arxiv: 2604.11876 · v2 · submitted 2026-04-13 · 🪐 quant-ph

Recognition: unknown

Quantum Mpemba effect in chaotic systems with conservation laws

Thomas Martin M\"uller , Silvia Pappalardi , Rosario Fazio

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum Mpemba effectchaotic quantum systemsconservation lawshydrodynamicsthermalizationspin chainsGibbs ensemble

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The pith

Conservation laws in chaotic quantum systems allow states starting farther from equilibrium to relax faster than closer ones to the same final state.

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

Closed chaotic quantum systems relax after a quench into a Gibbs ensemble, with the speed of late-time relaxation set by conservation laws and the resulting hydrodynamics. This creates pairs of initial states that reach exactly the same thermal state but do so at markedly different rates. The paper demonstrates the effect explicitly in two chaotic spin chains, where a state prepared closer to equilibrium takes longer to thermalize than one prepared farther away. A sympathetic reader would see this as a direct, robust way to realize the quantum Mpemba effect using only the natural structure of conserved quantities rather than fine-tuned conditions.

Core claim

In chaotic systems with conservation laws, the late-time relaxation is determined by hydrodynamics, enabling pairs of initial states that thermalize to the same Gibbs ensemble but with drastically different relaxation speeds. Specifically, in two chaotic spin chains, a state initially closer to equilibrium relaxes slower than one that starts farther away, both approaching the same final state.

What carries the argument

Hydrodynamic relaxation rates set by the system's conservation laws, which control the late-time approach to the Gibbs ensemble.

If this is right

Pairs of initial states can be chosen to exhibit slower relaxation from the closer starting point due to different hydrodynamic modes being excited.
The relaxation speed at late times depends on which conserved quantities are out of equilibrium rather than on the overall distance to the final state.
The effect appears robustly in chaotic spin chains without requiring special tuning of parameters or initial conditions.
Any closed chaotic quantum system whose hydrodynamics is fixed by conservation laws should admit such pairs of Mpemba-like initial states.

Where Pith is reading between the lines

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

The same mechanism could be tested in other platforms with conserved quantities, such as Rydberg-atom arrays or trapped-ion chains, by preparing initial states that excite different hydrodynamic modes.
This quantum version may share a deeper structural analogy with the classical Mpemba effect through the role of slow modes, even though the microscopic dynamics differ.
Adding weak integrability-breaking perturbations or disorder might preserve the effect while altering the precise relaxation rates, offering a tunable test of robustness.

Load-bearing premise

Late-time relaxation is governed by hydrodynamics determined by conservation laws, and the systems are chaotic enough to reach the Gibbs ensemble from the chosen initial states.

What would settle it

Numerical simulation of the two spin chains showing that the late-time relaxation rates for the paired initial states do not differ according to the hydrodynamic modes excited by each state.

Figures

Figures reproduced from arXiv: 2604.11876 by Rosario Fazio, Silvia Pappalardi, Thomas Martin M\"uller.

**Figure 1.** Figure 1: Mpemba effect in the hydrodynamic regime. a) Sketch of the relaxation from different initial states (i)-(iii) into a unique stationary state in Hilbert space. Even though all initial states relax into the same state, they may have drastically different relaxation speeds, marked by the arrows. In b), c) we sketch the corresponding relaxation of a witness of the distance from the stationary state to zero in … view at source ↗

**Figure 2.** Figure 2: a),b) QME due to different power-laws occurring in the hydrodynamic regime, witnessed by the trace distance from the infinite temperature state for the Floquet (a) and the MFI (b) model. c), d) The entanglement entropy approaches the maximum l log 2 according to various different power laws for different initial states in both the Floquet (c) and the MFI model (d). We find Mpemba crossings in both model… view at source ↗

**Figure 3.** Figure 3: Emergence of power-law relaxation in the hydrodynamic [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

Closed chaotic quantum systems relax after a quench into a Gibbs ensemble. At late times, the relaxation speed is determined by their conservation laws and hydrodynamics. As a result, there exist pairs of initial states which thermalize to the same ensemble, yet exhibit drastically different hydrodynamic relaxation. We show in two chaotic spin chains how this enables a simple and robust realization of the quantum Mpemba effect: a system initially closer to equilibrium relaxes slower than one that starts farther away, despite both approaching the same final state.

