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

M. S\"uzen

arxiv: 2604.14224 · v4 · pith:UOQMA5EFnew · submitted 2026-04-14 · 🪐 quant-ph · cond-mat.stat-mech

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

M. S\"uzen This is my paper

Pith reviewed 2026-05-21 00:19 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords spread complexityquantum ergodicityentanglement scramblingRosenzweig-Porter ensemblefidelity decayintegrability to chaosmaximally entangled statesbootstrapped Hamiltonians

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The pith

A time-integrated spread complexity measure shows a monotonic inverse relationship with fidelity decay when diagnosing entanglement scrambling across ergodic regimes.

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

The paper proposes using a time-integrated measure of spread complexity together with fidelity to diagnose both quantum ergodicity and how entanglement scrambles in quantum systems. It employs bootstrapped realizations from an ensemble of Hamiltonians shown to be equivalent to perturbing unitary evolutions. With Rosenzweig-Porter ensembles this integrated spread complexity gives a detailed view of different ergodic regimes. It displays a monotonic inverse relationship with how quickly the integrated fidelity decays for maximally entangled states. This offers a complementary tool for understanding the transition from integrable to chaotic behavior in entanglement dynamics.

Core claim

Using Rosenzweig-Porter ensembles and bootstrapped Hamiltonian realizations equivalent to perturbed unitary evolutions, the integrated spread complexity provides a fine-grained resolution of ergodic regimes by exhibiting a monotonic inverse relationship with the decay of integrated fidelity for maximally entangled states, serving as a complementary diagnosis for quantum ergodicity in the scrambling of entanglement.

What carries the argument

Bootstrapped time-integrated spread complexity computed from realizations of Hamiltonians in the Rosenzweig-Porter ensemble, which quantifies spreading of quantum information to resolve levels of ergodicity in entanglement scrambling.

If this is right

The measure distinguishes multiple levels of ergodicity beyond a simple integrable-versus-chaotic split.
It directly connects growth of spread complexity to loss of fidelity in maximally entangled states.
The equivalence to perturbing unitaries enables efficient numerical exploration of scrambling dynamics.
The diagnostic applies to the transition from integrability to chaos in quantum many-body systems.

Where Pith is reading between the lines

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

If the inverse relationship holds generally, the measure could help identify regimes where scrambling can be tuned for quantum information tasks.
Similar time-integrated probes might be applied to other random-matrix ensembles to test universality of the ergodicity resolution.
Implementation in quantum simulators could provide experimental checks by tracking complexity growth against fidelity loss over time.

Load-bearing premise

Bootstrapped realizations drawn from an ensemble of Hamiltonians are mathematically equivalent to perturbing unitary evolutions.

What would settle it

A simulation in which the integrated spread complexity fails to show a monotonic inverse relationship with integrated fidelity decay for maximally entangled states under Rosenzweig-Porter ensemble Hamiltonians would falsify the central claim.

Figures

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

**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: FIG. 2. Integrated spread complexity of maximally en [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**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 new general analytical relationship between spread complexity and fidelity of quantum dynamics is established with time-integrated quantities under operator perturbation. This approach diagnoses the degree of quantum ergodicity and the behavior of scrambling of entanglement simultaneously. The equivalence between the perturbation scheme and bootstrapped Hamiltonian realizations is shown to be a rigorous testbed. Using bootstrapped Rosenzweig-Porter ensembles for unitary dynamical perturbation, we demonstrated that the integrated spread complexity provides a fine-grained resolution across different ergodic regimes, exhibiting a monotonic inverse relationship with the decay of integrated fidelity for maximally entangled states.

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.

Desk Editor's Note private letter to a colleague

The paper combines time-integrated spread complexity with bootstrapped Rosenzweig-Porter ensembles to track entanglement scrambling via an inverse link to fidelity decay, but the claimed equivalence to perturbed unitaries needs explicit support.

read the letter

The paper's core move is to define a time-integrated spread complexity and apply it to bootstrapped samples drawn from Rosenzweig-Porter Hamiltonians. It reports that this quantity resolves different ergodic regimes and shows a monotonic inverse relationship with the decay of integrated fidelity for maximally entangled states. That combination is the main new piece relative to existing spread-complexity work. The numerical pipeline gives a practical way to interpolate between localized and ergodic behavior while keeping track of entanglement dynamics, which is a reasonable extension for people already using random-matrix ensembles in quantum many-body studies. The results appear to rest on concrete sampling rather than analytic derivation, so the figures presumably demonstrate the claimed monotonicity across the tunable parameter. One clear soft spot is the asserted mathematical equivalence between bootstrapped Hamiltonian realizations and perturbed unitary evolutions. The abstract treats this as the foundation for generating the statistics, yet without a derivation showing that the ensemble-averaged evolution operators match the perturbed case to the relevant order, it is not obvious whether the inverse relationship is robust or partly an artifact of the sampling procedure. If the full text supplies a clear justification or approximation argument, that concern shrinks; otherwise it remains the load-bearing step that referees would examine first. The work is aimed at researchers in quantum information and condensed-matter many-body physics who run numerics on chaos and scrambling diagnostics. It is the sort of concrete numerical proposal that can be checked directly once the code and ensemble details are available. I would send it to peer review. The idea is focused enough that referees can evaluate the numerics and the equivalence claim on their own terms.

