= Solution
Apply the modulo operation to the <eigenvalues> of $Z_A+Z_B$. The product <eigenstates> $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ have ordinary eigenvalues $2,0,0,-2$, respectively, and residues $2,0,0,2$ modulo $4$. The resulting <observable> is
$$
\boxed{O=2P_{\rm e}=I+Z_A\otimes Z_B}.
$$
Use the <entanglement-assisted nondemolition parity measurement> from part (a). Equal local meter records give $O=2$; unequal records give $O=0$. Its conditional <quantum measurement> maps are $\rho\mapsto P_{\rm e}\rho P_{\rm e}$ and $\rho\mapsto P_{\rm o}\rho P_{\rm o}$, with normalization by their probabilities. The <quantum nondemolition measurement> preserves every vector within each degenerate eigenspace, including superpositions of $|00\rangle$ and $|11\rangle$. Measuring the two system spins separately would destroy that even-sector coherence and would therefore not realize the same <Lüders rule> instrument. The nonlocal eigenvalue again becomes known only after <local operations and classical communication> compares the local records.
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