= Solution
{c}
The output of a state of Bloch radius $u$ has <eigenvalues> $(1\pm ru)/2$. These are nonnegative for every $0\leq u\leq1$ exactly when
$$
\boxed{-1\leq r\leq1.}
$$
If $|r|>1$, a pure input already gives a negative eigenvalue. Conversely, $|r|\leq1$ preserves the <Bloch ball>, hence positivity for states and, by positive scaling, for all positive operators. This is positivity of the map alone; physical implementation on part of an entangled system needs <complete positivity>.
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