Joint parity operator measurement

The joint parity operator on $n$ qubits is

$P = Z_1 Z_2 \cdots Z_n.$

Spectrum. Since each $Z_i$ has eigenvalues $\pm1$, $P$ has eigenvalue $(-1)^{|x|}$ on basis state $|x\rangle$, where $|x|$ is the Hamming weight of the bit string $x$: $+1$ on even-weight strings, $-1$ on odd-weight strings. $P^2 = I$ since $Z_i^2=I$ for every factor, so $P$ is both Hermitian and unitary — a valid observable with only two possible measurement outcomes.

Non-destructive measurement circuit. To measure $\langle P\rangle$ without destroying superpositions within a fixed-parity eigenspace (essential for stabilizer codes, where you want to detect an error without collapsing the encoded logical information), introduce one ancilla in $|0\rangle$ and apply $\mathrm{CNOT}_{i\to\text{ancilla}}$ for every data qubit $i$. Since CNOT targeting the ancilla implements $|x\rangle_{\text{data}}|0\rangle_{\text{anc}} \mapsto |x\rangle_{\text{data}}|{\oplus_i x_i}\rangle_{\text{anc}}$, measuring the ancilla in the $Z$ basis returns the parity bit $\oplus_i x_i$ (equivalently, $\langle P\rangle = 1-2\langle \text{ancilla}\rangle$) while leaving the data register's relative phases and any coherence within a parity sector completely undisturbed — only the parity value is extracted, exactly the property a syndrome measurement needs.

Where this shows up. GHZ-state and cluster-state verification protocols measure exactly this operator (or products of it, e.g. the stabilizer generators $K_i$ of the graph states elsewhere in this catalog are all built from parity-type $X$-and-$Z$-string measurements) to certify entanglement without full state tomography. In stabilizer error correction, syndrome bits are joint-parity measurements of $Z$- or $X$-type check operators, and the non-destructive ancilla trick above is the literal circuit used to extract each syndrome bit without collapsing the encoded logical qubit.