Referee Report

1 major / 1 minor

Summary. The paper proposes time-integrated measures of spread complexity and fidelity to diagnose quantum ergodicity and entanglement scrambling simultaneously. It employs bootstrapped realizations from Rosenzweig-Porter ensembles of Hamiltonians, asserting that this approach is mathematically equivalent to perturbing unitary evolutions, and reports a monotonic inverse relationship between integrated spread complexity and the decay of integrated fidelity for maximally entangled states across ergodic regimes.

Significance. If the asserted equivalence is rigorously established and the numerical results hold, the work could supply a useful complementary diagnostic for distinguishing ergodic regimes in quantum many-body systems, with the RP ensemble providing interpolation between localized and chaotic behaviors. The focus on maximally entangled states and time-integrated quantities offers a potentially practical extension of existing complexity and fidelity tools.

major comments (1)

[Abstract] Abstract: the claim that 'bootstrapped realizations drawn from an ensemble of Hamiltonians are mathematically equivalent to perturbing unitary evolutions' is asserted without derivation, explicit proof, or order-by-order matching of observables. This equivalence is presented as the foundation for generating the statistics of time-integrated spread complexity and fidelity; its absence means the reported monotonic inverse relationship with fidelity decay could be an artifact of the ensemble sampling procedure rather than a robust feature of the dynamics.

minor comments (1)

The abstract would be strengthened by specifying the system sizes, number of bootstrapped realizations, and the precise definition of the time-integrated quantities to allow immediate assessment of the numerical claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive criticism. We address the single major comment below and will incorporate the requested clarification into the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'bootstrapped realizations drawn from an ensemble of Hamiltonians are mathematically equivalent to perturbing unitary evolutions' is asserted without derivation, explicit proof, or order-by-order matching of observables. This equivalence is presented as the foundation for generating the statistics of time-integrated spread complexity and fidelity; its absence means the reported monotonic inverse relationship with fidelity decay could be an artifact of the ensemble sampling procedure rather than a robust feature of the dynamics.

Authors: We agree that the original manuscript asserted the equivalence without a self-contained derivation. In the revised version we will add an explicit section deriving the correspondence. Briefly, a bootstrapped realization H = H_0 + V, where V is drawn from the Rosenzweig-Porter ensemble, generates a unitary U(t) = exp(-iHt) whose statistics match those obtained by perturbing a reference unitary evolution U_0(t) with a random operator drawn from the same ensemble at each time step, to first order in the perturbation strength. We will show this order-by-order by expanding the Dyson series for both constructions and verifying that the ensemble-averaged moments of the time-evolution operator coincide. With this derivation in place, the observed monotonic inverse relation between integrated spread complexity and integrated fidelity decay follows directly from the shared statistics rather than from sampling artifacts; we will also add a short numerical cross-check confirming that the two procedures produce indistinguishable distributions for the observables of interest. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent numerical evaluation

full rationale

The paper defines time-integrated spread complexity and fidelity as diagnostic measures, then applies them via bootstrapped sampling from Rosenzweig-Porter Hamiltonian ensembles to observe a monotonic inverse relationship across ergodic regimes. The stated equivalence between bootstrapped realizations and perturbed unitaries is presented as a supporting foundation for the numerics rather than a definitional loop that forces the reported relationship. No steps in the provided claims reduce the central numerical finding to its inputs by construction, self-citation, or renaming; the results are framed as empirical outcomes from ensemble computations that remain falsifiable against the chosen observables.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that bootstrapped Hamiltonian ensembles are equivalent to perturbed unitaries and on the modeling choice of Rosenzweig-Porter ensembles as faithful representatives of ergodic regimes; no free parameters or invented entities are declared in the abstract.

axioms (1)

domain assumption Bootstrapped realizations from an ensemble of Hamiltonians are mathematically equivalent to perturbing unitary evolutions.
Explicitly invoked in the abstract as the justification for the numerical protocol.

pith-pipeline@v0.9.0 · 5630 in / 1230 out tokens · 121058 ms · 2026-05-21T00:19:20.971075+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define a time-integrated quantity C_A = ∫_0^t C(τ) dτ ... bootstrapped realizations from an ensemble of Hamiltonians ... Rosenzweig-Porter ensembles ... monotonic inverse relationship with the decay of integrated fidelity
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Krylov subspace ... Lanczos algorithm ... spread complexity C(t) = (1/n) Σ ⟨K_i | ψ(t)⟩
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

H_rp = H_0 + N^{-γ/2} H_GOE ... γ controls ... chaotic (0≤γ<1), fractal, localized (γ≥2)

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