  Generic quantum many-body systems typically show a linear growth of the
entanglement entropy after a quench from a product state. While entanglement is
a property of the wave function, it is generated by the unitary time evolution
operator and is therefore reflected in its increasing complexity as quantified
by the operator entanglement entropy. Using numerical simulations of a static
and a periodically driven quantum spin chain, we show that there is a robust
correspondence between the entanglement entropy growth of typical product
states with the operator entanglement entropy of the unitary evolution
operator, while special product states, e.g. $\sigma_z$ basis states, can
exhibit faster entanglement production. In the presence of a disordered
magnetic field in our spin chains, we show that both the wave function and
operator entanglement entropies exhibit a power-law growth with the same
disorder-dependent exponent, and clarify the apparent discrepancy in previous
results. These systems, in the absence of conserved densities, provide further
evidence for slow information spreading on the ergodic side of the many-body
localization transition.
