Superoperator

A linear map between spaces of linear operators. A physical quantum channel is a superoperator that is trace preserving and completely positive; a general superoperator need not have either property.

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 $|r|\le 1$, 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 $-1/3\le r\le 1$. 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.