Genuine Quantum Mpemba Effect

• Genuine quantum Mpemba effect is an anomalous relaxation phenomenon where a state farther from equilibrium reaches the thermal state faster than a closer one.
• The effect arises from spectral mode suppression and thermodynamic criteria, leveraging metrics like relative entropy to differentiate genuine speedups from trivial acceleration.
• Experimental realizations in NMR and trapped-ion platforms demonstrate practical applications, including enhanced thermometric precision and metrological advantages.

Searching arXiv for papers on the genuine quantum Mpemba effect and closely related formulations. The genuine quantum Mpemba effect is a class of anomalous relaxation phenomena in which a quantum state that is initially farther from its stationary target reaches that target faster than a state that is initially closer. In its thermodynamic usage, the ordering of initial states is by nonequilibrium free energy, equivalently by relative entropy to the Gibbs state up to an additive constant, so the effect is “genuine” when the faster trajectory starts thermodynamically farther from equilibrium rather than merely farther in an arbitrary geometric metric (Chatterjee et al., 16 Sep 2025). Subsequent literature has retained this thermodynamic core while extending the label to spectrally certified open-system speedups, symmetry-restoration anomalies in isolated many-body dynamics, operator relaxation in the Heisenberg picture, and finite-time thermometric advantages, so the term now denotes a family of closely related but not fully identical notions (Moroder et al., 2024).

1. Definitions and terminological scope

The minimal Mpemba statement is that a state farther from equilibrium relaxes faster than a state closer to equilibrium. In the open-system setting, the standard spectral explanation is that the farther state has a much smaller overlap with the slowest decay mode of the Liouvillian than the nearer state, so its long-time approach is governed by faster modes (Chatterjee et al., 16 Sep 2025). The thermodynamic strengthening used in several papers requires more: the faster state must start with higher nonequilibrium free energy, or equivalently larger relative entropy to the thermal state, and the two relaxation curves must cross in time (Moroder et al., 2024).

This distinction separates mere acceleration of relaxation from the genuine quantum Mpemba effect. For a thermal fixed point $\tau_\beta$, one common thermodynamic formulation is

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$

with genuine Mpemba behavior when

$$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$

(Moroder et al., 2024).

The literature also contains non-thermodynamic usages. In closed many-body systems, the effect is formulated as an inversion in the decay ordering of symmetry-breaking diagnostics such as entanglement asymmetry or charge variance, so that a more symmetry-broken state restores symmetry faster than a less broken one (Yu et al., 3 Jul 2025). In the Heisenberg picture, an operator-level version is called genuine when a transformed observable relaxes faster to the same steady-state value as the original observable, after removal of its overlap with the slowest adjoint-Liouvillian mode (Bagui et al., 18 May 2026). A plausible implication is that “genuine” now functions less as a single universal definition than as a criterion that excludes trivial or purely metric-dependent speedups.

2. Spectral mechanism in Markovian open systems

The common structural basis is Liouvillian mode decomposition. For a Markovian master equation, the density matrix can be expanded as

$$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$

where $\tau$ is the unique steady state, $\lambda_k$ are Liouvillian eigenvalues, and $l_k,r_k$ are left and right eigenoperators (Moroder et al., 2024). The slowest nonzero mode, conventionally associated with $\lambda_2$, sets the asymptotic relaxation time. If a unitary $U$ can be chosen so that

$\operatorname{Tr}(l_2\,U\rho_i U^\dagger)=0,$

then the slowest mode is removed and the long-time decay is governed by the next eigenvalue (Moroder et al., 2024).

A particularly sharp version arises for Davies maps. Because populations and coherences decouple in the energy eigenbasis, transforming an initial state with coherences into a state diagonal in that basis eliminates all coherent overlaps. When the spectral gap is defined by a complex conjugate pair, this guarantees an exponential speedup, since the slow coherent pair is removed in one step (Moroder et al., 2024). This is the basic route from coherence suppression to accelerated equilibration.

A more constructive version was later formulated for general Davies dynamics through permutation unitaries. If

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$0

then the unitary

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$1

produces

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$2

which is diagonal in the energy basis with permuted spectrum (Caldas et al., 8 Dec 2025). In that basis the slowest decay eigenoperator can be triangularized to a single off-diagonal element, so the overlap with the slowest mode vanishes for any permutation matrix. The same construction then selects a permutation that maximizes the initial distance from equilibrium with respect to the Hilbert-Schmidt distance, quantum relative entropy, or trace distance, thereby producing a genuine crossing rather than a mere asymptotic speedup (Caldas et al., 8 Dec 2025).

This spectral picture also underlies the strong Mpemba effect. In the strongest case the slowest mode is exactly absent, so relaxation changes from the $$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$3 timescale to the $$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$4 timescale. The effect is therefore not only a prefactor reduction; it changes the asymptotic decay exponent itself (Caldas et al., 8 Dec 2025).

