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.

Two identical copies of an unknown Bell state suffice for perfect identification by LOCC. Alice and Bob each measure $Z$ on the first copy and $X$ on the second. Comparing their local signs supplies the eigenvalues of $Z\otimes Z$ and $X\otimes X$. These are respectively $(+,+)$ for ${\Phi }^{+}$, $(+,-)$ for ${\Phi }^{-}$, $(-,+)$ for ${\Psi }^{+}$, and $(-,-)$ for ${\Psi }^{-}$. Each parity is certain even though individual local outcomes are random. Different copies provide the two complementary measurements, so measuring the first copy does not erase the phase information needed on the second.