= Solution
The <Pauli matrices> act by $X|0\rangle=|1\rangle$, $X|1\rangle=|0\rangle$, $Z|0\rangle=|0\rangle$, and $Z|1\rangle=-|1\rangle$. Consequently $X\otimes X$ swaps $00$ with $11$ and $01$ with $10$. The plus combinations are its $+1$ eigenvectors, and the minus combinations its $-1$ eigenvectors. Meanwhile $Z\otimes Z$ is $+1$ on the even-parity basis vectors and $-1$ on the odd-parity ones. Thus, with the paper's labels,
$$
\boxed{\begin{array}{c|cc}
\text{state}&X\otimes X&Z\otimes Z\\\hline
\Psi_+&+1&+1\\
\Psi_-&-1&+1\\
\Phi_+&+1&-1\\
\Phi_-&-1&-1
\end{array}}
$$
This verifies all four <Bell states> directly. The two observables commute: $XZ=-ZX$ on each factor, so the two minus signs cancel in their tensor products. Their joint eigenvalues distinguish the four states, as used in a <Bell-basis measurement>.
Back to article page
