Quantum Mpemba-like Effect: Anomalous Relaxation

• Quantum Mpemba-like effect is an anomalous relaxation behavior where states initially farther from equilibrium overtake closer ones, revealing spectral and symmetry dynamics.
• It spans both open and isolated systems, employing methods such as Lindblad mode suppression, entanglement asymmetry, and dressed distances to analyze acceleration.
• Experimental and theoretical studies demonstrate its relevance in systems from trapped ions to nuclear spins, with implications for quantum resources and enhanced thermometry.

The quantum Mpemba-like effect denotes anomalous relaxation in which a quantum preparation that is initially farther from equilibrium, stationarity, or symmetry restoration can become closer to the target state than a preparation that started nearer. In contemporary usage, the phenomenon spans open and isolated dynamics, ordinary and inverse thermalization, genuine thermodynamic formulations based on nonequilibrium free energy, and broader generalizations to operator relaxation, quantum resources, optical fields, relativistic detectors, and imaginary-time dynamics. A standard signature is the existence of two initial states, $\rho_1(0)$ and $\rho_2(0)$, such that $d(\rho_1(0))>d(\rho_2(0))$ but, after a crossover time $t_M$, $d(\rho_1(t))<d(\rho_2(t))$ for all later times (Ares et al., 12 Feb 2025).

1. Conceptual framework and diagnostics

In quantum settings, the relevant “distance” is model-dependent. Open-system studies commonly use the trace distance, $$D(\rho,\rho_{\mathrm{th}})=\frac{1}{2}\|\rho-\rho_{\mathrm{th}}\|_1$$, or the quantum relative entropy to equilibrium. The review by F. Ares, P. Calabrese, and S. Murciano also emphasizes entanglement asymmetry,

$$\Delta S_A^{(n)}=S^{(n)}(\rho_{A,Q})-S^{(n)}(\rho_A),$$

as a diagnostic for symmetry restoration in isolated dynamics, while later work employs Uhlmann fidelity, Hilbert-Schmidt distance, nonequilibrium free energy, and specialized metrics such as the dressed distance for operator evolution (Ares et al., 12 Feb 2025).

The literature distinguishes several related notions. The strong Mpemba effect refers to complete suppression of the slowest relaxation mode, yielding exponential speedup. The inverse Mpemba effect refers to anomalous heating rather than cooling. The genuine quantum Mpemba effect uses nonequilibrium free energy or relative entropy as the thermodynamic measure of distance. In operator dynamics, the term Mpemba-like effect is used because the Heisenberg-picture map is non-trace-preserving and the usual trace distance is not reliable (Bagui et al., 28 Oct 2025).

Setting	Diagnostic	Characteristic mechanism
Open Markovian states	Trace distance; relative entropy; nonequilibrium free energy	Suppression of the slowest Liouvillian mode
Isolated unitary systems	Entanglement asymmetry	Quasiparticle structure or operator spreading
Operator relaxation	Dressed distance	Non-trace-preserving Heisenberg evolution
Non-Markovian or correlated dynamics	Distance to steady state	Slippage, memory, or initial system-bath entanglement

This taxonomy shows that the phenomenon is not restricted to literal “hotter cools faster” scenarios. In the quantum literature it includes symmetry restoration, heating protocols, relaxation of observables, and resource depletion, provided that the farther preparation overtakes the nearer one under a suitable monotone or experimentally accessible proxy (Ares et al., 12 Feb 2025).

2. Open-system spectral mechanism

For Markovian open quantum systems governed by a Lindblad or GSKL master equation, the canonical mechanism is Liouvillian mode selection. If the state is expanded in decay modes, long-time relaxation is normally controlled by the smallest nonzero eigenvalue, but a suitably chosen initial state can have vanishing or negligible overlap with that slowest mode. In the nuclear-spin formulation, the population vector obeys

$$\vec{p}(t)=\vec{p}_{\mathrm{th}}+\sum_{n=1}^{3} a_n e^{\lambda_n t}\vec{v}_n,$$

and the effect arises when the farther state has $a_1\approx 0$; the strong effect corresponds to $a_1=0$ exactly (Chatterjee et al., 16 Sep 2025).

