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REVIEW 4 major objections 5 minor 38 references

(A)Symmetric Complexity and the Quantum Mpemba Effect

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that the zero-time value of a symmetrically projected Krylov complexity—the structural complexity—predicts whether a pair of tilted states will show a quantum Mpemba crossing, before any time evolution is run.

desk verdict The decomposition is genuinely new and the t=0 relation is clean, but the 'obviates time evolution' claim is contradicted by the paper's own Sec. 4.1, and the predictive evidence rests on a single realization without error bars. read the letter →

arxiv 2509.08078 v1 pith:X45OWHCY submitted 2025-09-09 hep-th

Cameron Beetar , Jeff Murugan , Hendrik J.R. van Zyl This is my paper
classification hep-th
keywords QuantumMpembaeffectKrylovcomplexitySpreadEntanglementasymmetryAubry-AndrémodelMany-bodylocalizationETH/MBLtransitionSymmetry-resolveddynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that Krylov spread complexity—how a quantum state spreads in the basis generated by repeatedly applying the Hamiltonian—can be refined into a quantitative diagnostic of the Quantum Mpemba Effect (QME), where a state prepared further from equilibrium relaxes faster than one prepared closer. The authors split spread complexity into a symmetric part (diffusion within conserved-charge sectors) and an asymmetric part (coherences between sectors), and show that the asymmetric part behaves like an asymmetry measure akin to entanglement asymmetry. Their central finding is that the static value at t=0—called structural complexity, |C_K,A(0)| = C_K,S(0)—already distinguishes tilted states that will display Mpemba-like crossings from those that will not, so the crossing can be anticipated without explicit time evolution. If correct, this gives a t=0 order parameter for anomalous relaxation in the Aubry-André model.

What carries the argument

Projected (a)symmetric complexity operators: Ĉ_K,S = Σ_q Σ_n n Π_q |K_n⟩⟨K_n| Π_q and Ĉ_K,A = Ĉ_K − Ĉ_K,S, which project the Krylov-position operator onto conserved-charge sectors and the coherences between them. Their expectation values partition spread complexity into intra-sector diffusion and inter-sector hopping; the structural complexity C_structural = |C_K,A(0)| = C_K,S(0) is the static t=0 magnitude that carries the predictive signal. The Lanczos coefficients, especially b1, provide a second microscopic static probe whose ordering shifts near the phase transition.

What would settle it

Compute the entanglement-asymmetry crossings for tilted Néel states over 240 disorder realizations at W ≈ 2.0 (where structural complexity is double-peaked) and W ≈ 3.0 (where it is single-peaked). If crossings appear at W ≈ 2.0, or fail to appear at W ≈ 3.0, the claim that structural complexity predicts the Mpemba effect would be falsified. A second check: test whether the single-realisation structural complexity surface (Fig. 8b) changes its peak structure across disorder realizations; if the single-peak onset wanders outside W ≈ 2.5–3.3 for typical realizations, the t=0 predictor is not rob

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Extended reading notes

Core claim

The paper's central claim is that a suitably decomposed Krylov complexity detects and predicts the quantum Mpemba effect in the Aubry-André model. Decomposing the Krylov complexity operator into its block-diagonal (symmetric) and off-block-diagonal (asymmetric) pieces relative to total magnetization, the asymmetric complexity starts at a negative value whose magnitude is exactly the symmetric complexity at t=0, and decays toward zero as symmetry is restored. When two tilted initial states are compared, Mpemba-like crossings appear not in the asymmetric complexity itself but in the shifted symmetric complexity C_K,S(t) − C_K,S(0), whose saturation ordering reverses across the ETH/MBL transiti

Load-bearing premise

That the tilt-induced crossing in the shifted symmetric complexity at a fixed disorder strength is a genuine quantum Mpemba effect and not just a reordering produced by the ETH/MBL phase structure; the authors themselves note this ambiguity.

