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A resource theoretical unification of Mpemba effects: classical and quantum

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that thermal and symmetry Mpemba effects are two faces of one resource-theoretic mechanism, resting on a relative-entropy splitting identity.

desk verdict A clean unifying decomposition with one honest proof gap in the asymptotic analysis; deserves a serious referee. read the letter →

arxiv 2507.16976 v2 pith:ITJ5QHPD submitted 2025-07-22 quant-ph cond-mat.stat-mechphysics.class-ph

Alessandro Summer , Mattia Moroder , Laetitia P. Bettmann , Xhek Turkeshi , Iman Marvian , John Goold This is my paper
classification quant-phcond-mat.stat-mechphysics.class-ph
keywords Mpembaeffectresourcetheoriesathermalityasymmetrymodesofrelativeentropysymmetryrestorationquantumthermodynamics
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 the thermal Mpemba effect and the symmetry Mpemba effect are not separate anomalies but two instances of one resource-theoretic phenomenon. The argument turns on a relative-entropy identity: for any symmetry group action with twirling map $G$, and any $G$-invariant steady state $\pi$, $S(\rho\|\pi)=S(\rho\|G[\rho])+S(G[\rho]\|\pi)$. This splits the resource behind thermal relaxation, athermality, into a symmetry-breaking part and a symmetry-preserving part, so that both effects become crossings of resource monotones. If the argument is right, the kind of Mpemba behavior seen in a given experiment is fixed by the chosen measure of distance from equilibrium rather than by the system's classical or quantum nature.

What carries the argument

The central object is the twirling (symmetrization) map $G[\rho]=\int_G dg\, U_g\,\rho\,U_g^\dagger$, together with the modes of asymmetry, a harmonic-analysis decomposition of the operator algebra into sectors labelled by irreducible representations $\mu$ of the symmetry group. The identity that carries the argument is $S(\rho\|\pi)=S(\rho\|G[\rho])+S(G[\rho]\|\pi)$, valid when $\pi$ is $G$-invariant. It does the work of separating the total resource into a symmetry-breaking contribution and a symmetry-preserving contribution, and the mode decomposition explains why the two contributions relax at different rates: a $G$-covariant map cannot mix modes, so the slowest mode in a nonzero-$\mu$ sector sets the asymmetry decay rate.

What would settle it

Numerically compute $\ln S(\rho(t)\|G[\rho(t)])$ at late times for a model in which the twirled state itself relaxes on a timescale comparable to the slowest symmetry-breaking mode, for instance the Z$_4$ quantum chain of Section III E with the initial state placed entirely on the slowest $\mu\neq 0$ eigenmode. The paper's asymptotic claim predicts slope $2\,\mathrm{Re}(\lambda_{\rm slow})$; if the observed slope differs, the replacement of the fixed steady state $\pi$ by the time-dependent $G[\rho(t)]$ in the Appendix C derivation is invalid.

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Extended reading notes

Core claim

The paper's central claim is that both thermal and symmetry Mpemba effects are governed by the overlap of the initial state with the slowest decaying mode appropriate to the resource being measured. For thermalization this is the slowest Liouvillian eigenmode; for symmetry restoration it is the slowest eigenmode living in a nonzero asymmetry sector, and the modes of asymmetry framework shows that every $G$-covariant map acts independently on those sectors. The load-bearing identity is Eq. (81): $S(\rho\|\pi)=S(\rho\|G[\rho])+S(G[\rho]\|\pi)$ for any $G$-invariant $\pi$. Consequently the relative entropy of athermality equals the relative entropy of asymmetry plus the relative entropy of athermality of the symmetrized state, and the same split applies to non-stationarity when $\pi$ is the nonthermal fixed point. The paper supports the unification with a classical Z$_4$ Markov chain, quantum Z$_4$ Lindblad dynamics, Davies maps, and Markovian and non-Markovian circuits with U(1) and SU(2) symmetries, including the first classical example of a symmetry Mpemba effect.

