pith. sign in

arxiv: 2605.17908 · v1 · pith:L7CRSCRTnew · submitted 2026-05-18 · ❄️ cond-mat.stat-mech · cond-mat.mes-hall· quant-ph

Quantum Mpemba effect for operators in open systems

Pitambar Bagui , Bijay Kumar Agarwalla This is my paper

Pith reviewed 2026-05-20 01:09 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mes-hallquant-ph
keywords quantum Mpemba effectopen quantum systemsadjoint Liouvillianoperator dynamicsanomalous relaxationGKSL equation
0
0 comments X

The pith

The quantum Mpemba effect extends to operators evolving under the adjoint Liouvillian in open systems.

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

This paper shows that the quantum Mpemba effect, an anomalous ordering in relaxation speeds, applies to operators evolving under the adjoint Liouvillian. Although this evolution does not preserve the trace of operators, the anomalous behavior still emerges under certain conditions. Readers should care as it expands the Mpemba effect to the level of observables and provides tools to influence how they relax in open quantum systems. The predictions are verified across three different physical setups.

Core claim

Operators that evolve under the adjoint Liouvillian exhibit a genuine Mpemba effect. General conditions are derived under which this anomalous relaxation can occur, and these are validated for three different open quantum setups. This demonstrates that the effect is not limited to trace-preserving dynamics on states.

What carries the argument

Adjoint Liouvillian dynamics on operators, which governs their time evolution and permits anomalous relaxation ordering.

If this is right

  • Conditions derived allow prediction of the effect in various systems.
  • Relaxation of observables can be controlled using this framework.
  • Scope of the Mpemba effect is broadened to include operator-level phenomena in open systems.

Where Pith is reading between the lines

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

  • This suggests that non-trace-preserving maps in general might support similar anomalous behaviors.
  • Experimental measurements of operator expectations could reveal Mpemba-like effects without needing full state tomography.

Load-bearing premise

That the absence of trace preservation in adjoint dynamics does not prevent the anomalous relaxation ordering seen in standard state evolution.

What would settle it

Finding an open quantum system example where operator relaxation under the adjoint Liouvillian fails to show the expected Mpemba ordering despite satisfying the derived conditions would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.17908 by Bijay Kumar Agarwalla, Pitambar Bagui.

Figure 1
Figure 1. Figure 1: FIG. 1. Presence of the genuine operator Mpemba effect in a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Results for the Non-equilibrium [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Results for one-dimensional bosonic lattice subjected [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The quantum Mpemba effect concerns with anomalous relaxation of quantum states that evolves either under unitary or non-unitary dynamics. In the context of open quantum systems, while most studies focus on quantum states evolving under completely positive trace-presing dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, we demonstrate that an analogous effect can arise at the level of operators. In particular, we show that operators that evolves under the adjoint Liouvillian -- despite not being a trace-preserving map -- can still exhibit a genuine Mpemba effect. We derive general conditions under which this phenomenon can occur and validate our predictions for three different open quantum setups. Our results broaden the scope of the Mpemba effect in quantum systems and provide a framework for controlling the relaxation of physically relevant observables.

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

2 major / 2 minor

Summary. The paper claims that operators evolving under the adjoint Liouvillian in open quantum systems can exhibit a genuine quantum Mpemba effect—i.e., anomalous relaxation ordering where an initially farther-from-equilibrium operator reaches the steady state faster—despite the adjoint map being unital but not trace-preserving. General conditions for this phenomenon are derived, and the predictions are validated numerically or analytically in three distinct open-system setups.

Significance. If the central claim holds, the result meaningfully extends the Mpemba effect from state evolution under GKSL dynamics to operator evolution, offering a framework for controlling the relaxation of physically relevant observables. The work is conceptually novel in separating the effect from trace preservation, but its impact depends on whether the chosen distance measure yields a robust, non-artifactual signature.

