Qubit transpose map ($\mathcal{T}(\rho )={\rho }^T$)

Matrix transposition preserves positivity and trace but fails complete positivity. For the Bell state $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$, partial transposition givesThe singlet $(|01⟩-|10⟩)/\sqrt{2}$ 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.