Load-bearing premise

The whole symmetry-Mpemba prediction rests on an unproved substitution: the asymptotic decay formula derived for a fixed steady state $\pi$ is applied with the time-dependent reference $G[\rho(t)]$, and if that step fails, the slowest-symmetry-mode explanation has no rigorous basis.

Editorial extensions

If this is right

  • For a Davies map, the quantum thermal Mpemba effect splits into a classical population part and a quantum coherence part (time-translation asymmetry), and either part can produce a crossing independently.
  • A crossing in either component entropy is necessary for a crossing in the total relative entropy; when both components cross, the total is guaranteed to cross.
  • Strong symmetry Mpemba effects can be engineered by preparing states with vanishing overlap with the slowest symmetry-restoring mode, as the tilted block-product states do in U(1)-symmetric circuits.
  • Because the split holds for any $G$-invariant steady state, the unification also covers non-stationarity with nonthermal fixed points, connecting the framework to entropy production.
  • The observed Mpemba phenomenon depends on the chosen resource monotone; the same thermalizing model can show a classical thermal, quantum thermal, or symmetry Mpemba effect under different choices.

Reading between the lines

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

  • The paper proves the additivity identity for the quantum relative entropy; extending the same split to Petz-Rényi $\alpha$-divergences would require a separate argument, so the unification is anchored to the $\alpha=1$ measure.
  • Because the mode-sector argument relies only on covariance, the same slowest-sector-mode recipe should predict Mpemba-speedup states for other group symmetries, such as lattice translation symmetry or parity, where the paper notes parity studies have not yet found an unambiguous effect.
  • A predictive use the authors leave implicit: given a generator and a symmetry, one could compute sector-resolved spectra and initial mode occupations to screen for fast-equilibrating states before running the full dynamics.
  • The classical Z$_4$ example suggests tabletop classical experiments with rotationally symmetric Markov systems could test symmetry Mpemba behavior without quantum hardware.
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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

2 major / 4 minor

Summary. The paper proposes a resource-theoretic unification of thermal and symmetry Mpemba effects. After reviewing resource theories, it derives the central identity Eq. (81): for any G-invariant state pi, S(rho||pi) = S(rho||G[rho]) + S(G[rho]||pi), splitting the relative entropy of athermality (or non-stationarity) into a symmetry-breaking part, the relative entropy of asymmetry, and a symmetry-respecting part. The paper then uses the modes-of-asymmetry formalism to argue that thermal Mpemba is governed by the slowest overall Liouvillian eigenmode while symmetry Mpemba is governed by the slowest mode in a nonzero symmetry sector. This is supported by several examples: a classical Z4 Markov chain, a Z4 Lindblad model, Davies-map thermalization with time-translation symmetry, Markovian and non-Markovian random circuits with U(1) and SU(2) symmetries, and an ETH-based unitary example. The paper also emphasizes that the presence and crossing time of the effect depend on the chosen monotone. The central identity is cleanly proved, but the theoretical derivation of the asymptotic decay of S(rho(t)||G[rho(t)]) is not fully established, and one spectral assumption in the random-circuit example is empirical rather than proven.

Significance. If the asymptotic step is made rigorous, the paper provides a genuinely unifying principle: Mpemba effects are crossings of resource monotones, and the relative entropy decomposes exactly into an asymmetry part and a symmetry-respecting athermality part. Equation (81) is a valuable parameter-free identity with broad applicability, and the use of modes of asymmetry to identify symmetry Mpemba with overlap with the slowest symmetry-restoring mode is a compelling and testable framework. The examples are consistent with the formalism, including what appears to be the first classical symmetry Mpemba effect, and the paper is careful to illustrate monotone dependence. However, the advertised governing principle currently rests on an unproved replacement of the fixed reference state by a time-dependent one in Appendix C, and on an observed rather than proven decay-rate ordering in random U(1) circuits. Both points are load-bearing for the central claim, so the paper is not yet fully self-contained, though the issues appear repairable.

