Universality in Exact Quantum State Population Dynamics and Control

We consider an exact population transition, defined as the probability of finding a state at a final time being exactly equal to the probability of another state at the initial time. We prove that, given a Hamiltonian, there always exists a complete set of orthogonal states that can be employed as time-zero states for which this exact population transition occurs. The result is general: it holds for arbitrary systems, arbitrary pairs of initial and final states, and for any time interval. The proposition is illustrated with several analytic models. In particular we demonstrate that in some cases, by tuning the control parameters a $\mathit{\text{complete}}$ transition might occur, where a target state, vacant at $t=0$, is fully populated at time $\tau$.