Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 58 / 4 / b

The Pauli matrices act by $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Z|0⟩=|0⟩$, and $Z|1⟩=-|1⟩$. 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,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.