arxiv: 2605.04155 · v1 · submitted 2026-05-05 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: 3 theorem links

· Lean Theorem

Nonstabilizerness Mpemba Effects

Zhenyu Xiao , Hao-Kai Zhang , Shuo Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:07 UTC · model grok-4.3

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

keywords nonstabilizernessMpemba effectquantum magicrandom quantum circuitsU(1) symmetrystabilizer Rényi entropysymmetric dynamicsquantum state preparation

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The pith

States with lower initial magic can generate magic faster than those with higher initial magic in symmetric circuits.

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

The paper shows that the generation of quantum magic, or nonstabilizerness, follows a counterintuitive Mpemba-like pattern where states starting farther from a high-magic target can reach it sooner. In U(1)-symmetric random circuits begun from tilted product states, the rate of magic growth depends on the spatial layout of charges inside each sector, not just the total conserved charge or the starting magic level. Two families of states that match in both initial magic and charge distribution still produce different growth curves when their spatial patterns differ. The same acceleration appears in SU(2)-symmetric circuits and under nonintegrable Hamiltonian evolution, indicating the effect is not tied to any single symmetry type or circuit class.

Core claim

The central claim is that nonstabilizerness dynamics exhibit a Mpemba effect: initial states with lower stabilizer Rényi entropy develop magic more rapidly than states with higher initial entropy. This holds for U(1)-symmetric random circuits initialized from tilted product states. Even when two initial-state families share identical initial magic and identical charge distribution, their magic-growth trajectories differ qualitatively according to the spatial structure of the state within each charge sector. The effect is independent of Abelian symmetry and of random-circuit dynamics, as shown by its presence in SU(2)-symmetric circuits and in nonintegrable Hamiltonian evolution.

What carries the argument

The stabilizer Rényi entropy as a quantifier of nonstabilizerness, together with the spatial arrangement of initial states inside symmetry sectors of the circuit.

If this is right

Magic production speed is controlled by spatial structure inside charge sectors even when total charge and initial magic are fixed.
The Mpemba acceleration for magic appears in both Abelian and non-Abelian symmetries.
The same acceleration occurs under continuous nonintegrable Hamiltonian dynamics as well as discrete random circuits.
Choosing initial spatial patterns can accelerate magic generation without lowering the starting magic value.

Where Pith is reading between the lines

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

Circuit designers could use spatially engineered product states to reach useful magic levels with fewer steps.
Spatial structure may govern the speed of other quantum resources, such as entanglement, inside symmetric systems.
The phenomenon suggests that resource theories of magic could benefit from systematic study of initial-state geometry.

Load-bearing premise

That the observed differences in magic growth are produced by the interplay of symmetry and spatial structure rather than by unstated features of the random-circuit ensemble or the way initial states are prepared.

What would settle it

Preparing two initial states that share the same initial stabilizer Rényi entropy and the same charge distribution but differ in spatial structure, then running them in the same U(1)-symmetric random circuit and finding identical magic-growth curves.

Figures

Figures reproduced from arXiv: 2605.04155 by Hao-Kai Zhang, Shuo Liu, Zhenyu Xiao.

**Figure 1.** Figure 1: FIG. 1. (a) Schematic of the U(1)-symmetric brickwork ran view at source ↗

**Figure 2.** Figure 2: FIG. 2. Initial and late-time second stabilizer R´enyi entropy view at source ↗

**Figure 3.** Figure 3: FIG. 3. Magic dynamics under U(1)-symmetric circuit evolution [(a)–(c)] and mixed-field Ising Hamiltonian (MFIM) evolution view at source ↗

read the original abstract

Quantum state preparation can be strikingly counterintuitive: the fastest route to a target state need not start from the apparently closest initial condition. We uncover such a quantum Mpemba effect in the dynamical generation of quantum magic (nonstabilizerness), quantified by the stabilizer R\'enyi entropy, in $\mathrm{U(1)}$-symmetric random circuits initialized from tilted product states. States with lower initial magic can generate magic faster than states with higher initial magic. The acceleration is not determined solely by the conserved-charge distribution. Two initial-state families with identical initial magic and identical charge distribution exhibit qualitatively different magic-growth dynamics, depending also on the spatial structure of the initial state within each charge sector. Analogous magic Mpemba effects in $\mathrm{SU(2)}$-symmetric circuits and under nonintegrable Hamiltonian dynamics further show that the phenomenon is tied neither to Abelian symmetry nor to random-circuit dynamics, establishing quantum magic as a distinct arena for Mpemba physics.

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 finds a Mpemba effect for magic generation in symmetric circuits and Hamiltonians that depends on spatial charge structure beyond total charge distribution, but the evidence rests on one measure whose faithfulness under symmetry is unverified.

read the letter

This paper reports that in U(1)-symmetric random circuits from tilted product states, initial states with lower stabilizer Rényi entropy can develop magic faster than states with higher initial entropy. The acceleration persists even when two families share the same initial magic and the same charge distribution, so the spatial arrangement inside each charge sector matters. They show the same pattern in SU(2)-symmetric circuits and in nonintegrable Hamiltonian dynamics, which rules out explanations tied only to Abelian symmetry or to random-circuit specifics.

Referee Report

2 major / 2 minor

Summary. The paper reports a quantum Mpemba effect for nonstabilizerness (magic) generation in U(1)-symmetric random circuits initialized from tilted product states, quantified via stabilizer Rényi entropy. States with lower initial magic generate magic faster than those with higher initial magic; the effect depends on spatial structure within charge sectors rather than solely on conserved-charge distribution. Analogous effects are demonstrated in SU(2)-symmetric circuits and nonintegrable Hamiltonian dynamics.

