  The preparation of quantum states lies at the foundation in the quantum
information processing. The convex mixing of some existing quantum states is
one of the effective candidate. In this paper, we mainly study how a target
quantum state can be optimally prepared by not more than three given pure
states. The analytic optimal distance based on the fidelity is found. We also
show that the preparation with more than four states can be essentially
converted to the case with not more than four states, which can be similarly
solved as the case with three states. The validity is illustrated by the
comparison of our analytical and numerical results.
