Consider a measurement operator ("observable") $\hat O$ which has ("a spectrum of") only two distinct eigenvectors; formally
$$\hat O |\bullet\rangle := r_{\bullet}~|\bullet\rangle, \qquad \hat O |\circ\rangle := r_{\circ}~|\circ\rangle, \qquad \langle \bullet |\circ\rangle = 0, $$
where the eigenvalues $r_{\bullet}$ and $r_{\circ}$ are (suitable) real numbers.
Is there a time-dependent unitary operator $\hat U[~t~]$ such that
$$ \frac{|\bullet\rangle + |\circ\rangle}{\sqrt 2} = \hat U[~t_{\text{blend}}~] ~ |\bullet\rangle $$ ?
If so, can this operator be expressed in the form $\hat U[~t~] := \text{Exp}[~\frac{-i~t}{\hbar}~\hat H[~t~]~]$ ?
If so, can the corresponding operator $\hat H[~t~]$ be expressed in terms of the given operator $\hat O$, or its eigenvectors or eigenvalues, and, as necessary, the coordinate value $t_{\text{blend}}$ and the coordinate variable $t$?
Edit
(along with correcting the above "evolution equation" stated within the bra-ket formalism):
Within the density matrix formalism it could be asked correspondingly whether there is a time-dependent unitary operator $\hat U[~t~]$ such that
$$ \sqrt{\frac{1}{2}} \left( \array{ |\bullet\rangle\langle\bullet | & |\bullet\rangle\langle\circ | \cr |\circ\rangle\langle\bullet | & |\circ\rangle\langle\circ | } \right) = \hat U[~t_{\text{blend}}~] ~ \left( \begin{array}{cc} |\bullet\rangle\langle\bullet | & 0 \\ 0 & |\circ\rangle\langle\circ | \end{array} \right) ~ \hat U[~t_{\text{blend}}~]^{\dagger}.$$