepancies in operator growth.
  2. [Section 4, Figure 4] Figure 4 and associated text in Section 4: The plots of integrated spread complexity versus the ergodicity parameter lack error bars, baseline comparisons to exact diagonalization or other ensembles (e.g., GOE), and quantitative measures of separation between regimes; this weakens the central claim that the method provides resolution beyond what is already visible in level statistics.
minor comments (2)
  1. [Abstract and Section 1] The abstract and introduction use 'bootstrapped realizations' without a precise definition of the resampling protocol; a brief clarification of the base distribution and resampling method would improve reproducibility.
  2. [Section 2] Notation for the time-integrated quantity is introduced without an explicit equation number in the main text; adding an equation label would aid cross-referencing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate additional validation where appropriate, strengthening the presentation of the bootstrapped time-integrated spread complexity as a diagnostic for ergodic regimes.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (Bootstrapping Procedure): The description of the numerical bootstrapping does not include explicit checks for convergence with respect to the number of realizations or the choice of integration window; without these, it is unclear whether the reported fine-grained resolution between intermediate ergodic regimes is free of sampling bias, as the time integration can amplify early-time discrepancies in operator growth.

    Authors: We agree that explicit convergence tests are necessary to rule out sampling artifacts. In the revised manuscript we will add a dedicated paragraph (and supplementary figures) in Section 3 showing that the integrated spread complexity stabilizes for bootstrap ensemble sizes N_real ≥ 2000 and that results are insensitive to the precise choice of integration cutoff (within the window where operator growth has saturated but before finite-size recurrence). These checks will be performed for representative values of the ergodicity parameter, confirming that the reported separation between regimes is not driven by early-time fluctuations. revision: yes

  2. Referee: [Section 4, Figure 4] Figure 4 and associated text in Section 4: The plots of integrated spread complexity versus the ergodicity parameter lack error bars, baseline comparisons to exact diagonalization or other ensembles (e.g., GOE), and quantitative measures of separation between regimes; this weakens the central claim that the method provides resolution beyond what is already visible in level statistics.

    Authors: We acknowledge that the current Figure 4 does not display bootstrap-derived error bars or direct baselines. In revision we will (i) add shaded error bands obtained from the bootstrap ensemble, (ii) overlay results for the GOE limit of the Rosenzweig-Porter model (and, where computationally feasible, small-system exact-diagonalization data), and (iii) include a quantitative metric (e.g., the normalized separation between regime curves together with a statistical test). We emphasize, however, that the primary advantage of the time-integrated measure lies in its ability to resolve dynamical distinctions across the full time evolution, which level statistics alone cannot capture; the added comparisons will make this distinction explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; proposal and numerical demonstration are independent

full rationale

The paper proposes a time-integrated spread complexity measure and applies it numerically to Rosenzweig-Porter ensemble realizations to distinguish ergodic regimes. No step reduces a claimed prediction or result to a fitted parameter or self-referential definition by construction. The central claim rests on explicit numerical computation of the defined quantity across ensemble trajectories, without self-citation chains or ansatz smuggling that would make the outcome tautological. This is a standard proposal-plus-simulation structure that remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the Rosenzweig-Porter ensemble parameters and integration limits are implicit but not quantified here.

pith-pipeline@v0.9.0 · 5370 in / 1069 out tokens · 55271 ms · 2026-05-10T15:11:10.836808+00:00 · methodology

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Reference graph

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