  We prove that every unitary acting on any multipartite system and having
operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This
then implies that every such multipartite unitary is locally equivalent to a
controlled unitary with every party but one controlling a set of unitaries on
the last party. We also prove that any bipartite unitary of Schmidt rank 2 is
locally equivalent to a controlled unitary where either party can be chosen as
the control, and at least one party can control with two terms, which implies
that each such unitary can be implemented using local operations and classical
communication (LOCC) and a maximally entangled state on two qubits. These
results hold regardless of the dimensions of the systems on which the unitary
acts.
