  In local quantum circuits with charge conservation, we initialize the system
in random product states and study the dynamics of the Renyi entanglement
entropy $R_\alpha$. We rigorously prove that $R_\alpha$ with Renyi index
$\alpha>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is
diffusive. Very recent numerical results of Rakovszky et al. show that this
upper bound is saturated (up to the sub-logarithmic correction) in random local
quantum circuits with charge conservation.
