On the Schmidt-rank-three bipartite and multipartite unitary operator

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 $n\ge4$.