= Solution
<Density operator controllability> cannot defeat unitary invariants: it grants access to the <unitary orbit of a density operator>, whose members have the same <eigenvalues>, rather than to every <density operator>. In the <Hilbert-Schmidt inner product> orthonormal normalization of this question,
$$
\|s_0\|^2=\frac{(0.5)^2+(0.5)^2}{2}=\frac14,
\qquad
\|s_1\|^2=\frac12.
$$
<Quantum Hamiltonian> evolution preserves this norm, so \b[the target is not reachable from the stated initial state]. Equivalently, their values of <purity of a density operator> are $1/2+1/4=3/4$ and $1/2+1/2=1$: the initial state is a <mixed state> and the target a <pure state>. Dissipation, an appropriate measurement protocol or an auxiliary system with additional resources would be needed to change the <spectrum>; freely controlling the closed-system <Quantum Hamiltonian> alone does not suffice.
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