Editorial extensions

If this is right

  • Mpemba-like crossings can be screened at t=0 by computing structural complexity for a pair of tilted states, without running long-time evolution.
  • The asymmetric complexity orders states by tilt exactly in regimes where entanglement-asymmetry crossings occur, giving a complexity-theoretic proxy for symmetry restoration.
  • Crossings in shifted symmetric complexity at fixed disorder W rule out the trivial explanation that the effect is an artifact of comparing different phases.
  • The sudden reordering of the first Lanczos coefficient near W ≈ 4 for Néel states acts as an early microscopic indicator of the effect.
  • Because spread complexity needs no arbitrary subsystem choice, it offers a global diagnostic complementary to entanglement asymmetry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The t=0 predictor is likely generic: any positive-definite operator projected onto symmetry sectors defines a structural complexity, so the same construction could diagnose Mpemba-like relaxation for other symmetries and models, such as XXZ chains or random circuits.
  • The paper's observation that larger tilts can increase both entanglement asymmetry and overlap with the thermal state suggests that entanglement asymmetry alone can mislabel which state is 'further from equilibrium'; structural complexity may be a more faithful ordering in strongly thermalizing regimes.
  • If the single-realisation robustness holds, structural complexity could be measured in cold-atom or trapped-ion experiments with quasiperiodic potentials by probing the Krylov expansion, turning a theoretical diagnostic into an observable.
  • The fact that crossings appear in the symmetric part, not the asymmetric part, invites reading coherence as a resource whose preparation cost is absorbed into the initial symmetric offset—an interpretation the paper sketches but does not develop.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript studies Krylov spread complexity in the Aubry-André model with a global U(1) symmetry. It defines projected symmetric and asymmetric complexity operators (3.1)-(3.2), shows C_K = C_K,S + C_K,A, and derives Eq. (3.3): at t=0 the symmetric and asymmetric expectation values are equal in magnitude and opposite in sign, with C_K,S(0) = -C_K,A(0) >= 0. The decomposition is numerically verified against the total complexity, and the asymmetric complexity is shown to decay toward zero. The paper then identifies Mpemba-like crossings in the shifted symmetric complexity C_K,S(t)-C_K,S(0), and defines the t=0 structural complexity C_structural = |C_K,A(0)| = C_K,S(0) (Eq. 4.1). The central new claim is that this t=0 quantity predicts whether a pair of tilted initial states exhibits a quantum Mpemba effect, based on a single-realization structural-complexity surface (Fig. 8) whose shape changes from two peaks to one peak near W≈2.5. Lanczos coefficients, especially b1, are also proposed as early predictors.

Significance. If the t=0 predictive claim were established, it would be a significant and practical result: a static state-functional diagnosing Mpemba-like inversions without time propagation. The decomposition itself is genuinely interesting, and Eq. (3.3) is a clean, parameter-free identity. The numerical consistency checks (C_K,S + C_K,A = C_K, and the bootstrap finite-size scaling in App. A) are strengths. However, the headline predictive claim is currently under-supported: it relies on a single disorder realization, is in-sample, and is only shown as a global shape change in the (θ,W) plane. The paper is also honest about important limitations—the phase-structure confound in Sec. 3.3 and the computational equivalence to time evolution in Sec. 4.1—but these acknowledgments undermine the abstract's stronger statements. The work is promising and publishable after substantial revision, but the central claim needs more evidence.

