Past exam of the introduction to quantum information course of the University of Oxford / 2017 / 3 / d / Solution

The universal NOT gate is the isotropic qubit map with factor $-1$. It sendsFor $|\mathbf{s}|\le 1$, the output has the same two nonnegative eigenvalues as the input, exchanged, and still has trace one. Thus it maps every single-qubit density matrix to a density matrix. Its linear extension to all operators is $\mathcal{N}(A)=\mathrm{tr}(A)I-A$, not $I-A$ for arbitrary unnormalized $A$. Positivity on one qubit does not establish complete positivity when an ancilla is present.