Exponentially accelerated relaxation and quantum Mpemba effect in open quantum systems

TL;DR

This work addresses the quantum Mpemba effect in open systems described by Davies maps and proposes a dressing protocol U = U1 P_pi Λ^† that both suppresses the slowest decay mode and maximizes the initial distance to equilibrium. By analyzing Hilbert-Schmidt distance, quantum relative entropy, and trace distance, the authors prove the existence of permutation matrices that optimize these measures, enabling a genuine Mpemba crossover where a system farther from equilibrium relaxes faster. The framework is demonstrated on a two-level system and on many-body spin models (TFI and XXZ), with clear crossovers indicating accelerated relaxation. The results offer a versatile, computationally efficient approach to engineer Mpemba-like speed-ups in Markovian open quantum dynamics and connect with prior QME literature.

Abstract

We investigate the quantum Mpemba effect in the relaxation of open quantum systems whose effective dynamics is described by Davies maps. We present a class of unitary transformations built from permutation matrices that, when applied to the initial state of the system, (i) suppress the slowest decaying modes of the nonunitary dynamics; (ii) maximize its distinguishability from the steady state. The first requirement guarantees exponentially accelerating convergence to the steady state, and the second implies that a quantum system initially farther from equilibrium approaches it more rapidly than one that starts closer. This protocol provides a genuine Mpemba effect, and its numerical simulation requires low computational effort. We prove that, for any initial state, one can always find a permutation matrix that maximizes its distance from equilibrium for a specified information-theoretic distinguishability measure. We illustrate our findings for a two-level system, and also for the nonunitary dynamics of the transverse field Ising chain and XXZ chain, each weakly coupled to a bosonic thermal bath, and demonstrate the quantum Mpemba effect as captured by the Hilbert-Schmidt distance, quantum relative entropy, and trace distance. Our results provide a versatile framework to engineer the genuine quantum Mpemba effect in Markovian open quantum systems.

Figures (4)

• Figure 1: (Color online) Overview of the quantum Mpemba effect (QME). (a) We consider a probe state $\rho({t_0})$ that is transformed into ${\rho'}({t_0}) = U\rho({t_0}){U^{\dagger}}$ by means of the unitary matrix $U = {U_1}{P_{\pi}}{\Lambda^{\dagger}}$. This state contributes to eliminate the slowest decaying mode in Davies maps with generator $\mathcal{L}[\bullet]$. Therefore, the relaxation to equilibrium is accelerated exponentially, controlled by the real part of eigenvalue $\lambda_4$. The unitary $U$ maximizes the distance between ${\rho'}({t_0})$ and the steady state ${\texttt{R}_1}$, such that $\mathcal{D}({\rho'}({t_0}),{\texttt{R}_1}) > \mathcal{D}({\rho}({t_0}),{\texttt{R}_1})$. QME occurs if there exist a time ${t_{\text{QME}}}$ such that for all $t > {t_{\text{QME}}}$, one finds $\mathcal{D}({e^{t\mathcal{L}}}[{\rho'}({t_0})],{\texttt{R}_1}) < \mathcal{D}({e^{t\mathcal{L}}}[\rho({t_0})],{\texttt{R}_1})$. (b) The protocol to accelerate convergence to equilibrium involves the following steps: (1) unitary matrix $\Lambda$ maps the probe state to its diagonal form; (2) permutation matrix ${P_{\pi}}$ rearranges the diagonal entries of ${\Lambda^{\dagger}}\rho({t_0}){\Lambda}$, so that the transformed state is as far away from equilibrium as possible; (3) the resulting state from the previous step is rotated by the unitary matrix $U_1$ to the eigenbasis of the Hamiltonian.
• Figure 2: (Color online) The quantum Mpemba effect in the dynamics of the two-level system described in Sec. \ref{['sec:00000000005A']}. Here we use $\delta = {\varepsilon_1} - {\varepsilon_2} = 1$, $T = 1$, and ${k_B} = 1$. To investigate the relaxation of the instantaneous states $\rho(t)$ and ${\rho'}(t)$ towards the steady state ${\texttt{R}_1}$, we consider the following figures of merit: (a) trace distance, ${\mathcal{D}_{\text{HSD}}}(\rho(t),{\texttt{R}_1})$; (b) quantum relative entropy, ${\mathcal{D}_{\text{QRE}}}(\rho(t),{\texttt{R}_1})$. In each panel, we plot each of these distinguishability measures as a function of the dimensionless parameter $\gamma{t}$. The blue solid lines refer to the initial state $\rho({t_0})$ of the system, while the dressed state ${\rho'}({t_0})$ is represented by red dashed lines.
• Figure 3: (Color online) The quantum Mpemba effect in the dynamics of the transverse field Ising (TFI) model with open boundary conditions [see details in Sec. \ref{['sec:00000000005B']}], where $N = 5$ spins, $h = J/2$, $\gamma = 1$, and $T = 0.1$. To investigate the relaxation of the instantaneous states $\rho(t)$ and ${\rho'}(t)$ towards the steady state ${\texttt{R}_1}$, we consider the following figures of merit: (a) Hilbert-Schmidt distance, ${\mathcal{D}_{\text{HSD}}}(\rho(t),{\texttt{R}_1})$; (b) quantum relative entropy, ${\mathcal{D}_{\text{QRE}}}(\rho(t),{\texttt{R}_1})$; (c) trace distance, ${\mathcal{D}_{\text{TD}}}(\rho(t),{\texttt{R}_1})$. In each panel, we plot each of these distinguishability measures as a function of the dimensionless parameter $Jt$. The blue solid lines refer to the initial state $\rho({t_0})$ of the system, while the dressed state ${\rho'}({t_0})$ is represented by red dashed lines.
• Figure 4: (Color online) The quantum Mpemba effect in the dynamics of the XXZ model with open boundary conditions [see details in Sec. \ref{['sec:00000000005B']}], where $N = 5$ spins, $\Delta = J/2$, $\gamma = 1$, and $T = 0.1$. To investigate the relaxation of the instantaneous states $\rho(t)$ and ${\rho'}(t)$ towards the steady state ${\texttt{R}_1}$, we consider the following figures of merit: (a) Hilbert-Schmidt distance, ${\mathcal{D}_{\text{HSD}}}(\rho(t),{\texttt{R}_1})$; (b) quantum relative entropy, ${\mathcal{D}_{\text{QRE}}}(\rho(t),{\texttt{R}_1})$; (c) trace distance, ${\mathcal{D}_{\text{TD}}}(\rho(t),{\texttt{R}_1})$. In each panel, we plot each of these distinguishability measures as a function of the dimensionless parameter $Jt$. The blue solid lines refer to the initial state $\rho({t_0})$ of the system, while the dressed state ${\rho'}({t_0})$ is represented by red dashed lines.