  We present a random-matrix analysis of the entangling power of a unitary
operator as a function of the number of times it is iterated. We consider
unitaries belonging to the circular ensembles of random matrices (CUE or COE)
applied to random (real or complex) non-entangled states. We verify numerically
that the average entangling power is a monotonic decreasing function of time.
The same behavior is observed for the "operator entanglement" --an alternative
measure of the entangling strength of a unitary. On the analytical side we
calculate the CUE operator entanglement and asymptotic values for the
entangling power. We also provide a theoretical explanation of the time
dependence in the CUE cases.
