  Unitary operations are physically implementable. We further the understanding
of such operations by studying the possible forms of nonlocal unitary
operators, which are bipartite or multipartite unitary operators that are not
tensor product operators. They are of broad relevance in quantum information
processing. We prove that any nonlocal unitary operator of Schmidt rank three
on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled
unitary. This operator can be locally implemented assisted by a maximally
entangled state of Schmidt rank $\min\{d_A^2,d_B\}$ when $d_A\le d_B$. We
further show that any multipartite unitary operator $U$ of Schmidt rank three
can be controlled by one system or collectively controlled by two systems,
regardless of the number of systems of $U$. In the scenario of $n$-qubit, we
construct non-controlled $U$ for any odd $n\ge5$, and prove that $U$ is a
controlled unitary for any even $n\ge4$.