3. Thermodynamic criteria and distance measures

The thermodynamic formulation identifies nonequilibrium free energy and relative entropy as the natural ordering variables. For a Gibbs target $$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$5, the relative entropy

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$6

is proportional, up to an additive constant, to the nonequilibrium free energy (Moroder et al., 2024). In the NMR realization of natural thermalization, the same quantity is written as

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$7

and a genuine quantum Mpemba effect occurs when the state with larger $$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$8 nonetheless reaches equilibrium sooner (Chatterjee et al., 16 Sep 2025).

Several metrics coexist in the literature. Trace distance is often used as the operational distance from equilibrium,

$$F_{\mathrm{neq}}(\rho)=\operatorname{Tr}(H\rho)+\beta^{-1}\operatorname{Tr}(\rho\ln\rho) =\beta^{-1}D(\rho\|\tau_\beta)+F_{\mathrm{eq}},$$9

and a standard experimental signature is a crossing of $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$0 and $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$1 for a nearer and a farther state (Chatterjee et al., 16 Sep 2025). Davies-map constructions also optimize genuine crossings under Hilbert-Schmidt distance and quantum relative entropy, showing that the phenomenon is not tied to a single metric (Caldas et al., 8 Dec 2025). The same papers are explicit that an arbitrary observable crossing is insufficient; the farther state must be farther in the chosen information-theoretic or thermodynamic measure at the initial time (Caldas et al., 8 Dec 2025).

A stricter thermal interpretation appears in the canonical quantum Mpemba effect for a dissipative qubit. There the requirement is not only faster relaxation of a hotter thermal state, but also thermodynamic directionality: both initial states must satisfy

$$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$2

where $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$3 is the effective steady-state temperature defined by the thermal state closest to the nonequilibrium steady state in trace distance (Li et al., 21 Nov 2025). This excludes comparisons in which one trajectory is heating while the other is cooling. A plausible implication is that the “genuine” qualifier is often used to rule out formally faster but thermodynamically ambiguous comparisons.

4. Experimental realizations in thermalizing open systems

A direct experimental observation of both the quantum Mpemba effect and the genuine quantum Mpemba effect without bath engineering was reported in a two-spin NMR platform consisting of two homonuclear spin-$$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$4 $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$5 nuclei in 2-Chloroacrylonitrile dissolved in DMSO, at $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$6 and $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$7, measured with a $$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$8 Bruker spectrometer (Chatterjee et al., 16 Sep 2025). The Hamiltonian is

$$F_{\mathrm{neq}}(\rho'(0))>F_{\mathrm{neq}}(\rho(0)), \qquad F_{\mathrm{neq}}(\rho'(t))<F_{\mathrm{neq}}(\rho(t)) \ \text{for sufficiently late } t$$9

with CAN parameters $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$0, $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$1, and $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$2 (Chatterjee et al., 16 Sep 2025). Thermalization was described by a GSKL equation dominated by dipolar relaxation, and inhomogeneous dephasing was experimentally removed so that the observed relaxation reflected natural thermalization rather than field inhomogeneity (Chatterjee et al., 16 Sep 2025).

The theory exploited the zero-quantum block. For initial populations satisfying

$$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$3

and in the regime $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$4, $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$5, and $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$6, the dynamics reduces to a population master equation with decay rates

$$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$7

The preparation logic then constructs a nearer state $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$8 that overlaps with both slow and fast modes and a farther state $$\rho(t)=\tau+\sum_{k=2}^{D^2}\operatorname{Tr}(l_k\rho_i)\,r_k\,e^{\lambda_k t},$$9 that overlaps only with the fastest mode (Chatterjee et al., 16 Sep 2025). Experimentally,

$\tau$0

so $\tau$1 is initially farther from equilibrium for generic $\tau$2, yet its trace-distance curve crosses below that of $\tau$3 at finite time (Chatterjee et al., 16 Sep 2025). Using the relative-entropy/free-energy measure $\tau$4, the same experiment showed that $\tau$5 has higher thermodynamic distance and still reaches equilibrium faster, establishing the genuine quantum Mpemba effect under natural dipolar relaxation with no external control during the relaxation window (Chatterjee et al., 16 Sep 2025).

Another experimentally realized form is the quantum strong Mpemba effect in a single trapped ion, where a specially prepared initial state satisfies

$\tau$6

and therefore relaxes with the next Liouvillian timescale rather than the slowest one (Zhang et al., 2024). That work also identified a Liouvillian exceptional point as the boundary between a regime with real low-lying eigenvalues, where strong Mpemba acceleration exists, and a regime with a complex-conjugate pair, where the strong effect disappears (Zhang et al., 2024).

A distinct NMR implementation used a $\tau$7 working qubit thermalizing through a generalized amplitude damping/Davies map induced by a $\tau$8 auxiliary spin acting as an effective heat sink in $\tau$9 $\lambda_k$0-labeled CHCl$\lambda_k$1 dissolved in acetone-d$\lambda_k$2 (Schnepper et al., 18 Nov 2025). There, a unitary transformed the initial state into a diagonal, population-inverted QME state with higher nonequilibrium free energy, and the measured free-energy curves crossed during relaxation. The same platform was then incorporated into a quantum Otto refrigerator, where the Mpemba-prepared state reduced the duration of the cooling stroke and produced a reported peak cooling-power gain of about $\lambda_k$3 (Schnepper et al., 18 Nov 2025).

