 Solution

The input is the Bell state $\rho =|{\Phi }^{+}⟩⟨{\Phi }^{+}|$, where $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$. 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 isIts eigenvalues are $\boxed{-1/2,1/2,1/2,1/2}$: in particular, its expectation value in $|{\Phi }^{+}⟩$ 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 positive but not completely positive, so it cannot be a deterministic physical quantum channel on an arbitrary input.