A resource theoretical unification of Mpemba effects: classical and quantum

TL;DR

This work unifies classical and quantum Mpemba effects within a single resource-theoretic framework, recasting thermalization as a depletion of athermality and symmetry restoration as depletion of asymmetry under free operations. By employing Rényi-divergence-based monotones and the framework of modes of asymmetry, the authors show that the presence and timescale of Mpemba crossings hinge on overlaps with slowest-relaxing or slowest symmetry-restoring modes, respectively. Through concrete classical spin-chain and Davies-map examples, and extensive circuit-based explorations for abelian and non-Abelian symmetries, they demonstrate both thermal and symmetry Mpemba effects and their decomposition into symmetry-preserving and symmetry-breaking components. A key result is the exact entropy-splitting relation S(ρ||π)=S(ρ||G[ρ])+S(G[ρ]||π), which clarifies how the same dynamics can exhibit multiple Mpemba behaviors under different monotones. Overall, the paper provides a principled blueprint for diagnosing and engineering fast-relaxation or rapid symmetry restoration by harnessing resource-theoretic quantities across classical, quantum, Markovian, and non-Markovian settings.

Abstract

The Mpemba effect originally referred to the observation that, under certain thermalizing dynamics, initially hotter samples can cool faster than colder ones. This effect has since been generalized to other anomalous relaxation behaviors even beyond classical domains, such as symmetry restoration in quantum systems. This work demonstrates that resource theories, widely employed in information theory, provide a unified organizing principle to frame Mpemba physics. We show how the conventional thermal Mpemba effect arises naturally from the resource theory of athermality, while its symmetry-restoring counterpart is fully captured by the resource theories of asymmetry. Leveraging the framework of modes of asymmetry, we demonstrate that the Mpemba effect due to symmetry restoration is governed by the initial overlap with the slowest symmetry-restoring mode -- mirroring the role of the slowest Liouvillian eigenmode in thermal Mpemba dynamics. Through this resource-theoretical formalism, we uncover the connection between these seemingly disparate effects and show that the dynamics of thermalization naturally splits into a symmetry-respecting and a symmetry-breaking term.

Figures (16)

• Figure 1: In a resource-theoretic framework, the Mpemba effect occurs when an initially more resourceful state depletes that resource faster than a state with less, under the evolution by the same free operation, so that their resource monotones cross. This single picture unifies a variety of anomalous equilibration phenomena (for example, restoring thermal equilibrium or symmetry in classical and quantum systems). Mpemba physics then becomes the study of why different initial states dissipate resources at different rates, and how we can harness those differences to engineer exotic effects such as ultrafast cooling. In this article, we apply this analysis to the specific resource theories shown in the schematic.
• Figure 2: The thermal Mpemba effect in an open classical system . For a classical spin chain described by \ref{['eq:energy:classical:ising']}, we study the -divergence between the non-equilibrium distribution and the thermal state at $\beta_i$ during the thermalization dynamics. Upper-right inset: The crossing times between a thermal and an optimized state for the $\alpha$-divergence \ref{['eq:renyi:divergence']} as a function of $\alpha$. Lower-left inset: the ten eigenvalues of the classical Liouvillian $\hat{\mathcal{L}}^\mathrm{cl}$ closest to zero. The second and the fourth eigenvalue determine the asymptotic decay rate of the -divergence for the thermal and the optimized state, respectively. We chose the parameters $J=-0.4$, $h=0.2$, $N_s=7$, the initial temperature $\beta_i = 0.5$ and the bath temperature $\beta_e=1$.
• Figure 3: The thermal Mpemba effect in an open quantum system. We consider the thermalization of a single qubit described by the Davies map Davies1979. For the state rotated with the unitary transformation outlined in Moroder2024 (dashed line), $M(\hat{\rho}_\theta(t))$ (see \ref{['eq:delta:F']}) decreases exponentially faster than for a random initial state (full line), displaying an Mpemba crossing. Upper-right inset: for the same two states, the Mpemba crossing time $\tau_\alpha$ of the Rényi relative entropy $M_\alpha(\hat{\rho}_\theta(t))$ (\ref{['eq:quantum:Renyi:divergence']}) varies with $\alpha$. Lower-left inset: different from the classical Liouvillian considered in \ref{['fig:classical:open:KL']}, the spectrum of the Davies generator also features complex eigenvalues. We considered the qubit frequency $h=10$, the bath temperature $\beta_e=0.1$, a random initial state defined by the Bloch vector $\mathbf{r}= (0.221, 0.867, 0.206)$ and an optimized state with Bloch vector $\mathbf{r}'= (0,0,0.919)$.
• Figure 4: $\mathrm{U}(1)$ symmetry sectors and mode-occupancy distributions for a three-qubit tilted states. The computational basis elements labeled as $\ketbra{m,\alpha_1}{m'\!,\alpha_2}$ are associated to a magnetization difference $\mu=m-m'$. By reordering the eight basis operators into an $8\times8$ matrix by sorting rows and columns by ascendant $m$, we obtain the $\mathrm{U}(1)$ symmetry sectors shown in panel (a), with each block (colored in yellow) corresponding to a fixed $\abs{\mu}$. Panels (b–d) display the mode-occupancy histograms (defined in \ref{['eq:mode:occupancy']}) for the three tilted-ferromagnet states of \ref{['eq:tilted:state']} with $N_s=3$: in (b) a perfectly symmetric state (only $\mu=0$ appears), in (c) an partially asymmetric state (nonzero weight in several $\mu\neq0$-sectors), and in (d) a maximally asymmetric state (uniform weight across all sectors).
• Figure 5: (a) the generator of the classical Markov chain with the four sites arranged on a ring and three type of jumps making them interacts. (b) and (c) the distributions $\mathbf{p_1}$ and $\mathbf{p_2}$ which are $(0.5,0,0.5,0)^T$ and $(0.5,0.25,0,0.25)^T$ respectively.