5. Extensions beyond standard thermal equilibration

In closed quantum many-body systems, the effect has been reframed as symmetry restoration rather than bath-driven thermalization. For a subsystem $\lambda_k$4, the $\lambda_k$5-th Rényi entanglement asymmetry is

$\lambda_k$6

and the quantum Mpemba effect occurs when a state with larger initial asymmetry satisfies $\lambda_k$7 after a crossing time (Yu et al., 3 Jul 2025). A 12-ion trapped-ion simulator realized this scenario for a tilted ferromagnet evolving under a long-range $\lambda_k$8-symmetric XY Hamiltonian, with symmetry restoration monitored through entanglement asymmetry reconstructed from randomized measurements and classical shadows; the effect was corroborated by a Frobenius distance to the stationary diagonal ensemble, while pure dephasing produced symmetry restoration without the characteristic crossing (Joshi et al., 2024).

Charge-preserving random circuits supply a microscopic mechanism for the closed-system version. For tilted ferromagnets, more asymmetric states restore symmetry faster because nonconserved operators spread into strings containing conserved densities that then diffuse away; for tilted antiferromagnets, the same crossing does not occur (Turkeshi et al., 2024). The review literature further emphasizes that no universal mechanism is known across all closed settings, with quasiparticles in integrable systems, sector-size effects in chaotic dynamics, and localization in MBL systems all appearing as distinct microscopic routes (Yu et al., 3 Jul 2025).

Non-Markovian dynamics introduce a qualitatively different class. In that setting the propagator $\lambda_k$9 and the time-local generator $l_k,r_k$0 remain time-dependent over a finite memory interval, and the long-time dynamics is preceded by a slippage map $l_k,r_k$1 (Strachan et al., 2024). The resulting non-Markovian quantum Mpemba effect can be weak, strong, or extreme. In the extreme case,

$l_k,r_k$2

so a specially chosen initial state reaches the steady state within the memory time itself, a possibility explicitly described as having no Markovian analogue (Strachan et al., 2024).

The concept has also been extended from states to observables. For operator dynamics under the adjoint Liouvillian,

$l_k,r_k$3

a genuine operator Mpemba effect is realized by subtracting the slowest mode while preserving the same steady-state value,

$l_k,r_k$4

The transformed operator then relaxes faster to exactly the same asymptotic observable value as the original one (Bagui et al., 18 May 2026). In quantum optics, an analogous bosonic Mpemba effect was predicted for decay toward vacuum in leaky resonators or waveguides: coherent states alone do not show the effect, whereas crossings arise when at least one initial state is non-classical, such as a Fock, squeezed, or Schrödinger cat state (Longhi, 2024).

6. Applications, detection strategies, and open issues

The effect has increasingly been treated as a resource rather than only an anomaly. A symmetry-protected many-body construction was used for dissipative preparation of a lattice Bose-Einstein condensate in a one-dimensional Bose-Hubbard chain, where inversion symmetry forces a class of initial product states to have zero overlap with the slowest Lindblad mode and therefore relax with the next rate in the spectrum. In that setting the symmetrically localized state yields a reported speedup of about $l_k,r_k$5 over random product states in favorable cases (Westhoff et al., 7 Apr 2025).

Thermometry provides a second application domain. In a Davies model with an effective two-band topology, the initial state that maximizes the short-time distinguishability

$l_k,r_k$6

is the ground state, and this optimal thermometric state exceeds a Haar-random reference state in thermalization speed with probability at least $l_k,r_k$7 for $l_k,r_k$8 (Li et al., 16 Apr 2026). A related metrological formulation proves that Mpemba-type inversions can produce a finite-time enhancement of the quantum Fisher information for temperature estimation, converting anomalous relaxation into a “metrological Mpemba effect” for two-level and $l_k,r_k$9-level probes coupled to bosonic baths (Chattopadhyay et al., 8 Jan 2026).

Because full tomography is costly, observable-level detection has become a separate methodological problem. For open systems with a known unique steady state, a good observable is one with nonzero overlap with the slowest decay mode,

$\lambda_2$0

so its expectation value can certify state-level Mpemba behavior from known state preparations without reconstructing the full density matrix during the evolution (Bagui et al., 2 Dec 2025). In Davies maps this yields a clean population-versus-coherence selection rule, whereas in generic GKSL dynamics the relevant observable may need to be identified from an operator-basis decomposition of the slowest eigenmode (Bagui et al., 2 Dec 2025).

The literature also records substantive caveats. One review states explicitly that there is no universal microscopic mechanism yet for the closed-system effect and that the observable in which the crossing appears can depend strongly on symmetry regime and dynamics (Yu et al., 3 Jul 2025). A different open-system analysis argues that the Mpemba effect is best understood as an emergent property of relaxation-mode structure and is not fundamentally quantum in conceptual origin, even though the dynamics and observables may be quantum (Das, 10 Dec 2025). This suggests terminological caution: the genuine quantum Mpemba effect is a coherent research program centered on anomalous relaxation and mode suppression, but its precise meaning still depends on whether the emphasis is thermodynamic free energy, symmetry restoration, operator relaxation, or operational metrology.