= Solution
{c}
The <universal NOT gate> is the <isotropic qubit map> with factor $-1$. It sends
$$
\varrho=\frac12(I+\mathbf s\cdot\boldsymbol\sigma)
\quad\longmapsto\quad
\mathcal N(\varrho)=\frac12(I-\mathbf s\cdot\boldsymbol\sigma)=I-\varrho.
$$
For $|\mathbf s|\leq1$, the output has the same two nonnegative <eigenvalues> as the input, exchanged, and still has <trace> one. Thus \b[it maps every single-qubit density matrix to a density matrix]. Its linear extension to all operators is $\mathcal N(A)=\operatorname{tr}(A)I-A$, not $I-A$ for arbitrary unnormalized $A$. Positivity on one <qubit> does not establish <complete positivity> when an <ancilla> is present.
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