= Basis-independent conversion of qubit Bloch radii
{title2=$a\mathbf n\mapsto b\mathbf n\text{ for every unit }\mathbf n\iff -1/3\leq b/a\leq1$}
Fix $0<a\leq1$ and $-1\leq b\leq1$. 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.
Back to article page
