Nonlocal and controlled unitary operators of Schmidt rank three

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.

Figure 1

Any bipartite unitary $U$ on $d_A\times d_B$ system of Schmidt rank three is locally equivalent to a controlled unitary when $d_A=2,3$, where the controlling side may be $A$ or $B$. This is expressed as $U=(Q\otimes I)\left({\sum }_{k=1}^{d_A}\right|k\rangle \langle k|\otimes V_k)(R\otimes I)$ or $U=(I\otimes Q)({\sum }_{k=1}^{d_B}{V}_k\otimes |k\rangle \langle k\left|\right)(I\otimes R)$, where $V_k$, $Q$, and $R$ are local unitaries. The output systems $A^{\prime }$ and $B^{\prime }$ are assumed to be of the same size as $A$ and $B$, respectively.