  The task of changing the overlap between two quantum states can not be
performed by making use of a unitary evolution only. However, by means of a
unitary-reduction process it can be probabilistically modified. Here we study
in detail the problem of mapping two known pure states onto other two states in
such a way that the final inner product between the outcome states is different
from the inner product of the initial states. In this way we design an optimal
non-orthogonal quantum state preparation scheme by starting from an orthonormal
basis. In this scheme the absolute value of the inner product can be reduced
only probabilistically whereas it can be increased deterministically. Our
analysis shows that the phases of the involved inner products play an important
role in the increase of the success probability of the desired process.
