  We introduce a technique that allows one to connect any two arbitrary (pure
or mixed) superposition states of an N-state quantum system. The proposed
solution to this inverse quantum mechanical problem is analytical, exact, and
very compact. The technique uses standard and generalized quantum Householder
reflections (QHRs), which require external pulses of precise areas and
frequencies. We show that any two pure states can be linked by just a single
generalized QHR. The transfer between any two mixed states with the same
dynamic invariants (e.g., the same density matrix eigenvalues) requires in
general N QHRs. Moreover, we propose recipes for synthesis of arbitrary
preselected mixed states using a combination of QHRs and incoherent processes
(pure dephasing or spontaneous emission).
