  Operator entanglement is a well-established measure of operator complexity
across a system bipartition. In this work, we introduce a measure for the
ability of a unitary channel to generate operator entanglement, representing an
operator-level generalization of the state-space entangling power. This
operator space entangling power is demonstrated to be linked to the scrambling
properties of the unitary channel via the recently introduced concept of mutual
averaged non-commutativity of quantum operator algebras. An upper bound for the
operator space entangling power is identified, corresponding to unitary
channels with scrambling properties akin to those of typical unitaries.
Additionally, for Hamiltonian dynamics, we find that the short-time growth rate
of the operator space entangling power matches the Gaussian scrambling rate of
the bipartite out-of-time-order-correlator, establishing a direct link between
information scrambling and operator entanglement generation for short time
scales. Finally, we examine the average growth of local operator entanglement
across a symmetric bipartition of a spin-chain. For dual-unitary circuits, a
combination of analytical and numerical investigations demonstrates that the
average growth of local operator entanglement exhibits two distinct regimes in
relation to the operator space entangling power of the building-block gate.
