  We study the growth of the operator entanglement entropy (EE) of the time
evolution operator in chaotic, many-body localized and Floquet systems. In the
random field Heisenberg model we find a universal power law growth of the
operator EE at weak disorder, a logarithmic growth at strong disorder, and
extensive saturation values in both cases. In a Floquet spin model, the
saturation value after an initial linear growth is identical to the value of a
random unitary operator (the Page value). We understand these properties by
mapping the operator EE to a global quench problem evolved with a similar
parent-Hamiltonian in an enlarged Hilbert space with the same chaotic, MBL and
Floquet properties as the original Hamiltonian. The scaling and saturation
properties reflect the spreading of the state EE of the corresponding time
evolution. We conclude that the EE of the evolution operator should
characterize the propagation of information in these systems.
