= Universal NOT gate
{title2=$\mathcal N(A)=\operatorname{tr}(A)I-A$}
The ideal universal NOT map sends every <qubit> <Bloch vector> $\mathbf s$ to $-\mathbf s$, taking a <pure state> to its orthogonal <pure state>. It is a trace-preserving <positive map> but not a <quantum channel>: applied to one half of a <Bell state>, it produces an operator with a negative <eigenvalue>. A physical approximation can reverse all directions with factor $-1/3$ using the equal mixture of conjugations by the three <Pauli matrices>. The <isotropic qubit map> complete-positivity bound proves that a larger uniform reversal is impossible among such isotropic maps.
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