Trace distance of remote state preparation through noisy channels

In this paper, we study remote state preparation (RSP) by w state through noisy channels. The trace distance is used to describe how close the original state is to the output state. Studies show the trace distance is a function of decohenrence rates and angles of the state to be prepared. At the same time, we investigate the influence of different types of noises on the trace distance and find the various types of noises have different degrees of influence on the trace distance for a definite qubit. We also study changes of the trace distance against polar and azimuthal angles.