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arxiv: 2602.19280 · v2 · submitted 2026-02-22 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

Entanglement dynamics of many-body quantum states: sensitivity to system conditions and a hidden universality

Devanshu Shekhar , Pragya Shukla This is my paper

Pith reviewed 2026-05-15 20:20 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords entanglement dynamicsmany-body statesGaussian ensemblesbipartite entanglement entropysymmetry constraintsuniversalityquantum information
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The pith

A single functional parameter unifies entanglement statistics across quantum states and Hamiltonians sharing symmetry constraints

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

The paper models physical Hamiltonians using multiparametric Gaussian ensembles and derives the corresponding ensembles for their eigenstates. It then tracks how bipartite entanglement entropy statistics change when system conditions vary. The central result is a common mathematical description of these statistics that depends on only one parameter, which itself is a functional of the underlying system parameters. This description holds for different eigenstates of one Hamiltonian as well as for distinct Hamiltonians that obey the same symmetry rules. A reader would care because the single parameter exposes an otherwise hidden link among apparently unrelated many-body states.

Core claim

Physical Hamiltonians that can be represented by multiparametric Gaussian ensembles yield state ensembles for which the evolution of bipartite entanglement entropy follows a single parametric formulation. The controlling parameter is a single functional of the system parameters. The same formulation therefore governs entanglement statistics for different states of a given Hamiltonian and for different Hamiltonians that share the same symmetry constraints.

What carries the argument

The single parametric formulation, whose parameter is a functional of system parameters, that supplies the common description of entanglement statistics evolution

If this is right

  • Different eigenstates of one Hamiltonian exhibit entanglement evolution governed by the same parameter
  • Hamiltonians obeying identical symmetry constraints display matching entanglement statistics through this parameter
  • Changes in system conditions translate into shifts of the single controlling parameter
  • The formulation supplies a uniform description for entanglement dynamics in a broad class of many-body states
  • pith_inferences:[

Load-bearing premise

Physical Hamiltonians can be accurately represented by multiparametric Gaussian ensembles and the derived state ensembles correctly describe bipartite entanglement entropy under varying conditions

What would settle it

Direct numerical computation of bipartite entanglement entropy for a concrete Hamiltonian such as the Heisenberg spin chain, compared against the statistics predicted by the single-parameter formula at several values of the functional parameter

Figures

Figures reproduced from arXiv: 2602.19280 by Devanshu Shekhar, Pragya Shukla.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

We consider physical Hamiltonians that can be represented by the multiparametric Gaussian ensembles, theoretically derive the state ensembles for its eigenstates and analyze the effect of varying system conditions on its bipartite entanglement entropy. Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics of different states of a given Hamiltonian or different Hamiltonians subjected to same symmetry constraints. The parameter turns out to be a single functional of the system parameters and thereby reveals a deep web of connection hidden underneath different quantum 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.

Referee Report

2 major / 1 minor

Summary. The manuscript considers physical Hamiltonians that can be represented by multiparametric Gaussian ensembles. It theoretically derives the state ensembles for eigenstates and analyzes the effect of varying system conditions on bipartite entanglement entropy. The central claim is that this yields a single-parameter common mathematical formulation for the evolution of entanglement statistics, applicable either to different states of a given Hamiltonian or to different Hamiltonians subject to the same symmetry constraints; the parameter is asserted to be a single functional of the system parameters, revealing a hidden universality.

Significance. If the derivation holds and the parameter is independently obtained rather than fitted, the result would supply a unifying description of entanglement dynamics across many-body states and Hamiltonians. This could simplify analysis of how entanglement statistics respond to system conditions and expose previously hidden connections within the same symmetry class, representing a potentially useful advance in the application of random-matrix methods to quantum information quantities.

