= Solution
{c}
The input is the <Bell state> $\rho=|\Phi^+\rangle\langle\Phi^+|$, where $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$. Acting on the first <qubit> reverses the signs of all three terms containing its <Pauli matrices>, so in <computational basis> order $00,01,10,11$ the output is
$$
\boxed{(\mathcal N\otimes\operatorname{id})(\rho)
=\frac14(I\otimes I-X\otimes X+Y\otimes Y-Z\otimes Z)
=\begin{pmatrix}
0&0&0&-1/2\\
0&1/2&0&0\\
0&0&1/2&0\\
-1/2&0&0&0
\end{pmatrix}.}
$$
Its <eigenvalues> are $\boxed{-1/2,1/2,1/2,1/2}$: in particular, its <expectation value> in $|\Phi^+\rangle$ is $-1/2$. It therefore fails to be <positive semidefinite>, although its <trace> remains one. This single <entangled state> proves that the <universal NOT gate> is \b[positive but not completely positive], so it cannot be a deterministic physical <quantum channel> on an arbitrary input.
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