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arxiv: 2603.28870 · v2 · submitted 2026-03-30 · 🪐 quant-ph · cond-mat.stat-mech

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Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems

Daniele Iannotti , Angelo Russotto , Barbara Jasser , Jovan Odavi\'c , Alioscia Hamma
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Pith reviewed 2026-05-14 21:21 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-stabilizernessU(1) symmetryHaar random statesstabilizer entropymany-body chaosconserved chargemagic monotone
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The pith

A conserved U(1) charge substantially suppresses non-stabilizerness in Haar random quantum states.

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

The paper computes exact averages and variances of stabilizer entropy for pure states drawn from the Haar measure restricted to a fixed U(1) charge sector. It finds that the conserved charge reduces the typical amount of non-stabilizerness relative to the fully unconstrained Haar ensemble. In the large-system limit the leading scaling of stabilizer entropy near zero relative charge density differs from the scaling of entanglement entropy, indicating that magic is more robust against small charge-density fluctuations. Numerical checks on the complex SYK model confirm the formulas while a local spin chain shows systematic departures tied to interaction locality.

Core claim

We present exact, closed-form results for the non-stabilizerness of random pure states subject to a U(1) symmetry constraint. Using stabilizer entropy as our non-stabilizerness monotone, we derive the average and the variance for U(1)-constrained Haar random states. We show that the presence of a conserved charge leads to a substantial suppression of non-stabilizerness (magic) compared to the unconstrained case, and identify a qualitative difference between entanglement and magic response. In the thermodynamic limit, stabilizer entropy exhibits a different leading-order scaling close to a vanishing relative charge density, implying that magic is more robust to charge density fluctuations.

What carries the argument

The U(1)-constrained Haar ensemble, which restricts random pure states to a fixed total charge sector and yields closed-form expressions for the moments of stabilizer entropy.

Load-bearing premise

The quantum states are distributed exactly according to the Haar measure restricted to one U(1) symmetry sector.

What would settle it

A direct numerical computation of the average stabilizer entropy in the midspectrum of the complex SYK model that deviates from the derived closed-form expression at large system size.

Figures

Figures reproduced from arXiv: 2603.28870 by Alioscia Hamma, Angelo Russotto, Barbara Jasser, Daniele Iannotti, Jovan Odavi\'c.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of the random-state ensemble: states [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Since m(s) < 1 for s ̸= 0, the difference in Eq. (10) for generic charge density is always divergent as (1 − m(s))L. The behaviour in Eq. (9) is also pecu￾liar in terms of the dependence of the charge density s, as the quadratic corrections are vanishing. This effect is not present, for example, in the average Von Neumann entanglement entropy of a subsystem for the same en￾semble. In that case, at leading … view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Disorder-averaged stabilizer entropy of cSYK eigen [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Stabilizer entropy of XXZ-NNN [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Graphical representation of [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical check of the validity of the asymptotic behaviour in Eq. (S.32). We compute numerically, as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between analytical (dashed lines) and numerical (markers with errorbars) predictions for the average of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical check of the asymptotic result in Eq. (S.46). We numerically compute, using the exact formula Eq. (S.39), [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Analytical result of the variance of the 2-SP, ∆ [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Numerical (empirical) variance convergence as a function of the number of sample realizations for states with [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Numerical checks demonstrating the response of Haar random states subjected to constraints across different Pauli [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Average stabilizer entropy [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Magnitude of the Hamiltonian matrix elements [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Relative variance signaling self-averaging with respect to disorder across different charge sectors. Panels in the left [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
read the original abstract

We present exact, closed-form results for the non-stabilizerness of random pure states subject to a U(1) symmetry constraint. Using stabilizer entropy as our non-stabilizerness monotone, we derive the average and the variance for U(1)-constrained Haar random states. We show that the presence of a conserved charge leads to a substantial suppression of non-stabilizerness (magic) compared to the unconstrained case, and identify a qualitative difference between entanglement and magic response. In the thermodynamic limit, stabilizer entropy exhibits a different leading-order scaling close to a vanishing relative charge density, implying that magic is more robust to charge density fluctuations than entanglement entropy. We test our analytical predictions against midspectrum eigenstates of two chaotic many-body systems with conserved U(1) charge: the complex-fermion Sachdev-Ye-Kitaev (cSYK) model and a Heisenberg XXZ chain with next-to-nearest-neighbour couplings and conserved magnetization. We find an excellent agreement for the non-local cSYK model and systematic deviations for the local XXZ chain, highlighting the role of interaction locality.

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

0 major / 2 minor

Summary. The paper derives exact closed-form expressions for the average and variance of stabilizer entropy (a non-stabilizerness monotone) over the U(1)-constrained Haar ensemble on fixed-charge subspaces. It shows that conserved charge substantially suppresses magic relative to the unconstrained case, with a distinct leading-order thermodynamic scaling near vanishing relative charge density that differs from entanglement entropy. Numerical comparisons are presented for midspectrum eigenstates of the non-local cSYK model (excellent agreement) and a local XXZ chain (systematic deviations attributed to interaction locality).

Significance. If the closed-form results hold, the work supplies a precise, parameter-free characterization of symmetry-induced suppression of magic in chaotic many-body systems. The exact derivation for the constrained Haar measure, the identification of qualitatively different scaling for magic versus entanglement, and the clean match to non-local chaotic models constitute a substantive advance in understanding non-stabilizerness under conservation laws.

minor comments (2)
  1. [§3] §3 (or equivalent derivation section): the explicit integration steps over the constrained manifold should be expanded with one intermediate identity to make the passage from the Haar measure to the final variance formula fully self-contained for readers.
  2. [Figure 2] Figure 2 caption: the error bars on the XXZ data points are not described; adding a brief statement on their origin (e.g., disorder averaging or finite-size effects) would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as their recommendation to accept. We are pleased that the exact results on U(1)-constrained non-stabilizerness, the distinct scaling behavior relative to entanglement, and the numerical comparisons were viewed as a substantive advance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents exact closed-form derivations of the average and variance of stabilizer entropy for U(1)-constrained Haar random states, obtained by direct integration over the symmetry sector using standard properties of the restricted Haar measure. No load-bearing steps reduce to self-definitions, fitted inputs renamed as predictions, or self-citation chains; the central results for suppression of magic and distinct scaling near zero charge density follow independently from the ensemble definition without circular reduction. Numerical comparisons to cSYK and XXZ models serve as external validation rather than internal justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard mathematical definition of the Haar measure restricted to fixed-charge sectors and the definition of stabilizer entropy; no free parameters, new entities, or ad-hoc assumptions are mentioned.

axioms (1)
  • standard math Haar measure on the U(1)-constrained subspace of the Hilbert space
    Used to compute averages and variances of stabilizer entropy over random states

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Forward citations

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