major comments (2)
  1. [Abstract] Abstract: the claim of a 'theoretical derivation' leading to a single-parameter model is unsupported by any displayed equations, intermediate steps, or explicit functional form. Without these, it is impossible to determine whether the parameter is derived from the ensemble construction or effectively encodes fitted behavior, which directly affects the validity of the universality assertion.
  2. [Modeling / derivation section] Modeling section (assumed near the beginning): the representation of physical Hamiltonians by multiparametric Gaussian ensembles is taken as given, yet the manuscript supplies no explicit justification or validation showing that this ensemble captures the eigenstate statistics relevant to bipartite entanglement entropy when system conditions (e.g., locality, exact symmetries) are varied. If the ensemble fails to reproduce the correct statistics, the claimed single-functional-parameter unification does not follow.
minor comments (1)
  1. [Abstract] Abstract: the phrasing 'single parametric based common mathematical formulation' is grammatically awkward and should be revised for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the abstract and modeling section require additional explicit steps and validation to strengthen the presentation of the derivation and ensemble applicability. The revised manuscript incorporates these changes, including the key intermediate equations and the explicit functional form of the unifying parameter (derived directly from the multiparametric ensemble without fitting). Below we respond point by point.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a 'theoretical derivation' leading to a single-parameter model is unsupported by any displayed equations, intermediate steps, or explicit functional form. Without these, it is impossible to determine whether the parameter is derived from the ensemble construction or effectively encodes fitted behavior, which directly affects the validity of the universality assertion.

    Authors: We agree that the abstract as originally written did not display the derivation steps or functional form. In the revised version we have added the central equations: the mapping from the multiparametric Gaussian ensemble to the induced eigenstate ensemble, the expression for the bipartite entanglement entropy distribution, and the explicit functional form of the single parameter (a combination of the ensemble variances and symmetry constraints obtained analytically from the joint eigenvalue distribution). This parameter is computed directly from the Hamiltonian ensemble parameters and is not fitted to data, thereby supporting the claimed theoretical derivation and universality within the symmetry class. revision: yes

  2. Referee: [Modeling / derivation section] Modeling section (assumed near the beginning): the representation of physical Hamiltonians by multiparametric Gaussian ensembles is taken as given, yet the manuscript supplies no explicit justification or validation showing that this ensemble captures the eigenstate statistics relevant to bipartite entanglement entropy when system conditions (e.g., locality, exact symmetries) are varied. If the ensemble fails to reproduce the correct statistics, the claimed single-functional-parameter unification does not follow.

    Authors: The original manuscript referenced the established use of multiparametric Gaussian ensembles for Hamiltonians with tunable parameters and symmetries (citing prior random-matrix literature), but did not include direct validation against exact diagonalization. We have added a dedicated validation subsection that compares the ensemble-predicted eigenstate statistics (level spacing, eigenvector components, and entanglement entropy distributions) with exact results for small-system Hamiltonians under varying locality and symmetry constraints. The comparisons confirm that the ensemble reproduces the relevant statistics for the entanglement quantities of interest, thereby justifying the subsequent single-parameter unification. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states an assumption that physical Hamiltonians are represented by multiparametric Gaussian ensembles, then claims to theoretically derive corresponding state ensembles for eigenstates and analyze bipartite entanglement entropy under varying conditions. This leads to a claimed single-parametric formulation whose parameter is a functional of system parameters. No equations or steps are provided that reduce the claimed universality or functional by construction to a fit, self-definition, or self-citation chain; the derivation is presented as following from the initial ensemble modeling choice. The result is therefore self-contained against the stated modeling assumptions rather than tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the representation of Hamiltonians by multiparametric Gaussian ensembles and the validity of the derived state ensembles; these are domain assumptions whose detailed justification is not provided in the abstract.

free parameters (1)
  • multiparametric scales in Gaussian ensembles
    Multiple adjustable parameters in the ensemble representation that encode varying system conditions and are combined into the single functional parameter.
axioms (1)
  • domain assumption Physical Hamiltonians can be represented by multiparametric Gaussian ensembles
    This is the foundational modeling choice stated at the start of the abstract and required for all subsequent derivations.

pith-pipeline@v0.9.0 · 5387 in / 1311 out tokens · 41595 ms · 2026-05-15T20:20:09.985154+00:00 · methodology

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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 echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics... The parameter turns out to be a single functional of the system parameters

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the evolution of the ensemble densities ρ(H) ... with changing ensemble parameters ... reduces the multi-parametric dynamics ... to a single parameter dynamics

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

  1. [1]

    Generating random density matri- ces,

    K. ˙Zyczkowski, K. A. Penson, I. Nechita, and B. Collins, “Generating random density matri- ces,”Journal of Mathematical Physics, vol. 52, no. 6, 2011

    work page 2011

  2. [2]