This spectral picture underlies several experimentally and theoretically distinct variants. The single trapped-ion qubit realization of the inverse effect showed that a colder qubit can heat faster than a hotter one, and that sufficiently coherent dynamics can produce the strong version through interference effects (Shapira et al., 2024). The first experimental observation of the quantum strong Mpemba effect in a single trapped ion used an optimal initial state with zero overlap with the slowest decaying mode and established a connection to a Liouvillian exceptional point, where eigenvalues and eigenmodes coalesce (Zhang et al., 2024). A later dissipative-qubit study formulated a canonical quantum Mpemba effect, in which both initial states are thermal, both undergo genuine cooling, and the acceleration is determined solely by the initial temperature (Li et al., 21 Nov 2025).

Natural rather than engineered dissipation was demonstrated in nuclear-spin thermalization. For two homonuclear spin-$\rho_2(0)$0 nuclei in solution, dipolar relaxation was identified as the dominant decoherence channel, and initial states satisfying

$\rho_2(0)$1

were used to keep the dynamics in the population sector. The farther states with negligible overlap on the slowest mode relaxed faster under natural thermalization, without bath engineering, and the same platform also exhibited the genuine quantum Mpemba effect when nonequilibrium free energy was used instead of trace distance (Chatterjee et al., 16 Sep 2025).

Quantum dots provide a complementary analytic setting. In a single-level quantum dot coupled to a thermal bath, the occurrence of the effect depends strongly on the sign and magnitude of the electron-electron interaction $\rho_2(0)$2. The analysis uses fermionic duality to interpret charge and parity decay modes, and finds that strongly attractive interaction $\rho_2(0)$3 produces a pronounced separation between the slow charge mode and the fast parity mode, making the Mpemba effect particularly visible in both nonequilibrium free energy and dot energy (Graf et al., 2024).

3. Isolated, integrable, and chaotic unitary dynamics

In isolated quantum systems, the effect is not driven by environmental dissipation but by quantum fluctuations generated after a quench. The review literature identifies faster local equilibration or faster symmetry restoration for initial states that are farther from equilibrium, with entanglement asymmetry serving as a sensitive probe when the Hamiltonian preserves a symmetry that the initial state breaks (Ares et al., 12 Feb 2025).

For integrable systems, the microscopic criterion is phrased in terms of quasiparticle excitations emitted during the quench. If the initial state with greater symmetry breaking generates faster-moving symmetry-breaking quasiparticles, then symmetry is restored earlier in that state, producing the effect. The same review stresses that for non-integrable or chaotic closed systems the mechanism is less clear and remains an active research direction (Ares et al., 12 Feb 2025).

A concrete chaotic realization was obtained in charge-preserving random circuits on qudits. There, tilted ferromagnets display the effect, whereas tilted antiferromagnets do not. The observable is entanglement asymmetry, and the mechanism is the spreading of nonconserved operators in terms of conserved densities. Numerical and analytical results show a characteristic Mpemba time scaling as $\rho_2(0)$4, with $\rho_2(0)$5 the subsystem size (Turkeshi et al., 2024).

This body of work establishes that anomalous overtaking does not require a bath. In isolated dynamics, locality, unitarity, and symmetry are sufficient ingredients for state-dependent acceleration of symmetry restoration, and the relevant “distance” may be entanglement- or symmetry-based rather than thermometric (Turkeshi et al., 2024).

4. Memory effects, slippage, and initial system–bath correlations

Non-Markovian dynamics introduce new classes of effects with no analogue in Markovian quantum dynamics. In a general open setting with finite memory time and a unique steady state, even the steady state itself can be perturbed away from stationarity by the non-Markovian slippage map and may then relax back only slowly. The key construction is the fast state

$\rho_2(0)$6

which is mapped directly onto the steady state after the memory interval. This yields weak, strong, and extreme non-Markovian quantum Mpemba effects, including the possibility of reaching the steady state within the memory time itself (Strachan et al., 2024).

