= Solution
{c}
With the basis known, choose a <unitary operator> that interchanges its two basis states. On the Bloch sphere this is a rotation by $\pi$ about an axis perpendicular to $\mathbf n$, sending $\mathbf n$ to $-\mathbf n$. Follow it by $\mathcal E_{1/2}$, which is completely positive. Then
$$
0.8\mathbf n\longmapsto-0.8\mathbf n\longmapsto-0.4\mathbf n.
$$
Thus \b[$\rho_A$ can be converted to $\rho_B$] using the allowed operations and a suitable physically available depolarizing parameter.
The reverse direction would have to increase Bloch radius from $0.4$ to $0.8$. Unitaries preserve radius, and every completely positive $\mathcal E_r$ has $|r|\leq1$, so any composition of the allowed operations cannot increase it. Hence \b[$\rho_B$ still cannot be converted to $\rho_A$ using these resources]. This conclusion uses the stated unitary/depolarizing resources; an unrestricted state-preparation channel would be a different allowance.
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