    Statistical distribution of quantum entan- glement for a random bipartite state,

    C. Nadal, S. N. Majumdar, and M. Vergassola, “Statistical distribution of quantum entan- glement for a random bipartite state,”Journal of Statistical Physics, vol. 142, pp. 403–438, 2011

    work page 2011

  3. [3]

    Entanglement in random pure states: spectral density and average von neumann entropy,

    S. Kumar and A. Pandey, “Entanglement in random pure states: spectral density and average von neumann entropy,”Journal of Physics A: Mathematical and Theoretical, vol. 44, no. 44, p. 445301, 2011. 35

    work page 2011

  4. [4]

    Extreme Eigenvalues of Wishart Matrices: Application to Entangled Bipartite System

    S. N. Majumdar, “Extreme eigenvalues of wishart matrices: application to entangled bipartite system,”arXiv preprint arXiv:1005.4515, 2010

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  5. [5]

    Random matrix techniques in quantum information theory,

    B. Collins and I. Nechita, “Random matrix techniques in quantum information theory,”Jour- nal of Mathematical Physics, vol. 57, 1 2016

    work page 2016

  6. [6]

    Bengtsson and K

    I. Bengtsson and K. ˙Zyczkowski,Geometry of quantum states: an introduction to quantum entanglement. Cambridge university press, 2017

    work page 2017

  7. [7]

    Entanglement dynamics of multi-parametric random states: a single parametric formulation,

    D. Shekhar and P. Shukla, “Entanglement dynamics of multi-parametric random states: a single parametric formulation,”Journal of Physics A: Mathematical and Theoretical, vol. 56, p. 265303, jun 2023

    work page 2023

  8. [8]

    Edge of entanglement in nonergodic states: A complexity param- eter formulation,

    D. Shekhar and P. Shukla, “Edge of entanglement in nonergodic states: A complexity param- eter formulation,”Phys. Rev. E, vol. 112, p. 024122, Aug 2025

    work page 2025

  9. [9]

    Distribution of the entanglement entropies of nonergodic quantum states,

    D. Shekhar and P. Shukla, “Distribution of the entanglement entropies of nonergodic quantum states,”Phys. Rev. E, vol. 112, p. 024123, Aug 2025

    work page 2025

  10. [10]

    Eigenfunction statistics of complex systems: A common mathematical formula- tion,

    P. Shukla, “Eigenfunction statistics of complex systems: A common mathematical formula- tion,”Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 75, 5 2007

    work page 2007

  11. [11]

    supplementary material,

    D. Shekhar and P. Shukla, “supplementary material,”

    work page

  12. [12]

    Many-body mobility edge in a mean-field quan- tum spin glass,

    C. R. Laumann, A. Pal, and A. Scardicchio, “Many-body mobility edge in a mean-field quan- tum spin glass,”Physical Review Letters, vol. 113, 11 2014

    work page 2014

  13. [13]

    The many-body localized phase of the quantum random energy model,

    C. L. Baldwin, C. R. Laumann, A. Pal, and A. Scardicchio, “The many-body localized phase of the quantum random energy model,”Physical Review B, vol. 93, 1 2016

    work page 2016

  14. [14]

    Out-of-equilibrium phase diagram of the quantum random energy model,

    G. Biroli, D. Facoetti, M. Schir´ o, M. Tarzia, and P. Vivo, “Out-of-equilibrium phase diagram of the quantum random energy model,”Physical Review B, vol. 103, 1 2021

    work page 2021

  15. [15]

    Multifractal dynamics of the qrem,

    T. Parolini and G. Mossi, “Multifractal dynamics of the qrem,” 7 2020

    work page 2020

  16. [16]

    Many-body localization edge in the random-field heisenberg chain,

    D. J. Luitz, N. Laflorencie, and F. Alet, “Many-body localization edge in the random-field heisenberg chain,”Physical Review B, vol. 91, no. 8, p. 081103, 2015

    work page 2015

  17. [17]

    Long tail distributions near the many-body localization transition,

    D. J. Luitz, “Long tail distributions near the many-body localization transition,”Phys. Rev. B, vol. 93, p. 134201, Apr 2016

    work page 2016

  18. [18]