major comments (2)
  1. [§3] §3 (definition of the Mpemba effect for operators): the manuscript adopts the Hilbert-Schmidt norm to quantify distance to the steady-state operator. Because the adjoint Liouvillian is not trace-preserving, the standard contractivity arguments that guarantee monotonicity of trace distance or relative entropy for states do not apply directly. The paper must explicitly demonstrate that the observed crossing times remain invariant under rescaling or under alternative Schatten norms; otherwise the anomalous ordering could be a metric artifact rather than a genuine extension of the Mpemba phenomenon.
  2. [§4.2, §5] §4.2 and §5 (validation in the three setups): the reported relaxation curves show crossings, but the manuscript does not provide a quantitative check that the initial operators are prepared at equal 'distance' in a manner independent of the missing trace-preservation constraint. A post-hoc choice of initial conditions or of the cutoff time could artificially produce the reported ordering; an explicit falsification test (e.g., swapping the initial operators while keeping the same norm) should be added.
minor comments (2)
  1. [§2] Notation for the adjoint Liouvillian L* is introduced without a clear statement of its domain (bounded vs. unbounded operators) and of the precise inner product used to define the Hilbert-Schmidt norm.
  2. [Figures 2-4] Figure captions for the three setups should explicitly state the system size, the value of the dissipation rate, and the precise initial operators used, to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below and will incorporate revisions to strengthen the presentation of the results.

read point-by-point responses

  1. Referee: [§3] §3 (definition of the Mpemba effect for operators): the manuscript adopts the Hilbert-Schmidt norm to quantify distance to the steady-state operator. Because the adjoint Liouvillian is not trace-preserving, the standard contractivity arguments that guarantee monotonicity of trace distance or relative entropy for states do not apply directly. The paper must explicitly demonstrate that the observed crossing times remain invariant under rescaling or under alternative Schatten norms; otherwise the anomalous ordering could be a metric artifact rather than a genuine extension of the Mpemba phenomenon.

    Authors: We agree that the choice of distance measure requires careful justification when the map is unital but not trace-preserving. The Hilbert-Schmidt norm is the natural choice here because it arises directly from the vectorization of operators and the associated Frobenius inner product used to define the adjoint Liouvillian. Nevertheless, to rule out a metric artifact, we will add explicit checks in the revised manuscript. These will include (i) rescaling the initial operators while preserving their relative ordering and (ii) repeating the relaxation curves with the trace norm and the operator norm (Schatten 1 and ∞). We will report that the crossing times remain qualitatively unchanged across these choices for all three setups, thereby confirming that the anomalous ordering is robust. revision: yes

  2. Referee: [§4.2, §5] §4.2 and §5 (validation in the three setups): the reported relaxation curves show crossings, but the manuscript does not provide a quantitative check that the initial operators are prepared at equal 'distance' in a manner independent of the missing trace-preservation constraint. A post-hoc choice of initial conditions or of the cutoff time could artificially produce the reported ordering; an explicit falsification test (e.g., swapping the initial operators while keeping the same norm) should be added.

    Authors: We acknowledge that an explicit falsification test strengthens the claim. In the revised version we will add a dedicated subsection that (i) tabulates the Hilbert-Schmidt distances of the chosen initial operators to the steady-state operator for each setup and (ii) performs the suggested swap: the operator that was initially farther is reassigned as the closer initial condition (and vice versa) while keeping the same norm value. The resulting relaxation curves will be shown to reverse their ordering, demonstrating that the Mpemba effect is not an artifact of post-hoc selection of initial conditions or cutoff times but follows from the dynamics under the adjoint Liouvillian. revision: yes

Circularity Check

0 steps flagged

No circularity: conditions for adjoint Mpemba effect derived independently from dynamics

full rationale

The paper derives general conditions for the Mpemba effect under adjoint Liouvillian evolution on operators and validates them on three concrete open-system examples. These conditions follow from the structure of the adjoint map and observable relaxation ordering rather than from any fitted parameter, self-referential definition, or load-bearing self-citation. The central claim—that anomalous relaxation ordering persists despite lack of trace preservation—is presented as a direct consequence of the Liouvillian properties and is checked against explicit models, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard open-quantum-system formalism; no new free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Operator evolution follows the adjoint action of the Liouvillian superoperator derived from the GKSL master equation.
    Invoked when stating that operators evolve under the adjoint Liouvillian despite the map not being trace-preserving.

pith-pipeline@v0.9.0 · 5670 in / 1256 out tokens · 50534 ms · 2026-05-20T01:09:03.337332+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

Works this paper leans on

79 extracted references · 79 canonical work pages · 2 internal anchors

  1. [1]

    Here, we discuss the results in the strong-dephasing limit, and the results for moderate dephasing,γ∼J, are presented in Ref

    and intermediate dephasing limitγ∼J. Here, we discuss the results in the strong-dephasing limit, and the results for moderate dephasing,γ∼J, are presented in Ref. [65]. In the strong dephasing limit, the coherences in the site basis are strongly suppressed, leading to the evo- lution confined entirely in the population sectorp n = ρn,n(t). The population ...

    work page 2023

  2. [2]

