Qubit transpose map
= Qubit transpose map
{title2=$\mathcal T(\rho)=\rho^T$}
Matrix transposition preserves positivity and trace but fails <complete positivity>. For the <Bell state> $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$, partial transposition gives
$$
(I\otimes\mathcal T)(|\Phi^+\rangle\langle\Phi^+|)=\frac12\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}.
$$
The singlet $(|01\rangle-|10\rangle)/\sqrt2$ has eigenvalue $-1/2$ for this matrix, so the output is not a <density operator>. Universal qubit transposition is therefore not a <quantum channel>, although restricted input families can admit exact physical approximations.