Dynamics of entanglement fluctuations and quantum Mpemba effect in the $ν=1$ QSSEP model

TL;DR

This work addresses how entanglement fluctuations evolve in a stochastic quantum diffusion model, the $\nu=1$ QSSEP. It develops a generalized diffusive quasiparticle (QP) picture that incorporates correlations between quasiparticle pairs and yields the full-time distribution of the entanglement entropy, via quantities like $P_{\Theta}(S_A)$ and the generating function $\chi_{\Theta}(\lambda)$. Applying this to entanglement asymmetry, it shows dynamical restoration of particle-number symmetry with long-time decay $\langle \Delta S_A^{(2)}(\Theta)\rangle \sim \ell/\sqrt{\Theta}$, and finds the quantum Mpemba effect is strongly inhibited by the momentum averaging in the random-hopping dynamics. The results provide analytic control over quantum fluctuations in diffusive regimes and point to future extensions to higher $\nu$ and interacting stochastic dynamics.

Abstract

We study the out-of-equilibrium dynamics of entanglement fluctuations in the $ν=1$ Quantum Symmetric Simple Exclusion Process, a free-fermion chain with hopping amplitudes that are stochastic in time but homogeneous in space. Previous work showed that the average entanglement growth after a quantum quench can be explained in terms of pairs of entangled quasiparticles performing random walks, leading to diffusive entanglement spreading. By incorporating the noise-induced statistical correlations between the quasiparticles, we extend this description to the full-time probability distribution of the entanglement entropy. Our generalized quasiparticle picture allows us to compute the average time evolution of a generic function of the reduced density matrix of a subsystem. We also apply our result to the entanglement asymmetry. This allows us to investigate the restoration of particle-number symmetry in the dynamics from initial states with no well-defined particle number, and the emergence of the quantum Mpemba effect.

Figures (8)

• Figure 1: Sketch of the diffusive quasiparticle picture for the out-of-equilibrium distribution of the entanglement entropy of a subsystem $A$ in the $\nu = 1$ QSSEP model \ref{['model_dH']}. At $t=0$, entangled pairs of quasiparticles, each one associated with a momentum $k$, are emitted and propagate diffusively. The distance $\xi_k(t)$ between the two quasiparticles of a pair describes a real Brownian motion. Since $\langle \xi_k(t)\xi_q(t)\rangle\neq 0$, quasiparticles with different momenta are statistically correlated. These correlations are key to describe the fluctuations of the entanglement via Eq. \ref{['qpS']}, giving access to the moments $\langle S_A(t)^n\rangle$ of the entanglement entropy at any time, and to the out-of-equilibrium evolution of the average entanglement asymmetry.
• Figure 2: Time evolution of the rescaled variance of the entanglement entropy $\sigma_S^2/\ell^2$ under the dynamics in Eq. \ref{['model_dH']} starting from the Néel state, for different subsystem sizes $\ell$ in the thermodynamic limit $L\to\infty$. The symbols correspond to the variance of the exact entanglement entropy, computed from the time-evolved correlation matrix as explained in Appendix \ref{['appA']}, for 1000 realizations. The red curve is the prediction in Eq. \ref{['s2dqp']} obtained from the diffusive quasiparticle picture.
• Figure 3: Out-of-equilibrium probability distribution function of the entanglement entropy at different times $\Theta = D t/\ell^2$ after a quench with the Hamiltonian \ref{['model_dH']}, starting from the Néel state and taking a subsystem of length $\ell=30$ and $L\to\infty$. The entanglement entropy $S_A$ is rescaled by its asymptotic value $\ell \log2$ in the stationary state. The dashed curves represent the prediction \ref{['eq:prob_dist']} of the diffusive quasiparticle picture. The histograms are the result of sampling the exact entanglement entropy, computed numerically from the time-evolved two-point correlation matrix for different noise realizations. The number of samples for each time considered is $\sim 10^4$.
• Figure 4: Numerical check of the identity in Eq. \ref{['ident']} for different values of the rescaled time $\Theta=Dt/\ell^2$. The blue circles correspond to the right-hand side of Eq. \ref{['ident']} obtained by evaluating numerically the integral in the square bracket of Eq. \ref{['trtrsq']}. The red diamonds are the value obtained for the average $\langle x_{\xi_k} x_{\xi_{k'}} \rangle = \langle x_{\xi_0} x_{\xi_{k'-k}} \rangle$ by sampling over $4\cdot10^4$ noise realizations.
• Figure 5: Time evolution of the average of different products of traces of the reduced correlation matrix $G_A$ under the dynamics in Eq. \ref{['model_dH']} starting from the Néel state. The red, green, and blue symbols are the average obtained by exactly evolving the Néel state for different realizations of the noise. We take a subsystem of size $\ell = 20$ and $L\to \infty$. The gray symbols joined by a line correspond to the predictions \ref{['generaldqp']} of the diffusive quasiparticle picture. The higher point functions of $x_{\xi_k}$ that appear when applying Eq. \ref{['generaldqp']} are estimated by numerical sampling. In both cases, the error bars are the standard deviation of the mean over the samples considered.