Persistent Ballistic Entanglement Spreading with Optimal Control in Quantum Spin Chains

Abstract

Entanglement propagation provides a key routine to understand quantum many-body dynamics in and out of equilibrium. The entanglement entropy (EE) usually approaches to a sub-saturation known as the Page value $\tilde{S}_{P} =\tilde{S} - dS$ (with $\tilde{S}$ the maximum of EE and $dS$ the Page correction) in, e.g., the random unitary evolutions. The ballistic spreading of EE usually appears in the early time and will be deviated far before the Page value is reached. In this work, we uncover that the magnetic field that maximizes the EE robustly induces persistent ballistic spreading of entanglement in quantum spin chains. The linear growth of EE is demonstrated to persist till the maximal $\tilde{S}$ (along with a flat entanglement spectrum) is reached. The robustness of ballistic spreading and the enhancement of EE under such an optimal control are demonstrated, considering particularly perturbing the initial state by random pure states (RPS's). These are argued as the results from the endomorphism of the time evolution under such an entanglement-enhancing optimal control for the RPS's.

Figures (13)

• Figure 1: (Color online) The illustration of different EE's (with equal bipartition) reachable by the 1D quantum systems with some typical examples listed in the right-hand side. The wavy lines among the arrows indicate the strength of entanglement.
• Figure 2: (Color online) The EE $S(t)$ against time $t$ with zero field $h^{\alpha}_n(t)=0$ (black dashed line), the constant field $h^x_n(t)=0.9045$ and $h^z_n(t)=0.8090$ where the system is nonintegrable PhysRevLett.111.127205_2013 (red dash-dot line), random $h^{\alpha}_n(t)$ (green solid line), and VEEF ($T=1.8$ by purple solid line and $T=10$ by blue solid line). We here consider the $N=10$ quantum Ising chain with periodic boundary condition. The inset shows that EE converges to the Page value [Eq. (\ref{['eq-Page']})] in the long-time limit PhysRevLett.71.1291. The results with the random $h^{\alpha}_n(t)$ are estimated by implementing ten independent simulations, where the variance is indicated by the error bars.
• Figure 3: (Color online) The EE $S(t)$ for the $N=10$ XXZ model obeys the logarithmic (diffusive) spreading with zero or random field bardarson_unbounded_2012znidaric_many-body_2008, but the linear (ballistic) spreading with VEEF. The inset shows the zoom-in of the area for $t \leq T_{S}$.
• Figure 4: (Color online) The EE $S(t)$of the 1D QIM ($N=10$) against the time $t$ with different total evolution time $T$. The inset shows the $S(t)$ for $T<T_{S}$, which satisfies the linear relation $S(t)=2.76t$.
• Figure 5: (Color online) (a) The EE $S'(t)$ of 1D QIM against time $t$ for the initial states $\vert\psi'_{0}\rangle$ with with different $\mu$ controlling the strength of random perturbation [Eq. (\ref{['eq-psi1']})]. $\vert\psi_{0}\rangle$ is the unperturbed initial state for obtaining the VEEF, and $\vert\psi_{r}\rangle$ is a RPS. The results are the average of $10$ independent simulations, and the error bars ($\sim O(10^{-3})$) are given by the variances. The insets show the $S'(T)$ versus $\mu$. (b) The EE $S'(0)$ of the perturbed initial state verses $\mu$ for different system sizes $N$, obtained by numerical simulations (symbols) and by Eq. (\ref{['eq-EEtime']}) (solid lines). The violation of the orthogonal conditions is characterized by $\epsilon$. (c) The $S'(t)$ obtained by numerical simulations (symbols) and by Eq. (\ref{['eq-EEtime']}) (solid lines) for $N=12, 14$ and $\mu = \pi/6, \pi/8, \pi/16$.