Quantum Mpemba Effect: Accelerated Equilibration

• Quantum Mpemba effect is a non-equilibrium phenomenon where a 'hotter' or more asymmetric quantum state relaxes faster than a cooler one by bypassing the slowest decay mode.
• Experimental studies using trapped-ion platforms show that initial state engineering can suppress the slow Liouvillian mode, leading to exponential acceleration in relaxation.
• The effect spans open systems governed by Lindblad dynamics and isolated systems monitored via entanglement asymmetry and quasiparticle propagation, offering insights into advanced quantum control.

Searching arXiv for papers on the Quantum Mpemba Effect to ground the article in the cited literature. The quantum Mpemba effect (QMPE) is an anomalous non-equilibrium relaxation phenomenon in which an initial quantum state that is “hotter” or otherwise farther from the steady state can equilibrate faster than a “cooler” or closer initial state. In quantum systems, the effect appears in two distinct settings: open systems governed by dissipative dynamics, typically of Lindblad form, and isolated systems undergoing unitary quenches, where equilibration is understood locally through reduced states, effective ensembles, or symmetry-restoration proxies. A central distinction is between weak QMPE, where the slowest relaxation channel is reduced but not removed, and strong QMPE, where the overlap with the slowest mode vanishes and relaxation is governed by a faster mode, yielding exponential acceleration (Ares et al., 12 Feb 2025).

1. Definitions, diagnostics, and basic phenomenology

In open quantum systems, the effect is defined relative to a steady state $\rho_{\mathrm{ss}}$, while in isolated dynamics it is defined relative to a local equilibrium proxy, such as a diagonal ensemble, generalized Gibbs ensemble, or a symmetry-restored reduced state. Operationally, QMPE is identified when the distance to equilibrium for two initial preparations crosses in time, so that the initially farther state becomes closer at later times (Ares et al., 12 Feb 2025).

Several distance measures are used. For a density matrix $\rho(t)$ and target state $\rho_{\mathrm{ss}}$, common choices are the trace distance

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$

the quantum relative entropy

$$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$

and the Frobenius or Hilbert–Schmidt distance

$$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$

The equilibration or mixing time is defined as the minimal time $\tau_\epsilon$ such that a chosen distance satisfies $D(t)\le \epsilon$; the Mpemba effect corresponds to a non-monotonic dependence of $\tau_\epsilon$ on initial temperature or preparation (Ares et al., 12 Feb 2025).

For isolated systems with conserved symmetry, a particularly important diagnostic is entanglement asymmetry. For a subsystem $A$ with reduced density matrix $\rho(t)$0 and symmetry generator $\rho(t)$1, one defines the symmetry-projected state $\rho(t)$2 by block-diagonalizing $\rho(t)$3 in $\rho(t)$4 sectors. The Rényi entanglement asymmetry is

$\rho(t)$5

with

$\rho(t)$6

For $\rho(t)$7 symmetry, a convenient representation uses charged moments

$\rho(t)$8

with $\rho(t)$9, giving

$\rho_{\mathrm{ss}}$0

This quantity is nonnegative and vanishes if and only if $\rho_{\mathrm{ss}}$1 respects the symmetry, making it a robust proxy for local symmetry restoration (Ares et al., 12 Feb 2025).

The review literature distinguishes weak and strong QMPE. In weak QMPE, the slower mode remains present, but its overlap is reduced; distances cross at a finite time, yet both trajectories remain asymptotically controlled by the same slow spectral rate. In strong QMPE, the overlap with the slowest mode is exactly zero, and relaxation proceeds with the next-fastest eigenmode, giving exponentially faster late-time decay (Ares et al., 12 Feb 2025).

