= Solution
The <Von Neumann equation> integrates to $\rho(t)=U(t)\rho(0)U(t)^\dagger$, so the <spectrum> and <purity of a density operator> are invariant under any allowed <open-loop control>. These are the kinematic restrictions.
There is no additional orbit restriction here: the generators $i\sigma_z$ and $i\sigma_x$, together with their <matrix commutator>, span the <special unitary Lie algebra>:
$$
[i\sigma_z,i\sigma_x]=-2i\sigma_y,\qquad \mathfrak g=\operatorname{span}_{\mathbb R}\{i\sigma_x,i\sigma_y,i\sigma_z\}=\mathfrak{su}(2).
$$
For freely chosen durations and piecewise continuous real control values, this gives <density operator controllability>. The connected reachable group is $SU(2)$; its action by conjugation is the same as that of $U(2)$, because a <global phase> cancels. Thus \b[exactly the pairs $\rho_0\leftrightarrow\rho_2$ and $\rho_1\leftrightarrow\rho_3$ can be interconverted]. No coherent control can turn either <mixed state> into either <pure state>, even though every orientation within each spectral orbit is reachable.
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