  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.