2. Open-system QMPE and Liouvillian mode suppression

In Markovian open systems, the dynamics is generated by a Lindblad master equation

$\rho_{\mathrm{ss}}$2

The Liouvillian $\rho_{\mathrm{ss}}$3 is generally non-Hermitian and admits a biorthogonal spectral decomposition in terms of right eigenoperators $\rho_{\mathrm{ss}}$4 and left eigenoperators $\rho_{\mathrm{ss}}$5,

$\rho_{\mathrm{ss}}$6

so that

$\rho_{\mathrm{ss}}$7

with

$\rho_{\mathrm{ss}}$8

Here $\rho_{\mathrm{ss}}$9 corresponds to the steady state, and $$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$0 for decaying modes (Ares et al., 12 Feb 2025).

This decomposition makes the mechanism of strong QMPE explicit. If the slowest decaying mode has eigenvalue $$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$1, then the long-time mixing time scales as

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$2

when $$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$3. If instead

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$4

then the slowest mode is removed and

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$5

so the relaxation rate is set by the next eigenvalue and is exponentially faster (Ares et al., 12 Feb 2025).

A particularly important design principle is initial-state engineering through a unitary rotation that makes the initial state orthogonal to the slowest left eigenoperator. This strategy was proposed in the Markovian setting and later realized experimentally in a single trapped ion (Zhang et al., 2024). In that experiment, a three-level system was engineered with Hamiltonian

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$6

and jump operators

$$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$7

with $$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$8 and $$D_{\mathrm{tr}}(t)=\frac{1}{2}\|\rho(t)-\rho_{\mathrm{ss}}\|_1,$$9. The strong Mpemba state $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$0 was prepared so that

$$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$1

and the observed relaxation was governed by $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$2 rather than $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$3, demonstrating exponential acceleration (Zhang et al., 2024).

The same experiment identified a Liouvillian exceptional point (LEP) as the boundary between strong and weak QMPE. As a control parameter was tuned, $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$4 and $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$5 merged and then formed a complex-conjugate pair; beyond that point, the two slowest contributions could no longer be simultaneously eliminated, and only weak Mpemba speedups remained (Zhang et al., 2024). This establishes a concrete link between strong QMPE and non-Hermitian spectral structure, but later work also showed that exceptional points are not universally necessary or sufficient in more general non-Hermitian settings (Ma et al., 25 Aug 2025).

Open-system QMPE also extends beyond equilibrium relaxation. In nonequilibrium open quantum systems coupled to two baths, the target state is a nonequilibrium steady state rather than a Gibbs state. In that setting, anomalous Mpemba and inverse Mpemba effects can appear in populations, entanglement, and quantum mutual information, and nonequilibrium conditions can enlarge the parameter regime where crossings occur (Wang et al., 2024). The same work emphasizes that nonequilibrium-induced coherence can substantially contribute to the effect, and that conventional secular Lindblad treatments may fail to capture it (Wang et al., 2024).

A thermodynamic formulation is available for Davies generators. For a Gibbs state $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$6, the non-equilibrium free energy is

$$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$7

with excess

$$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$8

If the Liouvillian gap is defined by a complex coherence mode, then dephasing an initial state in the energy basis suppresses the slow coherent modes and guarantees exponential speedup. When the transformed state also has higher non-equilibrium free energy, the resulting crossing in $$D(\rho(t)\Vert \rho_{\mathrm{ss}})=\mathrm{Tr}\!\left[\rho(t)\big(\ln\rho(t)-\ln\rho_{\mathrm{ss}}\big)\right],$$9 is interpreted as a genuine quantum Mpemba effect (Moroder et al., 2024).

3. Isolated dynamics, integrability, and symmetry restoration

In isolated systems, there is no bath and no global steady state. The dynamics is unitary,

$$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$0

and equilibration is understood locally, for reduced states or observables. In this setting, the quantum Mpemba effect concerns local thermalization or local symmetry restoration rather than global relaxation (Ares et al., 12 Feb 2025).