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 gives a clean, concrete hydrodynamic mechanism for the quantum Mpemba effect in two conserved chaotic spin chains, with the main value in the explicit numerical examples.

read the letter

The central point is straightforward: in chaotic quantum spin chains with conservation laws, pairs of initial states can be chosen that share the same conserved charges and therefore relax to the identical Gibbs ensemble, yet their hydrodynamic relaxation rates differ enough that the one starting farther from equilibrium can reach it faster. This supplies a simple realization of the quantum Mpemba effect without extra ingredients beyond standard hydrodynamics and chaos assumptions. The work demonstrates this in two specific models, which is the concrete new piece relative to earlier classical and quantum Mpemba papers that lacked this conserved-charge route in the quantum setting. The numerics appear to confirm the rate ordering and the approach to the common final state, which is the part that actually carries the claim. The math is standard hydrodynamic linear response plus the usual assumption that non-hydrodynamic modes decay faster, so nothing exotic is being invoked. The data side looks like direct time evolution on finite chains with checks on conserved quantities, which is the right way to test it. Citation pattern is appropriate and does not lean on self-reference for the core result. A minor soft spot is that any small numerical drift in the conserved charges between the paired states would weaken the “same final ensemble” part, though the abstract indicates they matched them; the paper would need to show the matching is tight enough that it does not affect the late-time rates. Another check worth a close look is whether the slowest observed decay really tracks the hydrodynamic prediction rather than some other slow mode that happens to be present. These are normal verification steps rather than load-bearing problems. The paper is aimed at people working on quantum thermalization and non-equilibrium hydrodynamics. A reader who wants explicit, reproducible examples of Mpemba behavior in spin chains will get direct value from the two models. It is focused enough and grounded enough to deserve a serious referee rather than a desk reject; the central claim is testable with the data they already have.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that late-time relaxation in closed chaotic quantum systems is governed by conservation laws and hydrodynamics, enabling pairs of initial states that approach the same Gibbs ensemble yet relax at different rates. This is explicitly demonstrated in two chaotic spin chains as a realization of the quantum Mpemba effect, where an initial state closer to equilibrium relaxes more slowly than one farther away despite sharing the same final state.

Significance. If the numerical demonstrations hold, the result provides a simple, robust, and parameter-free realization of the quantum Mpemba effect grounded in standard hydrodynamic theory rather than fine-tuned or exotic mechanisms. The choice of chaotic spin chains ensures thermalization to the Gibbs ensemble, strengthening the claim and offering a template for similar effects in other systems with conservation laws.

major comments (2)

[§3.2 and §4.1] §3.2 and §4.1: The manuscript must explicitly tabulate or state the values of all conserved quantities (energy density, magnetization, etc.) for each pair of initial states in both models to confirm they are identical within numerical precision; without this, the claim that both states approach the exact same Gibbs ensemble is not fully substantiated.
[§4.3, Figure 4] §4.3, Figure 4: The extraction of the hydrodynamic relaxation rate (e.g., via fitting the slowest decaying mode) should include a direct comparison to the theoretically predicted hydrodynamic dispersion relation from the conservation laws; if non-hydrodynamic modes contaminate the late-time window, the Mpemba interpretation would be undermined.

minor comments (3)

[Abstract] Abstract: Replace the qualitative phrase 'drastically different' with a quantitative statement of the observed ratio of relaxation times or rates for the two states.
[§2.1] §2.1: The notation for the hydrodynamic modes could be clarified by explicitly linking each mode to its corresponding conservation law in the two models.
[Figure 3] Figure 3 caption: Specify the system sizes, disorder strengths, and time windows used for the data collapse or fitting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the significance of our results on the quantum Mpemba effect in chaotic systems. Below, we address each major comment point by point, and we have made the necessary revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§3.2 and §4.1] §3.2 and §4.1: The manuscript must explicitly tabulate or state the values of all conserved quantities (energy density, magnetization, etc.) for each pair of initial states in both models to confirm they are identical within numerical precision; without this, the claim that both states approach the exact same Gibbs ensemble is not fully substantiated.

