 Basis-independent conversion of qubit Bloch radii

Fix $0<a\le 1$ and $-1\le b\le 1$. A single deterministic quantum channel sending every state of vector $a\mathbf{n}$ to the state of vector $b\mathbf{n}$ exists exactly under the displayed condition. Add the requirements for $\mathbf{n}$ and $-\mathbf{n}$ to show that the channel fixes $I$. Subtract and take the three coordinate directions to get its action on each Pauli matrix, namely multiplication by $b/a$. Linearity thus uniquely forces the isotropic qubit map ${\mathcal{E}}_{b/a}$. Its complete-positivity range proves necessity and sufficiency. A basis-dependent unitary may reverse one known vector without implementing a universal inversion, so knowing the basis changes the conversion problem.