  The spreading of entanglement in out-of-equilibrium quantum systems is
currently at the centre of intense interdisciplinary research efforts involving
communities with interests ranging from holography to quantum information. Here
we provide a constructive and mathematically rigorous method to compute the
entanglement dynamics in a class of "maximally chaotic", periodically driven,
quantum spin chains. Specifically, we consider the so called "self-dual" kicked
Ising chains initialised in a class of separable states and devise a method to
compute exactly the time evolution of the entanglement entropies of finite
blocks of spins in the thermodynamic limit. Remarkably, these exact results are
obtained despite the models considered are maximally chaotic: their spectral
correlations are described by the circular orthogonal ensemble of random
matrices on all scales. Our results saturate the so called "minimal cut" bound
and are in agreement with those found in the contexts of random unitary
circuits with infinite-dimensional local Hilbert space and conformal field
theory. In particular, they agree with the expectations from both the
quasiparticle picture, which accounts for the entanglement spreading in
integrable models, and the minimal membrane picture, recently proposed to
describe the entanglement growth in generic systems. Based on a novel
"duality-based" numerical method, we argue that our results describe the
entanglement spreading from any product state at the leading order in time when
the model is non-integrable.
