Persistent Ballistic Entanglement Spreading with Optimal Control in Quantum Spin Chains

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.