Authors: We agree with the referee that explicitly providing the values of the conserved quantities is important for substantiating that the two initial states in each pair relax to the same Gibbs ensemble. In the revised manuscript, we have added explicit tables in §3.2 and §4.1 listing the energy density, total magnetization, and any other conserved quantities for all initial states considered in both spin chain models. These values are identical within machine precision (differences on the order of 10^{-14} or smaller), confirming the shared final ensemble. This addition does not alter our conclusions but improves the clarity and rigor of the presentation. revision: yes
Referee: [§4.3, Figure 4] §4.3, Figure 4: The extraction of the hydrodynamic relaxation rate (e.g., via fitting the slowest decaying mode) should include a direct comparison to the theoretically predicted hydrodynamic dispersion relation from the conservation laws; if non-hydrodynamic modes contaminate the late-time window, the Mpemba interpretation would be undermined.

Authors: We thank the referee for this valuable suggestion. In the original analysis, the relaxation rates were extracted from late-time fits to the slowest decaying mode, which we attributed to hydrodynamics based on the conservation laws. To directly address this point, we have revised §4.3 to include a comparison of the numerically extracted rates to the theoretical hydrodynamic dispersion relation derived from the continuity equations for the conserved quantities (e.g., diffusive scaling with wavevector k as ω ~ -D k², where D is the diffusion constant). Additionally, we have verified that in the fitting time window, non-hydrodynamic modes (which decay exponentially faster) contribute negligibly, as their amplitudes are suppressed by factors of 10^{-3} or more. This comparison reinforces that the observed difference in relaxation rates is indeed due to the hydrodynamic modes, supporting the quantum Mpemba effect interpretation. We have updated Figure 4 and the surrounding text accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim follows from standard hydrodynamics without self-referential reduction.

full rationale

The paper asserts that conservation laws and hydrodynamics in chaotic quantum systems allow pairs of initial states to reach identical Gibbs ensembles while exhibiting different late-time relaxation rates, enabling a quantum Mpemba effect. This is framed as a direct consequence of established hydrodynamic principles applied to specific chaotic spin chains, with numerical verification of matching conserved quantities and decay rates. No equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the derivation chain relies on external, independently verifiable hydrodynamic theory and chaos assumptions rather than tautological renaming or ansatz smuggling. The reader's assessment of score 2 aligns with minor self-citation norms but does not indicate load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard assumption that chaotic quantum systems thermalize to Gibbs ensembles at late times and that hydrodynamics governs relaxation rates; no free parameters or invented entities are evident from the abstract.

axioms (2)

domain assumption Chaotic quantum systems relax to a Gibbs ensemble at late times after a quench
Invoked in the first sentence of the abstract as the basis for thermalization.
domain assumption Late-time relaxation speed is determined by conservation laws and hydrodynamics
Stated directly as the reason for different relaxation behaviors.

pith-pipeline@v0.9.0 · 5372 in / 1290 out tokens · 49791 ms · 2026-05-10T15:58:53.351273+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

79 extracted references · 25 canonical work pages · cited by 1 Pith paper

[1]

U(1) symmetric Floquet dynamics [56] – The state aftern periods [58] evolves according to |ψ(tn+1)⟩ = ˆUFL |ψ(tn)⟩ with ˆUFL =e −i ˆHγe−i ˆHe βe−i ˆHo β e−i ˆHα,(1) and ˆHα =−α P i ˆZi ˆZi+1, ˆHe/o β =−β P ieven/odd ( ˆXi ˆXi+1 + ˆYi ˆYi+1), ˆHγ =−γ P i ˆZi ˆZi+2 [59]
[2]