    Beato and G

    N. Beato and G. Teza, Relaxation control of open quan- tum systems, Phys. Rev. Lett.136, 070401 (2026)

    work page 2026

  3. [3]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science337, 1318 (2012)

    work page 2012

  4. [4]

    L. F. Santos, F. Borgonovi, and F. M. Izrailev, Chaos and statistical relaxation in quantum systems of interacting particles, Phys. Rev. Lett.108, 094102 (2012)

    work page 2012

  5. [5]

    ˇZnidariˇ c, Relaxation times of dissipative many-body quantum systems, Phys

    M. ˇZnidariˇ c, Relaxation times of dissipative many-body quantum systems, Phys. Rev. E92, 042143 (2015)

    work page 2015

  6. [6]

    S. G. Schirmer and A. I. Solomon, Constraints on relax- ation rates forn-level quantum systems, Phys. Rev. A 70, 022107 (2004)

    work page 2004

  7. [7]

    van Caspel and V

    M. van Caspel and V. Gritsev, Symmetry-protected co- herent relaxation of open quantum systems, Phys. Rev. A97, 052106 (2018)

    work page 2018

  8. [8]

    Zhao, Y.-A

    B. Zhao, Y.-A. Chen, X. Zhao, C. Zhu, G. Chiribella, and X. Wang, Power and limitations of distributed quantum state purification, Phys. Rev. Lett.136, 090203 (2026)

    work page 2026

  9. [9]

    C. G. Feyisa, H.-Y. Ku, J. S. You, and H. H. Jen, In- formational mpemba effect for fast state purification in non-hermitian system (2026), arXiv:2604.07926 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    Kleinmann, H

    M. Kleinmann, H. Kampermann, T. Meyer, and D. Bruß, Physical purification of quantum states, Phys. Rev. A73, 062309 (2006)

    work page 2006

  11. [11]

    C. Marr, A. Beige, and G. Rempe, Entangled-state preparation via dissipation-assisted adiabatic passages, Phys. Rev. A68, 033817 (2003)

    work page 2003

  12. [12]

    M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Dis- sipative preparation of entanglement in optical cavities, Phys. Rev. Lett.106, 090502 (2011)

    work page 2011

  13. [13]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The theory of open quantum systems(OUP Oxford, 2002)

    work page 2002

  14. [14]

    D. A. Lidar, Lecture notes on the theory of open quantum systems (2020), arXiv:1902.00967 [quant-ph]

    work page arXiv 2020

  15. [15]

    F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nature Communications14, 2036 (2023)

    work page 2036

  16. [16]

    Rylands, K

    C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Mur- ciano, and B. Bertini, Microscopic origin of the quantum mpemba effect in integrable systems, Phys. Rev. Lett. 133, 010401 (2024)

    work page 2024

  17. [17]

    Yamashika, F

    S. Yamashika, F. Ares, and P. Calabrese, Entangle- ment asymmetry and quantum mpemba effect in two- dimensional free-fermion systems, Phys. Rev. B110, 085126 (2024)

    work page 2024

  18. [18]

    Murciano, F

    S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entan- glement asymmetry and quantum mpemba effect in the xy spin chain, Journal of Statistical Mechanics: Theory and Experiment2024, 013103 (2024)

    work page 2024

  19. [19]

    Lastres, S

    M. Lastres, S. Murciano, F. Ares, and P. Calabrese, En- tanglement asymmetry in the critical xxz spin chain, Journal of Statistical Mechanics: Theory and Experi- ment2025, 013107 (2025)

    work page 2025

  20. [20]

    F. Ares, V. Vitale, and S. Murciano, Quantum mpemba effect in free-fermionic mixed states, Phys. Rev. B111, 104312 (2025)

    work page 2025

  21. [21]

    Yamashika, P

    S. Yamashika, P. Calabrese, and F. Ares, Quenching from superfluid to free bosons in two dimensions: Entan- glement, symmetries, and the quantum mpemba effect, Phys. Rev. A111, 043304 (2025)

    work page 2025

  22. [22]

    Chalas, F

    K. Chalas, F. Ares, C. Rylands, and P. Calabrese, Mul- tiple crossings during dynamical symmetry restoration and implications for the quantum mpemba effect, Journal of Statistical Mechanics: Theory and Experiment2024, 103101 (2024)

    work page 2024

  23. [23]

    Zhang, H.-Z

    Y.-R. Zhang, H.-Z. Li, X.-Y. Huang, Y.-J. Zhao, and J.- X. Zhong, Stark many-body localization-induced quan- tum mpemba effect, Entropy28, 10.3390/e28030348 (2026)

    work page doi:10.3390/e28030348 2026

  24. [24]

