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arxiv: 2507.11674 · v2 · submitted 2025-07-15 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· cond-mat.str-el· hep-th· quant-ph

ν-QSSEP: A toy model for entanglement spreading in stochastic diffusive quantum systems

Vincenzo Alba This is my paper

Pith reviewed 2026-05-19 04:08 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elhep-thquant-ph
keywords entanglement dynamicsstochastic quantum systemsquasiparticle picturediffusive entanglement growthfree fermion chainnoise averagingQSSEP model
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The pith

In the ν-QSSEP model with homogeneous noise, entanglement dynamics follows a stochastic quasiparticle picture with random walk trajectories.

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

The paper examines entanglement spreading in a free-fermion chain where hopping amplitudes are stochastic and spatially modulated with ν-site invariance. It focuses on the case ν=1, corresponding to homogeneous noise, to show that the dynamics can be described using a stochastic extension of the quasiparticle picture. Entanglement is carried by pairs of quasiparticles that have the same content as in clean systems but follow random walk paths due to the noise. This leads to diffusive growth of entanglement instead of the usual linear growth. The model preserves Gaussianity for each noise realization, enabling calculation of averaged quantities, and the steady states show invariance properties under random matrix multiplications.

Core claim

For ν=1 the entanglement dynamics is describable by a stochastic generalization of the quasiparticle picture: entanglement is propagated by pairs of quasiparticles whose trajectories are random walks, giving rise to diffusive entanglement growth.

What carries the argument

Stochastic quasiparticle pairs with random walk trajectories that carry fixed entanglement content while diffusing through the chain.

If this is right

  • Entanglement entropy grows diffusively rather than ballistically with time.
  • Noise-averaged entanglement quantities follow from the statistics of the random walks.
  • Steady-state correlators obey topological relationships reflecting Haar invariance under structured SU(ν) matrices.
  • The description extends to ν=2 with more complex structured random matrices.

Where Pith is reading between the lines

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

  • This suggests that in real diffusive quantum systems with noise, entanglement might spread according to random quasiparticle motion rather than ballistic propagation.
  • The model could be used to test how different noise correlations affect entanglement scaling in quantum many-body systems.

Load-bearing premise

The stochastic hopping amplitudes maintain ν-site translation invariance but are fully random in time to ensure the required invariance properties for the correlators.

What would settle it

Numerical evidence that the entanglement growth is not diffusive or does not match the predicted quasiparticle pair statistics in simulations of the ν=1 model.

Figures

Figures reproduced from arXiv: 2507.11674 by Vincenzo Alba.

Figure 1
Figure 1. Figure 1: FIG. 1. Definition of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Entanglement dynamics in the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Moments [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as in Fig. 3 for the quench from the dimer [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The same as in Fig. 5 for the quench from the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Entanglement dynamics after the quench from the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
read the original abstract

We investigate out-of-equilibrium entanglement dynamics in a generalization of the so-called $QSSEP$ model, which is a free-fermion chain with stochastic in space and time hopping amplitudes. In our setup, the noisy amplitudes are spatially-modulated satisfying a $\nu$-site translation invariance but retaining their randomness in time. For each noise realization, the dynamics preserves Gaussianity, which allows to obtain noise-averaged entanglement-related quantities. The statistics of the steady-state correlators satisfy nontrivial relationships that are of topological nature. They reflect the Haar invariance under multiplication with structured momentum-dependent random $SU(\nu)$ matrices. We discuss in detail the case with $\nu=1$ and $\nu=2$. For $\nu=1$, i.e., spatially homogeneous noise we show that the entanglement dynamics is describable by a stochastic generalization of the quasiparticle picture. Precisely, entanglement is propagated by pairs of quasiparticles. The entanglement content of the pairs is the same as for the deterministic chain. However, the trajectories of the quasiparticles are random walks, giving rise to diffusive entanglement growth.

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 / 3 minor

Summary. The manuscript introduces the ν-QSSEP model, a free-fermion chain with stochastic hopping amplitudes obeying ν-site translation invariance in space while remaining fully random in time. For each noise realization the dynamics remains Gaussian, permitting exact noise-averaged entanglement quantities. Steady-state correlators obey topological relations traceable to Haar invariance under structured, momentum-dependent SU(ν) matrices. The case ν=1 is treated in detail: entanglement is carried by quasiparticle pairs whose trajectories are random walks, producing diffusive rather than ballistic entanglement growth.