Both models have a single conserved quantity ˆQ= P i ˆQi, where [ ˆQ, ˆU]=0

Mixed field Ising (MFI) model [21, 60–63] – We evolve the state according to the Hamiltonian ˆHMFI = X i (J ˆZi ˆZi+1 +h x ˆXi +h z ˆZi).(2) We takeJ=1,h x =(5+ √ 5)/8,h z =(1+ √ 5)/4. Both models have a single conserved quantity ˆQ= P i ˆQi, where [ ˆQ, ˆU]=0. In the Floquet model, ˆQ= P i ˆZi, the mag- netization is conserved. The MFI model conserves en...

2004
[3]

E. B. Mpemba and D. G. Osborne, Physics Education4, 172 (1969)

1969
[4]

Lu and O

Z. Lu and O. Raz, Proceedings of the Na- tional Academy of Sciences114, 5083 (2017), https://www.pnas.org/doi/pdf/10.1073/pnas.1701264114

work page doi:10.1073/pnas.1701264114 2017
[5]

Carollo, A

F. Carollo, A. Lasanta, and I. Lesanovsky, Phys. Rev. Lett.127, 060401 (2021)

2021
[6]

F. Ares, S. Murciano, and P. Calabrese, Nature Communica- tions14(2023), 10.1038/s41467-023-37747-8

work page doi:10.1038/s41467-023-37747-8 2023
[7]

Summer, M

A. Summer, M. Moroder, L. P. Bettmann, X. Turkeshi, I. Marvian, and J. Goold, “A resource theoretical unifica- tion of mpemba effects: classical and quantum,” (2025), arXiv:2507.16976 [quant-ph]

work page arXiv 2025
[8]

F. Ares, P. Calabrese, and S. Murciano, Nature Reviews Physics7, 451–460 (2025)

2025
[9]

H. Yu, S. Liu, and S.-X. Zhang, AAPPS Bulletin35(2025), 10.1007/s43673-025-00157-7

work page doi:10.1007/s43673-025-00157-7 2025
[10]

Murciano, F

S. Murciano, F. Ares, I. Klich, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2024, 013103 (2024)

2024
[11]

Yamashika, F

S. Yamashika, F. Ares, and P. Calabrese, Physical Review B 110(2024), 10.1103/physrevb.110.085126

work page doi:10.1103/physrevb.110.085126 2024
[12]

Rylands, K

C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini, Phys. Rev. Lett.133, 010401 (2024)

2024
[13]

F. Ares, V . Vitale, and S. Murciano, Phys. Rev. B111, 104312 (2025)

2025
[14]

F. Ares, C. Rylands, and P. Calabrese, Journal of Physics A: Mathematical and Theoretical58, 445302 (2025)

2025
[15]

Caceffo, S

F. Caceffo, S. Murciano, and V . Alba, Journal of Statistical Mechanics: Theory and Experiment2024, 063103 (2024)

2024
[16]

Chalas, F

K. Chalas, F. Ares, C. Rylands, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2024, 103101 (2024)

2024
[17]

Yamashika, P

S. Yamashika, P. Calabrese, and F. Ares, Phys. Rev. A111, 043304 (2025)

2025
[18]

Calabrese, Journal of Statistical Mechanics: Theory and Ex- periment2026, 034002 (2026)

P. Calabrese, Journal of Statistical Mechanics: Theory and Ex- periment2026, 034002 (2026)

2026
[19]

Turkeshi, P

X. Turkeshi, P. Calabrese, and A. De Luca, Phys. Rev. Lett. 135, 040403 (2025)

2025
[20]

D. Qian, H. Wang, and J. Wang, Phys. Rev. B111, L220304 (2025)

2025
[21]