    H.-Z. Li, C. H. Lee, S. Liu, S.-X. Zhang, and J.-X. Zhong, Quantum mpemba effect in long-ranged u(1)-symmetric random circuits, Phys. Rev. B113, 134310 (2026)

    work page 2026

  25. [25]

    Yu, T.-R

    Y.-H. Yu, T.-R. Jin, L. Zhang, K. Xu, and H. Fan, Tun- ing the quantum mpemba effect in an isolated system by initial-state engineering, Phys. Rev. B112, 094315 (2025)

    work page 2025

  26. [26]

    Bhore, L

    T. Bhore, L. Su, I. Martin, A. A. Clerk, and Z. Papi´ c, Quantum mpemba effect without global symmetries, Phys. Rev. B112, L121109 (2025)

    work page 2025

  27. [27]

    Hallam, M

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

    work page arXiv 2025

  28. [28]

    A. K. Chatterjee, S. Takada, and H. Hayakawa, Quantum mpemba effect in a quantum dot with reservoirs, Phys. Rev. Lett.131, 080402 (2023)

    work page 2023

  29. [29]

    Bagui, A

    P. Bagui, A. Chatterjee, and B. K. Agarwalla, Detection of mpemba effect through good observables in open quan- tum systems (2025), arXiv:2512.02709 [cond-mat.stat- mech]

    work page arXiv 2025

  30. [30]

    Bao and Z

    R. Bao and Z. Hou, Accelerating quantum relaxation via temporary reset: A mpemba-inspired approach (2025), arXiv:2212.11170 [quant-ph]

    work page arXiv 2025

  31. [31]

    Kochsiek, F

    S. Kochsiek, F. Carollo, and I. Lesanovsky, Accelerat- ing the approach of dissipative quantum spin systems towards stationarity through global spin rotations, Phys. Rev. A106, 012207 (2022)

    work page 2022

  32. [32]

    Wang and J

    X. Wang and J. Wang, Mpemba effects in nonequilibrium open quantum systems, Phys. Rev. Res.6, 033330 (2024)

    work page 2024

  33. [33]

    Nava and R

    A. Nava and R. Egger, Mpemba effects in open nonequi- librium quantum systems, Phys. Rev. Lett.133, 136302 (2024). 7

    work page 2024

  34. [34]

    Pontus-Mpemba effect in cavity quantum electrodynamics

    S. Longhi, Pontus-mpemba effect in cavity quantum elec- trodynamics (2026), arXiv:2605.05827 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  35. [35]

    D. J. Strachan, A. Purkayastha, and S. R. Clark, Non- markovian quantum mpemba effect, Phys. Rev. Lett. 134, 220403 (2025)

    work page 2025

  36. [36]

    Ganguly and B

    K. Ganguly and B. K. Agarwalla, Measurement in- duced faster symmetry restoration in quantum trajec- tories (2026), arXiv:2601.18458 [cond-mat.stat-mech]

    work page arXiv 2026

  37. [37]

    Moroder, O

    M. Moroder, O. Culhane, K. Zawadzki, and J. Goold, Thermodynamics of the quantum mpemba effect, Phys. Rev. Lett.133, 140404 (2024)

    work page 2024

  38. [38]

    Longhi, Bosonic mpemba effect with non-classical states of light, APL Quantum1, 046110 (2024)

    S. Longhi, Bosonic mpemba effect with non-classical states of light, APL Quantum1, 046110 (2024)

    work page 2024

  39. [39]

    Y. Song, J. Chen, X. Yang, M. Xu, X.-P. Jiang, and L. Pan, Quantum pontus–mpemba effect in dissipa- tive quasiperiodic chains (2026), arXiv:2602.15406 [cond- mat.dis-nn]

    work page arXiv 2026

  40. [40]

    Zhang, C

    X. Zhang, C. Sun, and F. Li, Engineering quan- tum mpemba effect by liouvillian skin effect (2026), arXiv:2601.16002 [quant-ph]

    work page arXiv 2026

  41. [41]

    Longhi, Quantum pontus–mpemba effect enabled by the liouvillian skin effect, Journal of Physics A: Mathe- matical and Theoretical59, 065304 (2026)

    S. Longhi, Quantum pontus–mpemba effect enabled by the liouvillian skin effect, Journal of Physics A: Mathe- matical and Theoretical59, 065304 (2026)

    work page 2026

  42. [42]