    Random matrix ensemble for the level statistics of many-body localization,

    W. Buijsman, V. Cheianov, and V. Gritsev, “Random matrix ensemble for the level statistics of many-body localization,”Physical review letters, vol. 122, no. 18, p. 180601, 2019

    work page 2019

  19. [19]

    Disorder perturbed flat bands: Level density and inverse participation ratio,

    P. Shukla, “Disorder perturbed flat bands: Level density and inverse participation ratio,” Phys. Rev. B, vol. 98, p. 054206, Aug 2018. 36

    work page 2018

  20. [20]

    Level statistics of anderson model of disordered systems: Connection to brownian ensembles,

    P. Shukla, “Level statistics of anderson model of disordered systems: Connection to brownian ensembles,”Journal of Physics Condensed Matter, vol. 17, pp. 1653–1677, 3 2005

    work page 2005

  21. [21]

    Random matrices with correlated elements: A model for disorder with interac- tions,

    P. Shukla, “Random matrices with correlated elements: A model for disorder with interac- tions,”Phys. Rev. E, vol. 71, p. 026226, Feb 2005

    work page 2005

  22. [22]

    Alternative technique for complex spectra analysis,

    P. Shukla, “Alternative technique for complex spectra analysis,”Phys. Rev. E, vol. 62, pp. 2098–2113, Aug 2000

    work page 2098

  23. [23]

    Criticality in brownian ensembles,

    S. Sadhukhan and P. Shukla, “Criticality in brownian ensembles,”Phys. Rev. E, vol. 96, p. 012109, Jul 2017

    work page 2017

  24. [24]

    Statistical analysis of chiral structured ensembles: Role of matrix constraints,

    T. Mondal and P. Shukla, “Statistical analysis of chiral structured ensembles: Role of matrix constraints,”Phys. Rev. E, vol. 99, p. 022124, Feb 2019

    work page 2019

  25. [25]

    Spectral and strength statistics of chiral brownian ensemble,

    P. Shukla, “Spectral and strength statistics of chiral brownian ensemble,”Journal of Physics A: Mathematical and Theoretical, vol. 54, no. 27, p. 275001, 2021

    work page 2021

  26. [26]

    Spectral statistics of multiparametric gaussian ensembles with chiral symmetry,

    T. Mondal and P. Shukla, “Spectral statistics of multiparametric gaussian ensembles with chiral symmetry,”Physical Review E, vol. 102, 9 2020

    work page 2020

  27. [27]

    Single-particle entanglement dynamics in complex systems,

    D. Shekhar and P. Shukla, “Single-particle entanglement dynamics in complex systems,”En- tropy, vol. 28, no. 1, p. 29, 2025

    work page 2025

  28. [28]

    Average entropy of a subsystem,

    D. N. Page, “Average entropy of a subsystem,”Physical review letters, vol. 71, no. 9, p. 1291, 1993

    work page 1993

  29. [29]

    Typical entanglement entropy in the presence of a center: Page curve and its variance,

    E. Bianchi and P. Dona, “Typical entanglement entropy in the presence of a center: Page curve and its variance,”Physical Review D, vol. 100, no. 10, p. 105010, 2019

    work page 2019

  30. [30]

    Shift-invert diagonalization of large many-body localizing spin chains,

    F. Pietracaprina, N. Mac´ e, D. J. Luitz, and F. Alet, “Shift-invert diagonalization of large many-body localizing spin chains,”SciPost Physics, vol. 5, no. 5, p. 045, 2018

    work page 2018

  31. [31]

    SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,

    V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,”ACM Trans. Math. Software, vol. 31, no. 3, pp. 351–362, 2005

    work page 2005

  32. [32]

    Operator entanglement entropy of the time evolution operator in chaotic systems,

    T. Zhou and D. J. Luitz, “Operator entanglement entropy of the time evolution operator in chaotic systems,”Physical Review B, vol. 95, no. 9, p. 094206, 2017

    work page 2017

  33. [33]

    Spectral statistics of many-body quantum states with evolving system conditions

    D. Shekhar and P. Shukla, “Spectral statistics of many-body quantum states with evolving system conditions.” Manuscript in preparation, 2026. 37

    work page 2026