In integrable many-body systems with a conserved $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$1 charge, the key diagnostic is entanglement asymmetry. A more asymmetric initial state can restore symmetry faster than a less asymmetric one, even though it starts farther from the symmetry-restored regime. The microscopic explanation is given by a quasiparticle picture: initial states act as sources of entangled quasiparticle pairs with mode-dependent velocities, and the decay of entanglement asymmetry is controlled by how strongly symmetry-breaking fluctuations are loaded into slow or fast carriers (Rylands et al., 2023).

A central result in that framework is an exact expression for the entanglement asymmetry in terms of mode-resolved charged moments,

$$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$2

where the coefficients $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$3 are determined by the initial quasiparticle occupations and the propagation filter $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$4 with $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$5 (Rylands et al., 2023). At short times, the entanglement asymmetry depends on the variance of subsystem charge fluctuations, while at long times it depends on the slowest quasiparticles. This leads to a transparent criterion: a more asymmetric state relaxes faster when its extra symmetry-breaking fluctuations are carried more strongly by faster quasiparticles and less by the slowest modes (Rylands et al., 2023).

The review literature summarizes this mechanism by stating that, in integrable systems, QMPE occurs when the quasiparticle modes that most strongly carry the symmetry-breaking correlations have higher group velocities. This framework explains not only single crossings but also multiple crossings and model dependence, including behavior in the XXZ chain that echoes its zero-temperature phase diagram even though high-energy dynamics is involved (Ares et al., 12 Feb 2025).

Chaotic systems exhibit a different mechanism. In charge-preserving random circuits, the relevant quantity remains entanglement asymmetry, but its decay is linked to operator spreading of non-conserved operators in the presence of conserved densities. Tilted ferromagnets display crossings, whereas tilted antiferromagnets may not. The decisive difference is that, in tilted ferromagnets, larger asymmetry enhances the emission and diffusion of conserved densities, accelerating symmetry restoration, while the staggered structure of tilted antiferromagnets blocks that mechanism (Turkeshi et al., 2024). In that setting, the asymmetry decay obeys a diffusive scaling with Mpemba time

$$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$6

and the more asymmetric tilted ferromagnet restores symmetry faster to the grand-canonical ensemble (Turkeshi et al., 2024).

The isolated-system viewpoint is supported experimentally by a trapped-ion quantum simulator with $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$7 spin-$$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$8 ions and long-range $$D_F(t)=\sqrt{\mathrm{Tr}\!\left[(\rho(t)-\rho_{\mathrm{ss}})^\dagger(\rho(t)-\rho_{\mathrm{ss}})\right]}.$$9-preserving Hamiltonian dynamics. Initial tilted ferromagnets with different tilt angles $\tau_\epsilon$0 showed clear crossings in entanglement asymmetry for a four-ion subsystem: the more strongly symmetry-broken state restored symmetry faster. The same qualitative crossing appeared in the subsystem Frobenius distance to a stationary proxy, whereas control experiments with only environmental noise showed symmetry restoration without crossings, isolating the role of quantum fluctuations (Joshi et al., 2024).

4. Theoretical frameworks and genuinely quantum features

Several theoretical frameworks have been used to systematize QMPE. In open quantum systems, the dominant framework is non-Hermitian spectral theory of Lindblad generators, where strong and weak effects are cleanly characterized by the overlap with the slowest decay mode (Ares et al., 12 Feb 2025). In isolated integrable dynamics, the quasiparticle picture provides necessary and sufficient criteria in terms of occupation functions, slow-mode participation, and charge fluctuations (Rylands et al., 2023). In chaotic systems, operator spreading and hydrodynamics of non-conserved operators determine whether crossings occur (Turkeshi et al., 2024).

A notable development is the study of non-Markovian memory. In general non-Markovian open quantum dynamics, the effective evolution admits a finite memory time, and even a system initialized in the steady state can be driven away from it before returning. Within this framework, new classes of non-Markovian quantum Mpemba effects were identified, including an “extreme” case where a specially chosen initial state reaches the steady state within the memory time itself, which is the fastest possible relaxation compatible with the dynamics (Strachan et al., 2024). In that analysis, the effective post-memory dynamics is governed by a time-independent generator $\tau_\epsilon$1, but the initial slippage operator $\tau_\epsilon$2 reshapes which decay modes are excited, allowing memory-induced acceleration not available in Markovian systems (Strachan et al., 2024).

