Bell parity and phase observables ($B_{\mathrm{parity}}=(I-Z\otimes Z)/2,B_{\mathrm{phase}}=(I-X\otimes X)/2$)

The Bell basis diagonalizes the commuting Pauli operators $Z\otimes Z$ and $X\otimes X$. Two local anticommutations make the joint operators commute. Their signs encode parity and relative phase. The displayed projectors turn the signed eigenvalues into literal zero-or-one bit eigenvalues: even parity and plus phase have bit zero.