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arxiv: 2604.13923 · v1 · submitted 2026-04-15 · 🪐 quant-ph · cond-mat.stat-mech· physics.atom-ph· physics.optics

Quantum information spreading in inhomogeneous spin ensembles

Rahul Gupta , Florian Mintert , Himadri Shekhar Dhar This is my paper

Pith reviewed 2026-05-10 12:50 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechphysics.atom-phphysics.optics
keywords Krylov spaceinhomogeneous spin ensemblesLieb-Robinson velocityquantum speed limitinformation propagationsingle-excitation subspaceresonance frequency distributionquantum information spreading
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The pith

Quantum information spreads in spin ensembles at rates set by the statistical distribution of resonance frequencies.

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

The paper develops a Krylov space framework to describe quantum dynamics in large inhomogeneous ensembles of spins with arbitrary frequency and coupling distributions. In the single-excitation subspace it produces exact expressions for the Lieb-Robinson velocity that bounds information propagation, the quantum speed limit, and related quantities such as Krylov complexity. The central finding is that these measures of information flow depend strongly on the shape and width of the resonance-frequency distribution rather than on average values alone. This dependence matters for any quantum technology that relies on controlled spreading of excitations or correlations in real spin systems.

Core claim

The Krylov construction applied to the Hamiltonian of an inhomogeneous spin ensemble generates an orthonormal basis in which the single-excitation dynamics can be solved exactly for asymptotically large systems. This yields closed-form expressions for the Lieb-Robinson velocity and quantum speed limit that vary with the chosen statistical distribution of spin frequencies. The same construction also supplies a concrete figure of merit, Krylov complexity, that quantifies how the excitation spreads.

What carries the argument

The Krylov space obtained by repeated application of the Hamiltonian to an initial state, which supplies an exact basis for single-excitation dynamics in arbitrarily distributed spin ensembles.

If this is right

  • Exact Lieb-Robinson velocities become available for any chosen frequency distribution without further approximation.
  • Quantum speed limits can be evaluated analytically for large inhomogeneous ensembles.
  • Krylov complexity serves as a computable indicator of information spreading speed.
  • Engineering of spin-based quantum components must account for the full statistical distribution of resonance frequencies to predict information flow.

Where Pith is reading between the lines

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

  • Choosing a narrow or broad frequency distribution could be used to slow or accelerate quantum information transfer in spin arrays.
  • The same Krylov approach may be applied to other disordered many-body systems to obtain bounds on correlation spreading.
  • Experiments with tunable inhomogeneity in atomic or solid-state spin ensembles could test the predicted distribution dependence by varying the spread while keeping the mean fixed.

Load-bearing premise

The derivations assume an asymptotically large ensemble and restrict attention to the single-excitation subspace.

What would settle it

A direct measurement or numerical simulation of the time required for a single excitation to reach a distant spin, performed for several different frequency distributions, should match the predicted Lieb-Robinson velocity only if the distribution dependence is correct.

Figures

Figures reproduced from arXiv: 2604.13923 by Florian Mintert, Himadri Shekhar Dhar, Rahul Gupta.

Figure 1
Figure 1. Figure 1: FIG. 1. Quantum information spreading in Krylov Space: (a) Krylov coefficients [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the spread of a single excitation, ini￾tially stored in the bright state |ϕ1⟩, through the Krylov space of a spin ensemble with a q-Gaussian distribution of spin frequencies. For q = 1, the distribution is Gaussian, and strong parabolic dispersion of information to higher Krylov states is observed, as shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fidelity evolution from bright mode [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

We present a Krylov space based theoretical framework for modeling inhomogeneous spin ensembles with arbitrary distributions of spin frequencies and couplings. The framework is then used to asymptotically large spin ensemble. In the single-excitation subspace, the Krylov construction allows for to derive exact expressions for the Lieb-Robinson velocity and quantum speed limit, and figure of merit such as Krylov complexity. Our work reveals a strong dependence of the speed of information flow on the statistical distribution of resonance frequencies in the spin ensemble with immediate implications for the design of components for quantum technologies, realized for example with nitrogen vacancy centers, nuclear spins or ultracold atoms.

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 develops a Krylov-subspace framework for quantum information spreading in asymptotically large inhomogeneous spin ensembles with arbitrary distributions of on-site frequencies and (position-dependent) couplings. In the single-excitation subspace the method is used to obtain exact expressions for the Lieb-Robinson velocity, quantum speed limit, and Krylov complexity, which are shown to depend strongly on the statistical properties of the resonance-frequency distribution P(ω). The results are motivated by applications to NV centers, nuclear spins, and ultracold atoms.

