Quantum and classical complexity in coupled maps

We study a generic and paradigmatic two degrees of freedom system consisting of two coupled perturbed cat maps with different types of dynamics. The Wigner separability entropy (WSE) -- equivalent to the operator space entanglement entropy -- and the classical separability entropy (CSE) are used as measures of complexity. For the case where both degrees of freedom are hyperbolic, the maps are classically ergodic and the WSE and the CSE behave similarly, growing up to higher values than in the doubly elliptic case. However, when one map is elliptic and the other hyperbolic, the WSE reaches the same asymptotic value than that of the doubly hyperbolic case, but at a much slower rate. The CSE only follows the WSE for a few map steps, revealing that classical dynamical features are not enough to explain complexity growth.