= Solution
{c}
Conjugation by one <Pauli matrix> fixes that same matrix and reverses the other two. For example, $X X X=X$, $Y X Y=-X$, and $Z X Z=-X$. Applying this to each component of the <Bloch vector> gives
$$
\sum_{j=x,y,z}\sigma_j\varrho\sigma_j
=\frac12(3I-\mathbf s\cdot\boldsymbol\sigma).
$$
Hence the <quantum depolarizing channel> is
$$
\mathcal E_p(\varrho)
=\frac12\left[I+\left(1-\frac{4p}{3}\right)\mathbf s\cdot\boldsymbol\sigma\right],
\qquad
\boxed{\mathbf s'=\frac{3-4p}{3}\mathbf s.}
$$
The <Kraus operators> $\sqrt{1-p}\,I$ and $\sqrt{p/3}\,\sigma_j$ establish <complete positivity> for $0\leq p\leq1$. At $p=3/4$ every state becomes <maximally mixed>. For $p>3/4$ the factor is negative: it reverses the vector as well as reducing its length. Its most negative value is $-1/3$, consistent with the complete-positivity range of the <isotropic qubit map>.
Back to article page
