Implementation of non-local operations for arbitrary high-dimensional systems with qubit quantum channel

We propose a method to implement a kind of non-local operations between spatially separated two systems with arbitrary high-dimensions by using only low-dimensional qubit quantum channels and classical bit communications. The result may be generalized straightforwardly to apply for multiple systems, each of them with arbitrary dimensions. Compared with existed approaches, our method can economize classical resources and the needed low-dimensional quantum channels may be more easily established in practice. We also show the construction of the non-local quantum XOR gate for qutrit systems in terms of the obtained non-local operations as well as some single qutrit local gates.