major comments (2)
  1. [Appendix C, after Eq. (16) and Eq. (C14); Section IV] The asymptotic analysis of the asymmetry monotone is not proved. Appendix C derives the decay of D(p(t)||pi) and S(rho(t)||pi) for a fixed, full-rank reference state pi, and then states, after Eq. (16), that 'Replacing pi with G[p(t)] allows to extend this to the asymptotic behaviour of the relative entropy of asymmetry.' The quantum part similarly says, after Eq. (C14), that the proof extends 'by replacing pi with G[rho(t)].' This replacement is exactly the load-bearing step for the claim in Section IV that symmetry Mpemba is governed by the slowest symmetry-restoring mode. Since G[rho(t)] is itself relaxing and includes the full mu=0 sector, one must prove that the linear term in delta = rho - G[rho] vanishes (for example by G-invariance of ln G[rho]) and that the time dependence of (G[rho(t)])^{-1} changes only the prefactor and not the exponent 2 Re(lambda_j) for the slowest mu != 0 mode with nonzero overlap. Neither of these points is shown. Please provide a proof or state the required assumptions explicitly; without this, the central theoretical conclusion is an assertion rather than a theorem.
  2. [Section III G 1 and Fig. 9] The strong symmetry Mpemba effect in random U(1) circuits relies on the spectral ordering Re(lambda_mu) < Re(lambda_mu') for |mu| > |mu'|, which the text introduces as a numerical observation with heuristic mechanisms rather than as a proven statement. Because this ordering is a nontrivial property of the random circuit ensemble and is used to justify the claim that avoiding low-|mu| sectors produces a strong Mpemba effect, the conclusion should be labeled as conditional on the observed ordering, or the ordering should be supported by an analytic argument or by a precise statistical statement with a quantified failure probability over the circuit ensemble. The other examples in the paper do not depend on this assumption.
minor comments (4)
  1. [Appendix C 2, paragraph after Eq. (C9)] In the paragraph after Eq. (C9), the text refers to the slowest decaying eigenmode as 'j = 1' for athermality, but with the ordering 0 = Re(lambda_1) > Re(lambda_2) the slowest decaying mode is j = 2; lambda_1 is the steady state. Please correct this index.
  2. [Section V] The conclusion contains a typo: 'thte restoration' should read 'the restoration.'
  3. [Section III G 1, Eq. (76)] In the definition k(mu) = arg max_{k: ell_k in H_s^(mu)} s Re(lambda_k), the stray 's' before Re appears to be a typographical artifact and should be removed.
  4. [Fig. 9 caption and surrounding text] The text says the inset shows a 'clear, monotonic decrease' of Re(lambda_mu) with |mu|, but the panel displays the mean and spread over circuit realizations; please state explicitly that the monotonicity holds on average rather than for every realization, or quantify the probability of exceptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (81) is derived in-text from twirling properties, and the numerical demonstrations are self-contained; the Appendix C replacement step is an unproven extension, not a circular reduction.

full rationale

The paper's central identity Eq. (81) is derived directly from G-invariance of pi, idempotence and self-adjointness of the twirl G, and cyclicity of the trace; it is not assumed from the Mpemba conclusion. The resource-theoretic language (free states, monotones, and defining Mpemba as a monotone crossing) is definitional framing, but the specific claims about decay rates are supported by spectral expansions of the Liouvillian (Eqs. (15)-(16), (22)-(23)) and by numerical simulations. Citations to Marvian-Spekkens for modes of asymmetry and the relative entropy of asymmetry are to independent, published formalism, not to a result that presupposes the Mpemba effect. The only weak point is Appendix C, where the asymptotic formula for S(rho(t)||pi) is extended to S(rho(t)||G[rho(t)]) by the sentence 'Replacing pi with G[p(t)] allows to extend this...' without proof; since G[rho(t)] is time-dependent, this is a gap in the derivation. However, a proof gap or unjustified approximation is not circularity under the stated criteria: no equation is reduced to its own input and no fitted parameter is renamed as a prediction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameter is fitted to data; the numerical constants in the examples are illustrative Hamiltonian and circuit parameters. The main assumptions are the standard properties of twirling, diagonalizability of the generators, support conditions for relative entropies, and two ad hoc assumptions used to justify the symmetry-sector slowdown arguments.

