  Implementing nonlocal unitary operators is an important and hard question in
quantum computing and cryptography. We show that any bipartite nonlocal unitary
operator of Schmidt rank three on the $(d_A \times d_B)$-dimensional system is
locally equivalent to a controlled unitary when $d_A$ is at most three. This
operator can be locally implemented assisted by a maximally entangled state of
Schmidt rank $r=\min\{d_A^2,d_B\}$. We further show that stochastic-equivalent
nonlocal unitary operators are indeed locally equivalent, and propose a
sufficient condition on which nonlocal and controlled unitary operators are
locally equivalent. We also provide the solution to a special case of a
conjecture on the ranks of multipartite quantum states.