Noise can also induce or eliminate QMPE in open systems. For random telegraph noise, the dynamics can be embedded in an enlarged Markovian space, producing additional decay modes shifted by the noise correlation rate. In the long-correlation-time limit, these modes can dominate late-time relaxation and create anomalous slowdown for selected initial states, thereby inducing or eliminating QMPE; this can even lead to the counterintuitive slowdown of decoherence by noise (Zhao et al., 16 Jul 2025). This suggests that the spectral criterion generalizes beyond noiseless Markovian Liouvillians, although the relevant slow modes are then noise-induced rather than intrinsic to the original open system.

The role of genuinely quantum features depends on the setting. In open systems, populations may still dominate some relaxation pathways, but coherence and its decay can be decisive when unitary preprocessing bypasses long-lived coherent sectors (Ares et al., 12 Feb 2025). In isolated systems, by contrast, the anomalies are intrinsically quantum: they are governed by entanglement growth, interference, selection rules, quasiparticle propagation, and operator spreading. Entanglement asymmetry is especially important because it is defined at the reduced-density-matrix level and vanishes if and only if local symmetry is restored (Ares et al., 12 Feb 2025).

The relation to classical Mpemba physics is therefore twofold. The shared core is spectral-mode suppression: in both classical and quantum settings, crossings arise when initial conditions suppress or bypass slow relaxation pathways. The quantum setting adds operator-valued dynamics, coherence sectors, biorthogonal Liouvillian spectra, and diagnostics such as entanglement asymmetry and symmetry-resolved measures that have no direct classical counterpart (Ares et al., 12 Feb 2025).

5. Experimental observations and platforms

Experimental evidence for QMPE now spans both isolated and open trapped-ion platforms. In the isolated case, a trapped-ion simulator with $\tau_\epsilon$3 $\tau_\epsilon$4 ions implemented long-range XY dynamics with $\tau_\epsilon$5 symmetry. Tilted ferromagnets prepared by global rotations about the $\tau_\epsilon$6 axis showed crossings in subsystem entanglement asymmetry and Frobenius distance to the diagonal ensemble, establishing the effect in a many-body non-integrable setting (Joshi et al., 2024). The system used $\tau_\epsilon$7 ions, subsystem size $\tau_\epsilon$8, and tilt angles $\tau_\epsilon$9; the largest initial asymmetry relaxed fastest (Joshi et al., 2024).

In the open-system setting, a single trapped-ion qubit was used in an inverse Mpemba protocol: the bath temperature was suddenly increased, and colder initial states heated faster than warmer ones when the trace distance to the new steady state was monitored. In sufficiently coherent qubits, the speedup was exponential, constituting a strong inverse QMPE (Ares et al., 12 Feb 2025). A separate single-ion experiment realized strong QMPE in a three-level Markovian system by preparing one initial state as a unitary rotation of another so that the slowest Liouvillian mode was suppressed; the rotated, initially farther state relaxed exponentially faster (Zhang et al., 2024).

The following table summarizes the principal experimental settings explicitly described in the literature cited here.