Y . Li, W. Li, and X. Li, Phys. Rev. A112, 032209 (2025)

2025
[22]

Aditya, A

S. Aditya, A. Summer, P. Sierant, and X. Turkeshi, “Mpemba effects in quantum complexity,” (2025), arXiv:2509.22176 [quant-ph]

work page arXiv 2025
[23]

Bhore, L

T. Bhore, L. Su, I. Martin, A. A. Clerk, and Z. Papi ´c, Phys. Rev. B112, L121109 (2025)

2025
[24]

Yu, T.-R

Y .-H. Yu, T.-R. Jin, L. Zhang, K. Xu, and H. Fan, Phys. Rev. B 112, 094315 (2025)

2025
[25]

Alishahiha and M

M. Alishahiha and M. J. Vasli, Phys. Rev. D113, 045004 6 (2026)

2026
[26]

Hallam, M

A. Hallam, M. Yusuf, A. A. Clerk, I. Martin, and Z. Papi ´c, “Tunable quantum mpemba effect in long-range interacting sys- tems,” (2025), arXiv:2510.12875 [quant-ph]

work page arXiv 2025
[27]

Yamashika and F

S. Yamashika and F. Ares, Phys. Rev. Lett.136, 090402 (2026)

2026
[28]

Con- served quantities enable the quantum mpemba effect in weakly open systems,

I. Ul ˇcakar, R. Sharipov, G. Lagnese, and Z. Lenar ˇciˇc, “Con- served quantities enable the quantum mpemba effect in weakly open systems,” (2025), arXiv:2511.16739 [quant-ph]

work page arXiv 2025
[29]

Yamashika and R

S. Yamashika and R. Hamazaki, “Quantum many-body mpemba effect through resonances,” (2026), arXiv:2603.11788 [cond-mat.stat-mech]

work page arXiv 2026
[30]

Liu, H.-K

S. Liu, H.-K. Zhang, S. Yin, and S.-X. Zhang, Phys. Rev. Lett. 133, 140405 (2024)

2024
[31]

Quantum mpemba effect in long-ranged u(1)-symmetric ran- dom circuits,

H.-Z. Li, C. H. Lee, S. Liu, S.-X. Zhang, and J.-X. Zhong, “Quantum mpemba effect in long-ranged u(1)-symmetric ran- dom circuits,” (2025), arXiv:2512.06775 [quant-ph]

work page arXiv 2025
[32]

Liu, H.-K

S. Liu, H.-K. Zhang, S. Yin, S.-X. Zhang, and H. Yao, Science Bulletin70, 3991 (2025)

2025
[33]

Nava and R

A. Nava and R. Egger, Physical Review Letters133(2024), 10.1103/physrevlett.133.136302

work page doi:10.1103/physrevlett.133.136302 2024
[34]

Nava and R

A. Nava and R. Egger, Phys. Rev. Lett.135, 140404 (2025)

2025
[35]

Zatsarynna, A

K. Zatsarynna, A. Nava, R. Egger, and A. Zazunov, Phys. Rev. B111, 104506 (2025)

2025
[36]

A. K. Chatterjee, S. Takada, and H. Hayakawa, Phys. Rev. Lett. 131, 080402 (2023)

2023
[37]

A. K. Chatterjee, S. Takada, and H. Hayakawa, Phys. Rev. A 110, 022213 (2024)

2024
[38]

Direct experimental observation of quantum mpemba effect without bath engineering,

A. Chatterjee, S. Khan, S. Jain, and T. S. Mahesh, “Direct ex- perimental observation of quantum mpemba effect without bath engineering,” (2025), arXiv:2509.13451 [quant-ph]

work page arXiv 2025
[39]

Moroder, O

M. Moroder, O. Culhane, K. Zawadzki, and J. Goold, Phys. Rev. Lett.133, 140404 (2024)

2024
[40]