assumptions (5)
  • standard math Haar twirling G is a self-adjoint, trace-preserving projection onto G-invariant operators for compact groups.
    Used in Eqs. (34)-(35) and in the decomposition proof Eqs. (79)-(81). Self-adjointness under the Hilbert-Schmidt inner product is what licenses moving G from ρ onto ln π.
  • domain assumption The Lindblad generators in the examples are diagonalizable with a non-degenerate slowest eigenmode.
    The asymptotic formulas in Appendix C assume non-degenerate λ_j and a biorthonormal eigenmode expansion; degenerate or defective spectra would change the relaxation exponents.
  • domain assumption States have sufficient support (π full rank, compatible supports) for relative entropies and the quadratic late-time expansion to be finite.
    Required in Appendices C1 and C2 for S(ρ||π) and S(ρ||G[ρ]); otherwise the asymptotic expansion Eq. (C11) may diverge or need regularization.
  • ad hoc to paper In random U(1) circuits, sectors with larger |µ| decay faster, so Re λ_µ < Re λ_µ' for |µ| > |µ'|.
    This empirical ordering is the mechanism behind the strong symmetry Mpemba examples in Section III G 1, but the paper states that 'a more thorough explanation of this phenomenon requires further investigation'.
  • ad hoc to paper The time-dependent twirled state G[ρ(t)] can be replaced by the fixed point π when evaluating the asymptotic decay of S(ρ(t)||G[ρ(t)]).
    Stated in Appendix C after Eq. (16) without proof; this is the load-bearing extension linking symmetry Mpemba to the slowest asymmetry mode.
invented entities (1)
  • Resource theory of symmetry-respecting athermality
    purpose: To split the total athermality (or non-stationarity) monotone into a symmetry-breaking part S(ρ||G[ρ]) and a symmetry-preserving part S(G[ρ]||π), as defined in footnote [92].
    This is a mathematical construct introduced in the paper rather than a physically falsifiable entity; it has no observable outside the framework, but it is not used to smuggle in empirical parameters.

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Cite this review

Pith. "Pith review of A resource theoretical unification of Mpemba effects: classical and quantum." pith.science (2026). https://pith.science/paper/ITJ5QHPD

@misc{pith2026250716976,
  author       = {Pith},
  title        = {Pith review of: A resource theoretical unification of Mpemba effects: classical and quantum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ITJ5QHPD}},
  note         = {Machine review of arXiv:2507.16976}
}
read the original abstract

The Mpemba effect originally referred to the observation that, under certain thermalizing dynamics, initially hotter samples can cool faster than colder ones. This effect has since been generalized to other anomalous relaxation behaviors even beyond classical domains, such as symmetry restoration in quantum systems. This work demonstrates that resource theories, widely employed in information theory, provide a unified organizing principle to frame Mpemba physics. We show how the conventional thermal Mpemba effect arises naturally from the resource theory of athermality, while its symmetry-restoring counterpart is fully captured by the resource theories of asymmetry. Leveraging the framework of modes of asymmetry, we demonstrate that the Mpemba effect due to symmetry restoration is governed by the initial overlap with the slowest symmetry-restoring mode -- mirroring the role of the slowest Liouvillian eigenmode in thermal Mpemba dynamics. Through this resource-theoretical formalism, we uncover the connection between these seemingly disparate effects and show that the dynamics of thermalization naturally splits into a symmetry-respecting and a symmetry-breaking term.

Figures

Figures reproduced from arXiv: 2507.16976 by the authors.

Figure 1
Figure 1. FIG. 1. In a resource-theoretic framework, the Mpemba ef [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The thermal Mpemba effect in an open classical sys [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The thermal Mpemba effect in an open quantum [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. U(1) symmetry sectors and mode-occupancy distribu [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) the generator of the classical Markov chain with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Symmetry quantum Mpemba effect in a system with [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Symmetry Mpemba effect in the thermalization of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Circuit schematics for the protocols of Section [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Symmetry Mpemba effect for SU(2). The initial [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Splitting of the relative entropy of athermality for [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Different Mpemba effects in the same thermaliza [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

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Forward citations

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.