Platform	Setting	Principal observable
$D(t)\le \epsilon$0 trapped ions	Isolated quench, long-range $D(t)\le \epsilon$1-preserving dynamics	Entanglement asymmetry; Frobenius distance (Joshi et al., 2024)
Single trapped-ion qubit	Open inverse Mpemba protocol	Trace distance to heated steady state (Ares et al., 12 Feb 2025)
Single trapped ion, three-level system	Open Markovian strong QMPE	Trace distance; mode overlaps $D(t)\le \epsilon$2 (Zhang et al., 2024)

Beyond trapped ions, the literature identifies quantum dots coupled to reservoirs, photonic and bosonic systems, superconducting qubits, and self-contained quantum refrigerators as active or proposed platforms (Ares et al., 12 Feb 2025). In the qubit–qutrit self-contained refrigerator, for example, a unitary-transformed Mpemba state was shown numerically to accelerate the cooling of the target qubit by suppressing the slowest Liouvillian mode, and both global and local unitaries on the qubit–qutrit system could generate such a state (Mondal et al., 21 Jul 2025). In a quantum dot with reservoirs, anomalous crossings were identified in a dynamically defined dot temperature, with the slowest relaxation mode absent and the effect arising from the remaining modes (Chatterjee et al., 2023).

More recent work has broadened the experimental landscape further. A study of nuclear spins under dipolar relaxation reported direct observation of QMPE and genuine QMPE during natural thermalization without bath engineering, using trace distance and relative entropy as diagnostics, and showing that a state farther from equilibrium could relax faster under unforced thermalization (Chatterjee et al., 16 Sep 2025). This suggests that engineered dissipation is not required in every platform, although the original trapped-ion experiments remain the cleanest demonstrations of strong mode-selective acceleration.

6. Relation to other Mpemba phenomena and open problems

The quantum Mpemba effect is now understood as a family of related anomalous-relaxation phenomena rather than a single mechanism. In open Markovian systems, the decisive condition for strong QMPE is the exact suppression of the slowest Liouvillian mode,

$D(t)\le \epsilon$3

which shifts the asymptotic decay from $D(t)\le \epsilon$4 to $D(t)\le \epsilon$5 (Ares et al., 12 Feb 2025). In isolated integrable systems, the decisive criterion is instead how symmetry-breaking charge fluctuations are distributed over slow and fast quasiparticles (Rylands et al., 2023). In chaotic circuits, the criterion is whether the initial state seeds faster spreading of the relevant non-conserved operators (Turkeshi et al., 2024).

Several extensions remain under active investigation. One is the dependence on the chosen metric: trace distance, relative entropy, Frobenius distance, thermodynamic free energy, and fidelity can all reveal crossings, but their detailed ordering properties need not always coincide (Ares et al., 12 Feb 2025). Another is finite-size scaling and the many-body limit, especially the distinction between ergodic Hamiltonians satisfying ETH, integrable models with ballistic quasiparticles, and many-body-localized regimes where operator spreading is slow but structured (Ares et al., 12 Feb 2025).

There are also open questions about geometry and resource theory. A thermodynamic perspective based on non-equilibrium free energy shows that dephasing or unitary preprocessing can turn a farther state into a genuinely faster-relaxing one when the slowest coherent mode is removed (Moroder et al., 2024). Resource-theoretic extensions suggest analogous Mpemba phenomena for local coherence and imaginarity in random circuits, while non-Gaussianity and magic can exhibit related Pontus–Mpemba behavior under preheating even when ordinary QMPE is absent for the same initial families (Aditya et al., 26 Sep 2025).

A further frontier concerns measurement and scalability. Since full state tomography is expensive in many-body systems, recent work has shown that suitable observables can be sufficient to detect QMPE if they overlap with the slow Liouvillian mode. In open systems with a unique steady state, a carefully chosen observable can certify accelerated relaxation without reconstructing the full density matrix trajectory (Bagui et al., 2 Dec 2025). This suggests practical routes for extending QMPE studies to larger devices.

The field therefore presents two complementary unifying pictures. One is spectral: QMPE occurs when the relevant slow mode is weakly excited or fully suppressed. The other is dynamical: more asymmetric, hotter, or otherwise farther initial states can relax faster when the physical carriers of the relevant non-equilibrium content are also faster, or when coherence, symmetry, or memory reshapes the accessible relaxation pathways. Both pictures are now supported by theory and experiment across open and isolated quantum systems (Ares et al., 12 Feb 2025).