Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / e / Solution

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. ThenThus ${\rho }_A$ 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 $|r|\le 1$, so any composition of the allowed operations cannot increase it. Hence ${\rho }_B$ 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.