Probabilistically implementing nonlocal operations between two distant qutrits

We propose a method to probabilistically implement a nonlocal operation, $%exp {[}ixi U_{A}U_{B}{]}$, between two distant qutrits $A$ and $B$, where $%xi in [0,2pi ]$ and $U_{A}$, $U_{B}$ are local unitary and Hermitianoperations for qutrits $A$ and $B$ respectively. The consumptions ofresource for one performance of the method are a single non-maximallyentangled qutrit state and 1-trit classical communication. For a given $xi $%, the successful probability of the method depends on the forms of bothentanglement resource and Bob's partial-measurement basis. We systematicallydiscuss the optimal successful probabilities and their correspondingconditions for three cases: adjustable entanglement resource, adjustablepartial-measurement basis, adjustable entanglement resource andpartial-measurement basis. It is straightforward to generalize the methodfor producing nonlocal unitary operations between any two $N$-level systems.