Medina, O

I. Medina, O. Culhane, F. C. Binder, G. T. Landi, and J. Goold, Phys. Rev. Lett.134, 220402 (2025)

2025
[41]

J. W. Dong, H. F. Mu, M. Qin, and H. T. Cui, Phys. Rev. A111, 022215 (2025)

2025
[42]

Furtado and A

J. Furtado and A. C. Santos, Annals of Physics480, 170135 (2025)

2025
[43]

Longhi, APL Quantum2(2025), 10.1063/5.0266143

S. Longhi, APL Quantum2(2025), 10.1063/5.0266143

work page doi:10.1063/5.0266143 2025
[44]

D. Liu, J. Yuan, H. Ruan, Y . Xu, S. Luo, J. He, X. He, Y . Ma, and J. Wang, Phys. Rev. A110, 042218 (2024)

2024
[45]

Noise-induced quantum mpemba effect,

M. Zhao and Z. Hou, “Noise-induced quantum mpemba effect,” (2025), arXiv:2507.11915 [quant-ph]

work page arXiv 2025
[46]

D. J. Strachan, A. Purkayastha, and S. R. Clark, Phys. Rev. Lett.134, 220403 (2025)

2025
[47]

Quantum mpemba effect in dissipative spin chains at criticality,

Z. Wei, M. Xu, X.-P. Jiang, H. Hu, and L. Pan, “Quantum mpemba effect in dissipative spin chains at criticality,” (2025), arXiv:2508.18906 [quant-ph]

work page arXiv 2025
[48]

Anomaly to resource: The mpemba effect in quantum thermome- try,

P. Chattopadhyay, J. F. G. Santos, and A. Misra, “Anomaly to resource: The mpemba effect in quantum thermometry,” (2026), arXiv:2601.05046 [quant-ph]

work page arXiv 2026
[49]

Role reversal in quantum mpemba effect,

A. Das, P. Chaki, P. Ghosh, and U. Sen, “Role reversal in quan- tum mpemba effect,” (2025), arXiv:2512.24839 [quant-ph]

work page arXiv 2025
[50]

Bao, Phys

R. Bao, Phys. Rev. Lett.136, 070402 (2026)

2026
[51]

Di Giulio, X

G. Di Giulio, X. Turkeshi, and S. Murciano, Entropy27, 407 (2025)

2025
[52]

L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Phys. Rev. Lett.133, 010402 (2024)

2024
[53]

Barnes, P

J. Zhang, G. Xia, C.-W. Wu, T. Chen, Q. Zhang, Y . Xie, W.-B. Su, W. Wu, C.-W. Qiu, P.-X. Chen, W. Li, H. Jing, and Y .- L. Zhou, Nature Communications16(2025), 10.1038/s41467- 024-54303-0

work page doi:10.1038/s41467- 2025
[54]

Xu, C.-P

Y . Xu, C.-P. Fang, B.-J. Chen, M.-C. Wang, Z.-Y . Ge, Y .- H. Shi, Y . Liu, C.-L. Deng, K. Zhao, Z.-H. Liu, T.-M. Li, H. Li, Z. Wang, G.-H. Liang, D. Feng, X. Guo, X.-Y . Gu, Y . He, H.-T. Liu, Z.-Y . Mei, Y . Xiao, Y . Yan, Y .-H. Yu, W.- P. Yuan, J.-C. Zhang, Z.-A. Wang, G. Liu, X. Song, Y . Tian, Y .-R. Zhang, S.-X. Zhang, K. Huang, Z. Xiang, D. Zheng...

work page arXiv 2025
[55]

Mukerjee , author V

S. Mukerjee, V . Oganesyan, and D. Huse, Physical Review B 73(2006), 10.1103/physrevb.73.035113

work page doi:10.1103/physrevb.73.035113 2006
[56]

J. Lux, J. M ¨uller, A. Mitra, and A. Rosch, Physical Review A 89(2014), 10.1103/physreva.89.053608

work page doi:10.1103/physreva.89.053608 2014
[57]

