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The Bell basis consists ofLet $Z={\sigma }_z$, $X={\sigma }_x$. The Pauli operators $Z\otimes Z$ and $X\otimes X$ commute: each of the two local anticommutations contributes a minus sign, so they cancel. Their eigenvalue table isTo make the bit values literally zero or one rather than signed eigenvalues, the commuting Bell parity and phase observables areThey distinguish all four states jointly. The parity bit records whether the computational bits agree; the phase bit records the relative sign.