major comments (4)
  1. [§4.1, Fig. 8] The central predictive claim rests on a single realization (ϕ=0). The QME baseline in Fig. 1 is averaged over 240 realizations, but Fig. 8 is one realization with no error bars or distribution. The text says the behavior 'appears to be a robust feature of almost all realisations,' but no supporting plot or quantitative test is given; App. D.5 concedes that the data are 'susceptible to realisation-specific fluctuations.' This is load-bearing: the two-peak-to-one-peak transition could be an artifact of one phase realization. Please provide an ensemble-averaged structural-complexity surface with error bars or quantiles, and show that the crossover near W≈2.5 is statistically significant.
  2. [§4.1, 'necessary but insufficient'] The argument for predictive power is logically inconsistent. The text first says structural complexity may be predictive 'in the form of some necessary but insufficient condition,' then concludes 'This implies that the structural complexity may be used to predict whether a pair of states will exhibit a Mpemba effect.' A necessary-but-insufficient condition cannot predict occurrence; it can only rule out some cases. Moreover, the reported signal is a global shape change (two peaks → one peak) located near W≈2.5, the same W-regime where EA crossings begin. This is at best a phase/regime marker, not a demonstrated pair-level classifier. Show, for fixed W, how C_structural ranks crossing vs non-crossing tilt pairs and report a classification statistic (e.g., sensitivity/specificity or AUC).
  3. [Abstract and §4.1] The abstract's claim that the t=0 structure 'obviat[es] the need for explicit time evolution' is contradicted by §4.1, which states that obtaining the complete Krylov basis is 'effectively equivalent to computing the time evolution ... to arbitrarily late times' and 'places significant computational cost.' The structural complexity is dynamics-informed and expensive to evaluate; only the late-time propagation is avoided. Please reframe the claim or demonstrate a cheaper proxy (e.g., truncating the Krylov basis or using early Lanczos coefficients) that preserves predictive power.
  4. [§3.3] The crossings in C_K,S(t)-C_K,S(0) are the main dynamical 'Mpemba' signature, yet §3.3 acknowledges that it 'remains unclear whether the observed crossings are uniquely attributable to the quantum Mpemba effect, or whether they reflect a more generic sensitivity of operator dynamics to phase structure.' Since TFS cross at all W and TNS only near/above W_c, the pattern could track the ETH/MBL transition rather than the EA-defined QME. Please specify a null model (e.g., crossings among random same-phase pairs, or conditioning on W) and test whether symmetric-complexity crossings contain information beyond phase membership.
minor comments (5)
  1. [Eq. (3.6)] The summation index 'd_q' in the symmetry-resolved operator is not defined; presumably it is the dimension of sector q. Please define it explicitly.
  2. [References] References [2] and [20] appear to be the same work (The quantum Mpemba effects), and [19] lacks an arXiv identifier. Please consolidate and standardize.
  3. [Table 1] The footnote about unreliable entries is given in the caption, but the table body marks only the θ=0.1 column with '*'; make explicit which W/θ entries should be treated with caution.
  4. [Appendix D.2] Figures 19–21 use inconsistent labels ('K_diff(t)-K_diff(0)' vs 'C_K,S(t)-C_K,S(0)' and 'C(t)'). Align notation with the main text.
  5. [Fig. 8 caption] The single-realization nature (ϕ=0) is only in the caption; the main text should state it clearly where the predictive claim is made, since all other main-text data are averaged.

Circularity Check

0 steps flagged · score 2.0 of 10

No by-construction circularity: the projective-complexity decomposition is a definitional identity, and the t=0 structural complexity is an independent input. The predictive claim is weakened by in-sample calibration and the admitted Krylov/time-evolution equivalence, but these are support limitations, not circular reductions.

