{title2=$\mathcal E_r(A)=rA+(1-r)\operatorname{tr}(A)I/2$}

This one-parameter family maps every <Bloch vector> $\mathbf s$ to $r\mathbf s$. It is unital, trace preserving and Hermiticity preserving for every real $r$. It is a <positive map> exactly when $|r|\leq1$, because the output eigenvalues of a state are $(1\pm r|\mathbf s|)/2$. Its <Choi matrix> has eigenvalues $(1+3r)/2$ and $(1-r)/2$ with multiplicities one and three in the trace-two convention, so <Choi's theorem on completely positive maps> gives complete positivity exactly for $-1/3\leq r\leq1$. In that range the <Kraus representation> is a mixture of conjugations by $I,X,Y,Z$ with probabilities $(1+3r)/4,(1-r)/4,(1-r)/4,(1-r)/4$. The physically allowed maps are qubit <depolarizing channels>, including a limited range of negative Bloch scaling.


 Isotropic qubit map (source code)