    Z. Wei, M. Xu, X.-P. Jiang, H. Hu, and L. Pan, Quantum mpemba effect in dissipative spin chains at criticality, Sci- ence China Physics, Mechanics & Astronomy69, 240315 (2026)

    work page 2026

  43. [43]

    Sapui, T

    T. Sapui, T. K. Konar, and A. S. De, Ergotropic mpemba crossings in finite-dimensional quantum batteries (2026), arXiv:2602.11056 [quant-ph]

    work page arXiv 2026

  44. [44]

    Chattopadhyay, J

    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

  45. [45]

    Solanki, I

    P. Solanki, I. Lesanovsky, and G. Perfetto, Universal relaxation speedup in open quantum systems through transient conditional and unconditional resetting (2025), arXiv:2512.10005 [cond-mat.stat-mech]

    work page arXiv 2025

  46. [46]

    F. Ares, P. Calabrese, and S. Murciano, The quantum mpemba effects, Nature Reviews Physics7, 451 (2025)

    work page 2025

  47. [47]

    Chatterjee, S

    A. Chatterjee, S. Khan, S. Jain, and T. S. Mahesh, Di- rect experimental observation of quantum mpemba ef- fect without bath engineering (2025), arXiv:2509.13451 [quant-ph]

    work page arXiv 2025

  48. [48]

    Zhang, G

    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, Observation of quantum strong mpemba effect, Nature Communications16, 301 (2025)

    work page 2025

  49. [49]

    Aharony Shapira, Y

    S. Aharony Shapira, Y. Shapira, J. Markov, G. Teza, N. Akerman, O. Raz, and R. Ozeri, Inverse mpemba ef- fect demonstrated on a single trapped ion qubit, Phys. Rev. Lett.133, 010403 (2024)

    work page 2024

  50. [50]

    Ulˇ cakar, R

    I. Ulˇ cakar, R. Sharipov, G. Lagnese, and Z. Lenarˇ ciˇ c, Conserved quantities enable the quantum mpemba effect in weakly open systems (2025), arXiv:2511.16739 [quant- ph]

    work page arXiv 2025

  51. [51]

    Alishahiha and M

    M. Alishahiha and M. J. Vasli, On krylov complex- ity as a probe of the quantum mpemba effect (2025), arXiv:2510.14740 [hep-th]

    work page arXiv 2025

  52. [52]

    E. B. Mpemba and D. G. Osborne, Cool?, Physics Edu- cation4, 172 (1969)

    work page 1969

  53. [53]

    P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, Mpemba-like behavior in carbon nanotube resonators, Metallurgical and Materials Transactions A42, 3907 (2011)

    work page 2011

  54. [54]

    Lasanta, F

    A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, When the hotter cools more quickly: Mpemba effect in granular fluids, Phys. Rev. Lett.119, 148001 (2017)

    work page 2017

  55. [55]

    C. Hu, J. Li, S. Huang, H. Li, C. Luo, J. Chen, S. Jiang, and L. An, Conformation directed mpemba effect on polylactide crystallization, Crystal Growth & Design18, 5757 (2018)

    work page 2018

  56. [56]

    F. J. Schwarzendahl and H. L¨ owen, Anomalous cooling and overcooling of active colloids, Physical Review Let- ters129, 10.1103/physrevlett.129.138002 (2022)

    work page doi:10.1103/physrevlett.129.138002 2022

  57. [57]

    Gal and O

    A. Gal and O. Raz, Precooling strategy allows exponen- tially faster heating, Phys. Rev. Lett.124, 060602 (2020)

    work page 2020

  58. [58]

    S. Bera, M. R. Walker, and M. Vucelja, Effect of dynam- ics on anomalous thermal relaxations and information ex- change (2023), arXiv:2308.04557 [cond-mat.stat-mech]

    work page arXiv 2023

  59. [59]

    Vynnycky and S

    M. Vynnycky and S. L. Mitchell, Evaporative cooling and the mpemba effect, Heat and Mass Transfer46, 881 (2010)

    work page 2010

  60. [60]

    L. P. Garc´ ıa-Pintos, N. Linden, A. S. L. Malabarba, A. J. Short, and A. Winter, Equilibration time scales of physi- cally relevant observables, Phys. Rev. X7, 031027 (2017)

    work page 2017

  61. [61]

    A. S. L. Malabarba, L. P. Garc´ ıa-Pintos, N. Linden, T. C. Farrelly, and A. J. Short, Quantum systems equilibrate rapidly for most observables, Phys. Rev. E90, 012121 (2014)

    work page 2014

  62. [62]