Significance. If the central claims hold, the work supplies a clean, analytically tractable toy model that interpolates between deterministic quasiparticle pictures and diffusive entanglement dynamics in noisy quantum systems. The explicit stochastic generalization for ν=1 and the topological structure of the correlators for general ν are potentially useful benchmarks for more realistic open-system simulations.

major comments (2)
  1. [§3.1] §3.1 (model definition): the ν-site translation invariance is imposed on the noise while time randomness is retained; it is not shown explicitly how this structured noise still yields the claimed full Haar invariance of the steady-state correlators under the structured SU(ν) matrices, which is load-bearing for the topological relations.
  2. [§4.2] §4.2, paragraph after Eq. (22): the stochastic quasiparticle picture asserts that pair trajectories are random walks whose diffusion constant controls the entanglement growth; no explicit computation of the diffusion constant from the second moment of the noise or from the two-point correlator is provided, leaving the diffusive scaling claim without a quantitative anchor.
minor comments (3)
  1. Notation for the structured SU(ν) matrices is introduced only in the text; a compact definition or appendix would improve readability.
  2. Figure 3 caption does not state the system size or number of noise realizations used for the numerical curves.
  3. The abstract states 'topological nature' of the relations; the main text should clarify whether these are homotopy invariants or merely algebraic identities preserved by the Haar measure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3.1] §3.1 (model definition): the ν-site translation invariance is imposed on the noise while time randomness is retained; it is not shown explicitly how this structured noise still yields the claimed full Haar invariance of the steady-state correlators under the structured SU(ν) matrices, which is load-bearing for the topological relations.

    Authors: We agree that the connection between the imposed ν-site translation invariance of the noise and the resulting Haar invariance under momentum-dependent SU(ν) matrices deserves a more explicit derivation. The manuscript states the invariance but does not walk through the steps showing how time randomness combined with the spatial structure produces the full Haar measure on the relevant subgroup. In the revised manuscript we will expand the discussion in §3.1 (and, if needed, add a short appendix) to derive this explicitly, starting from the noise correlator and showing the invariance of the steady-state two-point functions under the structured unitary transformations. revision: yes

  2. Referee: [§4.2] §4.2, paragraph after Eq. (22): the stochastic quasiparticle picture asserts that pair trajectories are random walks whose diffusion constant controls the entanglement growth; no explicit computation of the diffusion constant from the second moment of the noise or from the two-point correlator is provided, leaving the diffusive scaling claim without a quantitative anchor.

    Authors: The referee correctly notes the absence of an explicit calculation. The diffusive scaling is asserted on the basis of the random-walk trajectories, but the diffusion constant itself is not extracted from the noise second moment or from the two-point correlator. In the revised version we will insert this computation immediately after Eq. (22), relating the diffusion constant D to the variance of the stochastic hopping amplitudes (specifically, D proportional to the second moment of the noise distribution) and confirming that the entanglement growth rate is set by this D. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper introduces the ν-QSSEP model with ν-site translation-invariant noise as an explicit assumption, then uses preservation of Gaussianity to compute noise-averaged quantities. Steady-state correlator statistics are attributed to standard Haar invariance under structured SU(ν) matrices, an external mathematical property rather than a result fitted or defined from the paper's own outputs. For ν=1 the stochastic quasiparticle picture is obtained by replacing deterministic trajectories with random walks while keeping the entanglement content per pair unchanged; this follows directly from the model definition without reducing to a self-citation chain or a parameter fit renamed as prediction. No load-bearing step equates an output to its input by construction, and the central claims retain independent content from the stochastic dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the preservation of Gaussianity under the stochastic dynamics and on the Haar invariance induced by the structured noise; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption For each noise realization the dynamics preserves Gaussianity.
    This property is stated to allow exact computation of noise-averaged entanglement quantities.
  • domain assumption The steady-state correlators obey nontrivial topological relationships reflecting Haar invariance under multiplication with structured momentum-dependent random SU(ν) matrices.
    Invoked to obtain the statistics of steady-state correlators.

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Forward citations

Cited by 2 Pith papers

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  1. Domain-wall melting in all-to-all QSSEP from random-matrix theory

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    In the thermodynamic limit the quantum and classical full-counting statistics of charge coincide exactly with no finite-time corrections, while the averaged von Neumann entanglement entropy admits a fully explicit exp...

  2. Dynamics of entanglement fluctuations and quantum Mpemba effect in the $\nu=1$ QSSEP model

    cond-mat.stat-mech 2025-10 unverdicted novelty 6.0

    Incorporating noise-induced quasiparticle correlations in the ν=1 QSSEP model yields the full-time distribution of entanglement entropy and shows the quantum Mpemba effect is extremely fine-tuned and hard to observe.

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