= Solution
{c}
The <Bell state> $|\Omega\rangle$ is an eigenvector of $J_r$ with eigenvalue $(1+3r)/2$. Its three-dimensional orthogonal complement has eigenvalue $(1-r)/2$. The <Choi criterion for complete positivity> therefore gives
$$
\boxed{-\frac13\leq r\leq1.}
$$
For sufficiency one can also write a <Kraus representation> explicitly:
$$
\mathcal E_r(\rho)=\frac{1+3r}{4}\rho+
\frac{1-r}{4}(X\rho X+Y\rho Y+Z\rho Z).
$$
In this range the four coefficients are nonnegative and sum to one, giving a physical mixture of Pauli conjugations. This is the complete positivity range of the <depolarizing channel> in the Bloch-scaling parameterization.
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