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

Write the unknown common eigenbasis as an unknown unit Bloch vector $\mathbf{n}$. Then ${\rho }_A$ has vector $0.8\mathbf{n}$, while ${\rho }_B$ has vector $-0.4\mathbf{n}$. A single deterministic channel working for every possible basis must therefore map $0.8\mathbf{n}$ to $-0.4\mathbf{n}$ for every unit $\mathbf{n}$.

By the basis-independent conversion of qubit Bloch radii, linearity forces that channel to be ${\mathcal{E}}_{-1/2}$. Explicitly, add the conditions for $\mathbf{n}$ and $-\mathbf{n}$ to get $\mathcal{E}(I)=I$, then use the three coordinate directions to fix its action on every Pauli matrix. But $-1/2<-1/3$, so part iv excludes complete positivity. Thus the forward conversion cannot be implemented reliably without knowing the basis.

The reverse conversion would require ${\mathcal{E}}_{-2}$, which is not even positive by part ii. Therefore the reverse conversion is also impossible. Unitary operations interspersed with physically allowed depolarizing maps cannot evade these restrictions: their composition remains a physical quantum channel. The question concerns one basis-independent operation on the supplied qubit, not an additional resource of many copies for learning its basis.