Probabilistically implementing nonlocal operations between two distant qutrits

We propose a method to probabilistically implement a nonlocal operation, % \exp\rm i\xi U_AU_B, between two distant qutrits A and B, where \xi \in 0,2\pi and U_A, U_B are local unitary and Hermitian operations for qutrits A and B respectively. The consumptions of resource for one performance of the method are a single non-maximally entangled qutrit state and 1-trit classical communication. For a given \xi , the successful probability of the method depends on the forms of both entanglement resource and Bob's partial-measurement basis. We systematically discuss the optimal successful probabilities and their corresponding conditions for three cases: adjustable entanglement resource, adjustable partial-measurement basis, adjustable entanglement resource and partial-measurement basis. It is straightforward to generalize the method for producing nonlocal unitary operations between any two N-level systems.