A distinct mechanism arises from initial system-reservoir entanglement alone. Most earlier open-system treatments assumed an initial product state, but a later study showed that initial entanglement can transiently violate trace-contractivity for the reduced dynamics and thereby generate a Mpemba effect that is different from strong-mode suppression. In a two-level atom undergoing spontaneous emission in a photonic waveguide at zero temperature, a time-reversed entangled initial state produces backflow and retarded thermalization, with

$\rho_2(0)$7

The reduced system can initially move away from equilibrium before decaying, so a preparation that starts closer can relax more slowly than one that starts farther (Longhi, 30 Apr 2025).

These results broaden the conceptual basis of the subject. In Markovian settings, anomalous relaxation is largely spectral; in non-Markovian and correlated settings, memory kernels, slippage, and initial correlations become dynamical resources or obstructions in their own right (Strachan et al., 2024).

5. Extensions beyond state thermalization

One major extension concerns operators rather than states. In the Heisenberg picture, open-system evolution is generally non-trace-preserving, so the trace distance of an operator need not decay monotonically. To address this, the operator-relaxation framework introduces the dressed distance, which is monotonic for Hermitian operators under arbitrary Lindblad dynamics and therefore supports a consistent definition of an operator Mpemba-like effect. The resulting classification is dimension-sensitive: in a single qubit only accelerated relaxation of operators is possible, while genuine Mpemba-like crossings appear in qutrits and higher-dimensional systems, as well as for nonlocal current operators in a double-quantum-dot setup (Bagui et al., 28 Oct 2025).

A second extension shifts the target from thermal equilibrium to the depletion of quantum resources. In random-circuit models framed by quantum resource theories, coherence and imaginarity display a quantum Mpemba effect when the system is initialized in resourceful product states, while non-Gaussianity and magic do not. At the same time, all four resources exhibit the Pontus-Mpemba effect, in which a preheating stage accelerates later relaxation relative to direct cooling (Aditya et al., 26 Sep 2025).

Quantum optics supplies several further realizations. In a leaky optical resonator or waveguide, coherent states of different amplitudes preserve their ordering relative to the vacuum and therefore do not show the effect, but certain non-classical states of light—Fock states, squeezed states, and Schrödinger cat states—can overtake coherent states when the full reduced density matrix is tracked. The effect is not visible at the level of mean photon number alone (Longhi, 2024). In the quantum theory of lasers, a single-mode laser above threshold exhibits a Mpemba-like effect in photon-number statistics: a Fock state at the mean photon number can relax faster than a near-equilibrium Poissonian state because the slowest mode is absent when the initial mean photon number matches the steady-state mean, so $\rho_2(0)$8 and faster modes dominate (Longhi, 19 Mar 2025).

Together these works show that “Mpemba-like” behavior has become a general language for anomalous overtaking in non-equilibrium quantum dynamics, even when the relaxing object is an operator, a resource monotone, or a photonic distribution rather than a density matrix approaching a Gibbs state (Bagui et al., 28 Oct 2025).

6. Relativistic, chaotic, imaginary-time, and continuous-space variants

In Unruh thermalization, a uniformly accelerated Unruh–DeWitt detector undergoing irreversible relaxation to a Gibbs state follows trajectory-dependent Bloch-sphere flows whose rates depend on the field type and spacetime dimension. Two-temperature heating and cooling protocols reveal a quantum Mpemba-like asymmetry: heating is faster than cooling in terms of Uhlmann fidelity change. The maximum fidelity difference serves as a diagnostic separating Unruh thermalization from classical bath-driven thermalization of an inertial detector (Wang et al., 6 Sep 2025).

Strongly chaotic dissipative many-body systems can also exhibit related phenomena. In Sachdev–Ye–Kitaev systems coupled strongly to SYK thermal baths, effective temperature oscillations and Mpemba crossings appear under strong coupling, whereas these effects are absent in the Lindbladian description. The work emphasizes that, in this regime, the anomaly is tied to genuine far-from-equilibrium dynamics rather than merely to initial-state alignment with Liouvillian modes (Wang et al., 2024).

