Entanglement propagation provides a key routine to understand quantum
many-body dynamics in and out of equilibrium. In this work, we uncover that the
``variational entanglement-enhancing'' field (VEEF) robustly induces a
persistent ballistic spreading of entanglement in quantum spin chains. The VEEF
is time dependent, and is optimally controlled to maximize the bipartite
entanglement entropy (EE) of the final state. Such a linear growth persists
till the EE reaches the genuine saturation $\tilde{S} = - \log_{2}
2^{-\frac{N}{2}}=\frac{N}{2}$ with $N$ the total number of spins. The EE
satisfies $S(t) = v t$ for the time $t \leq \frac{N}{2v}$, with $v$ the
velocity. These results are in sharp contrast with the behaviors without VEEF,
where the EE generally approaches a sub-saturation known as the Page value
$\tilde{S}_{P} =\tilde{S} - \frac{1}{2\ln{2}}$ in the long-time limit, and the
entanglement growth deviates from being linear before the Page value is
reached. The dependence between the velocity and interactions is explored, with
$v \simeq 2.76$, $4.98$, and $5.75$ for the spin chains with Ising, XY, and
Heisenberg interactions, respectively. We further show that the nonlinear
growth of EE emerges with the presence of long-range interactions.

通过 VEETF 强化量子自旋链中的纠缠传播，在饱和之前，纠缠熵以线性速度扩展，而存在长程相互作用时，纠缠熵的增长是非线性的。