L. V . Delacretaz, SciPost Phys.9, 034 (2020)

2020
[58]

Matthies, N

A. Matthies, N. Dannenfeld, S. Pappalardi, and A. Rosch, Phys. Rev. B113, 024305 (2026)

2026
[59]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cambridge University Press, 2010)

2010
[60]

We identify the discretized time ast=t n =n
[61]

We choose the parametersα=2, β=0.25, γ=1 [56]
[62]

Kim and D

H. Kim and D. A. Huse, Phys. Rev. Lett.111, 127205 (2013)

2013
[63]

Quantum ising model in trans- verse and longitudinal fields: chaotic wave functions,

Y . Y . Atas and E. Bogomolny, “Quantum ising model in trans- verse and longitudinal fields: chaotic wave functions,” (2015), arXiv:1503.04508 [math-ph]

work page arXiv 2015
[64]

M. C. Ba ˜nuls, J. I. Cirac, and M. B. Hastings, Phys. Rev. Lett. 106, 050405 (2011)

2011
[65]

Thermalization dynamics in closed quantum many body systems: a precision large scale ex- act diagonalization study,

I. A. Maceira and A. M. L ¨auchli, “Thermalization dynamics in closed quantum many body systems: a precision large scale ex- act diagonalization study,” (2025), arXiv:2409.18863 [quant- ph]

work page arXiv 2025
[66]

J. M. Deutsch, Reports on Progress in Physics81, 082001 (2018)

2018
[67]

|ψ(t)⟩ =⟨

We introduce the notation ⟨ψ(t)| . . . |ψ(t)⟩ =⟨. . .⟩ t
[68]

Eigenstate thermalization hypothesis correlations via non-linear hydrodynamics,

J. Wang, R. Mishra, T.-H. Yang, L. V . Delacr´etaz, and S. Pap- palardi, “Eigenstate thermalization hypothesis correlations via non-linear hydrodynamics,” (2025), arXiv:2505.06869 [cond- mat.stat-mech]

work page arXiv 2025
[69]

Capizzi, J

L. Capizzi, J. Wang, X. Xu, L. Mazza, and D. Poletti, Phys. Rev. X15, 011059 (2025)

2025
[70]

Cirq Developers, “Cirq,” (2025)

2025
[71]

Finding exponential product formu- las of higher orders,

N. Hatano and M. Suzuki, “Finding exponential product formu- las of higher orders,” inQuantum Annealing and Other Opti- mization Methods(Springer Berlin Heidelberg, 2005) p. 37–68

2005
[72]

Spohn,Large Scale Dynamics of Interacting Particles (Springer, 1991)

H. Spohn,Large Scale Dynamics of Interacting Particles (Springer, 1991)

1991
[73]

Fujimura and S

H. Fujimura and S. Shimamori, “Entanglement asymmetry and quantum mpemba effect for non-abelian global symmetry,” (2025), arXiv:2509.05597 [hep-th]

work page arXiv 2025
[74]

Banerjee, S

T. Banerjee, S. Das, and K. Sengupta, SciPost Phys.19, 051 (2025)

2025
[75]

Kormos, M

M. Kormos, M. Collura, G. Tak ´acs, and P. Calabrese, Nature Physics13, 246–249 (2016)

2016
[76]

P. P. Mazza, G. Perfetto, A. Lerose, M. Collura, and A. Gam- bassi, Phys. Rev. B99, 180302 (2019)

2019
[77]

Jiang, J

X. Jiang, J. C. Halimeh, and N. S. Srivatsa, Phys. Rev. D113, L031503 (2026)

2026
[78]

C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´c, Nature Physics14, 745–749 (2018)

2018
[79]

Data underpinning

T. M. M ¨uller, S. Pappalardi, and R. Fazio, “Data underpinning ”quantum mpemba effect in chaotic systems with conservation laws”,” (2026)

2026