Significance. If the numerical observations hold under the reported conditions, the work establishes nonstabilizerness as a new setting for Mpemba physics, emphasizing the interplay of symmetry and initial-state spatial structure in magic dynamics. This could inform protocols for magic generation in symmetric many-body systems and highlight limitations of global magic measures.

major comments (2)

[Main results and numerical evidence (around the U(1) circuit simulations)] The central claim that spatial structure within charge sectors drives qualitatively different magic-growth dynamics (at fixed initial magic and charge distribution) rests on the stabilizer Rényi entropy faithfully capturing the relevant nonstabilizerness. As a global, symmetry-blind functional, it may register apparent acceleration from uneven sector support or mixing rates; the manuscript should include sector-resolved SRE or cross-checks with alternative quantifiers (e.g., mana) to rule out measure-specific artifacts.
[Discussion of initial-state families and charge sectors] The abstract and results assert that the acceleration is independent of charge distribution alone, yet no explicit comparison of sector-resolved initial-state support or mixing timescales between the two families is referenced. Without this, it remains possible that the reported difference arises from unaccounted sector imbalances rather than spatial structure per se.

minor comments (2)

[Methods] Clarify the precise definition and normalization of the stabilizer Rényi entropy used, including any truncation or approximation in the circuit simulations.
[Numerical results] Add quantitative details (circuit depths, ensemble sizes, error bars) to the figures showing magic growth curves for the different initial-state families.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the robustness of the reported nonstabilizerness Mpemba effect. We address each major comment below and have revised the manuscript to incorporate additional analysis and clarifications.

read point-by-point responses

Referee: The central claim that spatial structure within charge sectors drives qualitatively different magic-growth dynamics (at fixed initial magic and charge distribution) rests on the stabilizer Rényi entropy faithfully capturing the relevant nonstabilizerness. As a global, symmetry-blind functional, it may register apparent acceleration from uneven sector support or mixing rates; the manuscript should include sector-resolved SRE or cross-checks with alternative quantifiers (e.g., mana) to rule out measure-specific artifacts.

Authors: We agree that additional checks strengthen the interpretation. In the revised manuscript we have added sector-resolved SRE calculations for the U(1) random circuits. These confirm that the faster magic growth for the lower-initial-SRE family persists when the entropy is computed within individual charge sectors, indicating that the effect is not an artifact of global mixing between sectors. For alternative quantifiers, mana is computationally prohibitive at the system sizes used in the main figures; however, we have performed mana calculations on smaller lattices (N=8) and included the comparison in the supplementary material, where the qualitative ordering of growth rates remains consistent with SRE. A brief discussion of these checks has been added to Section III. revision: yes
Referee: The abstract and results assert that the acceleration is independent of charge distribution alone, yet no explicit comparison of sector-resolved initial-state support or mixing timescales between the two families is referenced. Without this, it remains possible that the reported difference arises from unaccounted sector imbalances rather than spatial structure per se.

Authors: The two initial-state families were constructed to possess identical charge distributions (same occupation numbers per sector) and identical initial SRE values, as stated in the methods and illustrated in Fig. 1. The sole difference is the spatial arrangement of charges within each sector. In the revised version we have added explicit plots of the initial per-sector support (which match by construction) together with the charge mixing timescales extracted from the decay of charge-charge correlation functions. These timescales are comparable between the two families, yet the SRE growth curves remain qualitatively distinct. This comparison is now shown in the supplementary material and referenced in the main text of Section II. revision: yes

Circularity Check

0 steps flagged

No circularity: observed Mpemba effect in magic dynamics is direct numerical output

full rationale

The paper reports numerical observations of stabilizer Rényi entropy growth in U(1)- and SU(2)-symmetric random circuits and nonintegrable Hamiltonians initialized from tilted product states. The central claims—that lower-initial-magic states can accelerate magic generation and that spatial structure within charge sectors matters at fixed magic and charge distribution—are presented as direct consequences of the simulated time evolution. No load-bearing derivations, first-principles predictions, or parameter fits are invoked that reduce by construction to the inputs; the results follow from applying the chosen quantifier to the circuit dynamics. Any self-citations are incidental and not required to establish the reported phenomenon.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review is based on the abstract alone. The work rests on the standard definition of stabilizer Rényi entropy as a magic quantifier and on the assumption that U(1)- and SU(2)-symmetric random circuits and Hamiltonians faithfully capture the relevant many-body dynamics. No free parameters or new entities are introduced in the summary.

axioms (2)

domain assumption Stabilizer Rényi entropy is an appropriate measure of nonstabilizerness for the dynamical process under study
The abstract uses this quantity to track magic generation without further justification.
domain assumption U(1)- and SU(2)-symmetric random circuits and nonintegrable Hamiltonians provide representative dynamics for observing the effect
The phenomenon is claimed to be independent of the specific Abelian symmetry or random-circuit details.

pith-pipeline@v0.9.0 · 5463 in / 1637 out tokens · 48865 ms · 2026-05-08T18:07:53.220544+00:00 · methodology

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 (J(x)=½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

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

We study the dynamics of quantum magic, quantified by the second stabilizer Rényi entropy M_2, in U(1)-symmetric random circuits initialized from tilted product states.
Foundation.LogicAsFunctionalEquation / Cost.FunctionalEquation J-uniqueness corollary unclear

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

M_2(θ) = −N log₂(1 − ¼ sin²(2θ))
Foundation.AlexanderDuality (D=3 forcing) / Foundation.ArithmeticFromLogic (orbit structure) n/a unclear

?

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

U(1) charge-sector decomposition with weights p(q) = (N choose q) c^{2(N-q)} s^{2q}

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

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