Imaginary-time dynamics provide a non-thermal but computationally important variant. Numerically exact quantum Monte Carlo simulations show that in several interacting quantum many-body models, initial states with higher energy can relax faster than lower-energy initial states under imaginary-time projection. The mechanism is again slow-mode avoidance: states with less weight on low-energy excitations converge more rapidly to the ground state. Because imaginary-time evolution is central to projective QMC and related algorithms, this imaginary-time Mpemba effect suggests a route to faster many-body computation, especially when long projection times are costly (Chang et al., 2024).

A continuous-space open-system realization appears in the quantum tunneling Mpemba effect for a particle in a symmetric double-well potential with absorbing boundaries. Using the non-Hermitian effective Hamiltonian, the survival probability is decomposed as

$\rho_2(0)$9

The first non-trivial even-parity coefficient $d(\rho_1(0))>d(\rho_2(0))$0 is proven to have a universal non-monotonic thermal peak governed by the Sturm–Liouville oscillation theorem and independent of the global system size $d(\rho_1(0))>d(\rho_2(0))$1. Observable crossings in survival probability or trace distance require a strict timescale separation, $d(\rho_1(0))>d(\rho_2(0))$2 and $d(\rho_1(0))>d(\rho_2(0))$3 (Hayakawa et al., 4 Jul 2026).

7. Experimental realizations, applications, and open problems

Experimental work has moved from proof-of-principle demonstrations to naturally occurring and application-oriented realizations. The 2025 review identifies three landmark experiments: Joshi et al. observed the effect in a 12-site trapped-ion spin chain by monitoring entanglement asymmetry during unitary dynamics; Shapira et al. demonstrated the inverse and strong effects in a trapped-ion qubit using trace distance to the steady state; and Zhang et al. observed the strong effect in a single trapped ion, where off-diagonal elements engineered by unitary rotation suppressed the slowest Lindblad eigenmode and greatly accelerated relaxation (Ares et al., 12 Feb 2025).

A decisive step beyond engineered baths was the direct observation in nuclear spins undergoing natural thermalization. In two $d(\rho_1(0))>d(\rho_2(0))$4 spins in 2-chloroacrylonitrile dissolved in DMSO, the effect was observed under dipolar relaxation without external control during thermalization, and the same platform also showed the genuine quantum Mpemba effect based on non-equilibrium free energy rather than trace distance (Chatterjee et al., 16 Sep 2025). Another experiment realized the genuine effect in a spin-$d(\rho_1(0))>d(\rho_2(0))$5 system interacting with a heat sink and then embedded the protocol in a quantum Otto refrigerator, reporting an observed power gain $d(\rho_1(0))>d(\rho_2(0))$6 (Schnepper et al., 18 Nov 2025).

Metrological applications have become a second major front. One study proves that Mpemba-type inversions generically yield a finite-time enhancement of the quantum Fisher information for temperature estimation, establishing a metrological Mpemba effect in two-level and $d(\rho_1(0))>d(\rho_2(0))$7-level probes (Chattopadhyay et al., 8 Jan 2026). A related thermometric analysis shows that, in a Markovian multilevel probe model, the initial states that are optimal for thermometry exhibit the quantum Mpemba effect with high probability and thermalize faster than most initial states (Li et al., 16 Apr 2026). These results reframe anomalous relaxation as a design principle for nonequilibrium thermometry rather than a mere curiosity.

Several questions remain open. The review literature highlights universality and classical–quantum correspondence, the status of the effect in closed ergodic or chaotic systems, the role of monitored dynamics and measurement-induced entanglement transitions, the impact of disorder, inhomogeneities, and impurities, and the scope for non-adiabatic state preparation, quantum control, and quantum simulation (Ares et al., 12 Feb 2025). A plausible implication is that future progress will depend less on a single universal definition than on identifying which distance measure, spectral decomposition, or symmetry diagnostic is physically meaningful in each dynamical setting.