Nonlocal and controlled unitary operators of Schmidt rank three
Lin Chen, Li Zhi Yu · Physical Review A · 2014
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 $({d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B})$-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 $r=\mathrm{min}{{d}_{A}^{2},{d}_{B}}$. 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.