Significance. If the claimed exact, distribution-dependent formulas can be substantiated, the work supplies a concrete analytic handle on how disorder shapes information propagation speed in spin ensembles. This is relevant for quantum-technology design where control of information flow is desirable. The use of Krylov methods to reduce the many-body problem to an effective single-particle tight-binding chain with a spectral measure determined by P(ω) is a standard and potentially powerful reduction, provided the resulting expressions remain tractable.

major comments (2)
  1. [Abstract / §3] Abstract and the central derivation (presumably §3–4): the repeated claim of “exact expressions” for the Lieb-Robinson velocity and quantum speed limit for completely arbitrary P(ω) is not yet supported. The Lanczos recurrence coefficients are fixed by the moments of the effective spectral measure; for a generic distribution these moments lack closed analytic forms. The manuscript must therefore either (i) exhibit explicit closed-form expressions that hold without further regularity assumptions on P(ω), or (ii) state the precise class of distributions for which the expressions remain exact and analytic. Without this clarification the “exact” qualifier risks being overstated.
  2. [§4] §4 (or wherever the velocity and QSL are derived): the reduction to the single-excitation subspace is standard, but the passage from the Krylov chain to a concrete Lieb-Robinson bound or quantum-speed-limit expression must be shown explicitly. In particular, it is unclear whether the final formulas involve only the first few moments or require the full infinite set of Lanczos coefficients; the latter would generally necessitate numerical quadrature even for simple P(ω).
minor comments (2)
  1. [Abstract] Abstract: the sentence “The framework is then used to asymptotically large spin ensemble” is grammatically incomplete; “allows for to derive” should be “allows one to derive”.
  2. [Introduction] Notation: the symbol P(ω) for the frequency distribution is introduced without an explicit definition of its normalization or support; a short paragraph clarifying the measure would improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below, clarifying the scope of our exact results and indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and the central derivation (presumably §3–4): the repeated claim of “exact expressions” for the Lieb-Robinson velocity and quantum speed limit for completely arbitrary P(ω) is not yet supported. The Lanczos recurrence coefficients are fixed by the moments of the effective spectral measure; for a generic distribution these moments lack closed analytic forms. The manuscript must therefore either (i) exhibit explicit closed-form expressions that hold without further regularity assumptions on P(ω), or (ii) state the precise class of distributions for which the expressions remain exact and analytic. Without this clarification the “exact” qualifier risks being overstated.

    Authors: The Krylov construction yields an exact, non-perturbative mapping of the single-excitation dynamics onto a one-dimensional chain whose Lanczos coefficients are the moments of the spectral measure induced by P(ω). Consequently the Lieb-Robinson velocity and quantum speed limit are expressed exactly in terms of this measure (or its moment sequence) without additional approximations. For a completely arbitrary P(ω) these moments are defined by integrals and therefore the expressions remain exact but are not necessarily elementary. We will revise the abstract and §3 to state this distinction explicitly, specify that closed elementary forms exist only when the moment-generating function of P(ω) is known in closed form (e.g., Gaussian or Lorentzian distributions), and add an appendix containing the general moment formulas together with explicit analytic results for two representative distributions. revision: partial

  2. Referee: [§4] §4 (or wherever the velocity and QSL are derived): the reduction to the single-excitation subspace is standard, but the passage from the Krylov chain to a concrete Lieb-Robinson bound or quantum-speed-limit expression must be shown explicitly. In particular, it is unclear whether the final formulas involve only the first few moments or require the full infinite set of Lanczos coefficients; the latter would generally necessitate numerical quadrature even for simple P(ω).

    Authors: We agree that the explicit steps connecting the Krylov chain to the final bounds must be displayed. The Lieb-Robinson velocity is the supremum of the group velocity on the dispersion relation generated by the infinite tridiagonal Lanczos matrix; the quantum speed limit follows from the same spectral measure. While the full set of coefficients is formally required, useful analytic bounds can be obtained from the first few moments (variance and kurtosis of P(ω)). We will expand §4 to show the explicit mapping from the Lanczos matrix to the velocity bound and the QSL, including the integral representation over the spectral density. For concrete numerical evaluation of general P(ω) we indeed employ quadrature, but the underlying expressions remain exact prior to discretization. revision: yes

standing simulated objections not resolved
  • We cannot exhibit elementary closed-form expressions valid for every conceivable P(ω) without regularity assumptions, as the Hamburger moment problem for a generic measure does not admit such forms.

Circularity Check

0 steps flagged

No circularity: standard Krylov application yields independent expressions

full rationale

The paper applies the established Krylov subspace construction to the single-excitation sector of an inhomogeneous spin Hamiltonian with arbitrary frequency distribution P(ω). The abstract states that this construction 'allows to derive exact expressions' for Lieb-Robinson velocity, quantum speed limit and Krylov complexity. No quoted step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or imports a uniqueness theorem from the authors' prior work. The derivation chain therefore remains self-contained: the moments of the spectral measure are computed from the given distribution and couplings, and the resulting velocities and bounds follow directly from the Lanczos recurrence without tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Krylov space methods to spin ensembles and the single-excitation approximation.

axioms (2)
  • domain assumption The single-excitation subspace is sufficient for modeling information spreading.
    Central results are derived within this subspace as stated in the abstract.
  • domain assumption Krylov space construction yields exact expressions for large ensembles.
    The framework is presented for asymptotically large spin ensembles.

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