  We describe conditions for generating entanglement between two regions at the
optimal rate in a class of one-dimensional quantum circuits with Floquet
dynamics. The optimal value follows from subadditivity and Araki-Lieb
inequalities. A quantum circuit composed of parallel SWAP gates that act
periodically on entangled pairs is a simple system that saturates the bound. We
show that any other system that entangles at this maximal rate must act as a
generalized SWAP gate dynamics on the relevant states of the Hilbert space. We
further discuss some characterizations of states according to entropy
generation. States with multipartite entanglement generically fail to entangle
efficiently as time evolves. This suggests that chaos, which tend to produce
such entanglement patterns, is expected to work against the process of
spreading information efficiently. It also provides a simple intuition for why
the entangling tsunami velocity must be slower than the Lieb-Robinson velocity.