full rationale

The paper's formal chain is not circular. The projected symmetric and asymmetric complexity operators are defined in Eqs. (3.1)-(3.2), and the identity C_K = C_K,S + C_K,A together with Eq. (3.3) (⟨ψ|C_K,S|ψ⟩ = -⟨ψ|C_K,A|ψ⟩ at t=0) is a definitional consistency condition, not a derivation of the Mpemba effect. The structural complexity C_structural = |C_K,A(0)| = C_K,S(0) is a genuine t=0 quantity computed from the Hamiltonian, the initial tilted state, and the symmetry projectors; it is not fitted to the entanglement-asymmetry crossings. The claimed predictive power is an observed correlation: the TNS structural-complexity surface becomes singly peaked near W≈2.5, the same regime where EA crossings begin. That is an in-sample, post-hoc threshold, and Fig. 8 is a single realization, so the predictive claim is under-supported—but it is not equivalent to its input by construction. No equation maps EA crossing existence to a structural-complexity value. Self-citations (e.g., ref. [11]) are not load-bearing; the QME baseline [19] is reproduced independently, and the ETH/MBL critical point is verified via finite-size scaling. The paper itself flags the key limitations: Sec. 3.3 concedes that the observed crossings may reflect phase-structure sensitivity rather than uniquely the QME, and Sec. 4.1 admits that constructing the full Krylov basis is effectively equivalent to computing time evolution to arbitrarily late times, which undercuts the abstract's 'obviating the need for explicit time evolution' wording. These are overclaims and missing out-of-sample tests, but they do not amount to a circular derivation. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to the central claims: the model parameters V=1/2, N=12, N_A=3, and 240 realisations are fixed by the numerical setup, and W_c is estimated independently via finite-size scaling of level statistics. The projected symmetric/asymmetric operators and structural complexity are derived quantities, not new physical entities. The axioms are standard Krylov-space mathematics plus model-specific and interpretational assumptions, which are stated or cited.

assumptions (5)
  • standard math The Lanczos algorithm constructs a complete orthonormal basis for the Krylov subspace generated by H and the reference state.
    Invoked throughout to define spread complexity and the projected operators; standard linear algebra.
  • domain assumption The Aubry-André Hamiltonian conserves total magnetization, so the Hilbert space decomposes into U(1) charge sectors.
    Used in Eqs. (1.3), (3.1), (3.2) to define the symmetry projectors. This is a property of the model, not proven in the paper.
  • domain assumption The tilt operator U(theta) = exp(-i theta/2 sum sigma^y_i) monotonically increases the initial entanglement asymmetry for theta in [0, pi/2].
    Section 2 states this monotonicity and uses it to label states as more or less asymmetric. It is taken from prior work on entanglement asymmetry, not derived here.
  • domain assumption Vanishing entanglement asymmetry is necessary but not sufficient for thermal equilibrium, so asymmetry crossings can serve as a QME diagnostic.
    Section 2 and Section 4.3 discuss this; it underlies the choice of entanglement asymmetry as the baseline QME definition.
  • domain assumption The average over 240 phase realisations is representative, and realisation-averaged coherences vanish in the MBL phase due to random phases rather than thermalization.
    Section 3.2 uses this to interpret the decay of C_K,A in the MBL regime; it is an ensemble-averaging assumption about the model.

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Cite this review

Pith. "Pith review of (A)Symmetric Complexity and the Quantum Mpemba Effect." pith.science (2026). https://pith.science/paper/X45OWHCY

@misc{pith2026250908078,
  author       = {Pith},
  title        = {Pith review of: (A)Symmetric Complexity and the Quantum Mpemba Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X45OWHCY}},
  note         = {Machine review of arXiv:2509.08078}
}
abstract

The Quantum Mpemba Effect (QME) -- the counter-intuitive phenomenon where states further from equilibrium can relax faster than those closer to it -- challenges standard expectations of quantum thermalization. In this work, we introduce Krylov complexity as a sensitive diagnostic for the QME. We show that Krylov spread complexity encodes the asymmetry essential to the effect, and we define a new class of projective (a)symmetric complexities that sharpen this connection. Strikingly, the structure of these projective complexities at the initial moment ($t=0$) already carries predictive power for the onset of Mpemba-like inversions, obviating the need for explicit time evolution. Our results suggest that the geometry of states in Krylov space captures deep information about non-monotonic relaxation and provides a powerful framework for diagnosing and anticipating anomalous thermalization phenomena in quantum systems.

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Reference graph

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    Diagonalize the Hamiltonian to find the complete set of energy eigenstates{|E i⟩} and eigenvalues{E i}

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    The dotted line is used to indicateW= 1.0, which corresponds to the lowest value of the potential strength for which the N´ eel state has a monotonically decreasing divergence

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Reviewed August 4, 2026 · model on record in the stance chip above.