= Error syndrome
{wiki}
An <error syndrome> is the list of outcomes obtained by measuring a chosen ordered family of <stabilizer generators>. Write $s_j=0$ for <eigenvalue> $+1$ and $s_j=1$ for <eigenvalue> $-1$. For a definite <Pauli error> $E$ on a code state, the rule is
$$
g_jE=(-1)^{s_j}Eg_j,\qquad g_jE|\psi_L\rangle=(-1)^{s_j}E|\psi_L\rangle.
$$
The commuting checks admit joint <orthogonal projectors>
$$
\Pi_s=\prod_{j=1}^r\frac{I+(-1)^{s_j}g_j}{2},\qquad
\sum_{s\in\{0,1\}^r}\Pi_s=I,\qquad \Pi_0=P.
$$
For <density operator> $\rho$, the <Born rule> gives outcome <probability> $\operatorname{tr}(\Pi_s\rho)$ and conditional state $\Pi_s\rho\Pi_s/\operatorname{tr}(\Pi_s\rho)$ when that <probability> is nonzero. A definite <Pauli error> has one deterministic <error syndrome> independent of the encoded amplitudes. A coherent <linear combination> of <Pauli errors> may occupy several sectors, so measuring the <error syndrome> selects a sector; recovery preserves the <logical qubit> when the entire error family is jointly correctable. https://www.preskill.caltech.edu/ph229/notes/chap7.pdf[Preskill, §§7.2 and 7.9.1]{ref}.
An <error syndrome> locates an error only relative to an assumed family and a recovery rule. Two <Pauli errors> differing by a logical operator can share an <error syndrome>; see <stabilizer-syndrome coset>. In the <three-qubit bit-flip syndrome table>, $X_3$ and $X_1X_2$ both give $01$. Recovering with $X_3$ corrects the first but converts the second into the logical operator $X_1X_2X_3$. Likewise, $Z_1$ has <error syndrome> $00$ yet changes the encoded relative phase.
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