  For a quantum system governed by a non-Hermitian Hamiltonian, we studied the
problem of obtaining an optimum Hamiltonian that generates nonunitary
transformations of a given initial state into a certain final state in the
smallest time $\tau$. The analysis is based on the relationship between the
states of the two-dimensional subspace of the Hilbert space spanned by the
initial and final states and the points of the two-dimensional complex Bloch
sphere.