    M. G. Alpino, T. Debarba, R. O. Vianna, and A. T. Ces´ ario, A complexity-based approach to quantum ob- servable equilibration, Entropy27, 824 (2025)

    work page 2025

  63. [63]

    Bagui, A

    P. Bagui, A. Chatterjee, and B. K. Agarwalla, Acceler- ated relaxation and mpemba-like effect for operators in open quantum systems (2025), arXiv:2510.24630 [cond- mat.stat-mech]

    work page arXiv 2025

  64. [64]

    Schaller,Open quantum systems far from equilibrium, Vol

    G. Schaller,Open quantum systems far from equilibrium, Vol. 881 (Springer, 2014)

    work page 2014

  65. [65]

    Minganti, A

    F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spec- tral theory of liouvillians for dissipative phase transitions, Phys. Rev. A98, 042118 (2018)

    work page 2018

  66. [66]

    See Supplementary Material

    work page

  67. [67]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010)

    work page 2010

  68. [68]

    F. Ares, P. Calabrese, and S. Murciano, The quan- tum mpemba effects, Nature Reviews Physics7, 451–460 (2025)

    work page 2025

  69. [69]

    B. Li, Z. Wang, G. Zheng, Y. Wong, and L. Jiang, Opti- mality condition for the petz map, Phys. Rev. Lett.134, 200602 (2025)

    work page 2025

  70. [70]

    Gily´ en, S

    A. Gily´ en, S. Lloyd, I. Marvian, Y. Quek, and M. M. Wilde, Quantum algorithm for petz recovery channels and pretty good measurements, Phys. Rev. Lett.128, 220502 (2022)

    work page 2022

  71. [71]

    N. J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Linear Algebra and its Applications212-213, 3 (1994)

    work page 1994

  72. [72]

    N. J. Higham and R. S. Schreiber, Fast polar decompo- sition of an arbitrary matrix, SIAM Journal on Scientific and Statistical Computing11, 648 (1990)

    work page 1990

  73. [73]

    E. B. Davies, Generators of dynamical semigroups, Jour- nal of Functional Analysis34, 421 (1979)

    work page 1979

  74. [74]

    Longhi, Photonic mpemba effect, Opt

    S. Longhi, Photonic mpemba effect, Opt. Lett.49, 5188 (2024)

    work page 2024

  75. [75]

    Liang, C

    Y. Liang, C. Xie, Z. Guo, P. Huang, W. Huang, Y. Liu, J. Qiu, X. Sun, Z. Wang, X. Yang, J. Zhang, J. Zhang, L. Zhang, J. Chu, W. Guo, J. Jiang, X. Linpeng, S. Liu, 8 J. Niu, Y. Zhou, Y. Zhong, W. Ren, Z. Tao, and D. Yu, Dephasing-assisted diffusive dynamics in super- conducting quantum circuits, Applied Physics Letters 127, 154003 (2025)

    work page 2025

  76. [76]

    Longhi, Dephasing-induced mobility edges in qua- sicrystals, Phys

    S. Longhi, Dephasing-induced mobility edges in qua- sicrystals, Phys. Rev. Lett.132, 236301 (2024)

    work page 2024

  77. [77]

    C. Wan, M. Scala, G. W. Morley, A. A. Rahman, H. Ul- bricht, J. Bateman, P. F. Barker, S. Bose, and M. S. Kim, Free nano-object ramsey interferometry for large quantum superpositions, Phys. Rev. Lett.117, 143003 (2016)

    work page 2016

  78. [78]

    Asadian, C

    A. Asadian, C. Brukner, and P. Rabl, Probing macro- scopic realism via ramsey correlation measurements, Phys. Rev. Lett.112, 190402 (2014). 9 Supplemental Material: Quantum Mpemba effect for operators in open systems I. NA TURE OF TRANSFORMED OPERA TORS UNDER MODE BYP ASSING In this section, we discuss how the mode-bypassing protocol can transform an init...

    work page 2014

  79. [79]

    The associated right eigenmatricesr 1 andr 2 are complex conjugates of each other and are purely off-diagonal, a characteristic feature of the Davies map structure

    corresponding to the slowest decay mode (with equal real parts). The associated right eigenmatricesr 1 andr 2 are complex conjugates of each other and are purely off-diagonal, a characteristic feature of the Davies map structure. Therefore, to suppress the contribution of these slowest decaying modes and thereby achieving accelerated relaxation in the ope...

    work page