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Say I have the stabilizer $XXI$, and a phase-flipped state $|\psi\rangle$. What does it mean to measure the stabilizer $XXI$?

What is the math behind "measurement"?

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Measurements can be described by a Hermitian operator, $A$. In this case, $A=X\otimes X\otimes I$. That operator has a spectral decomposition $$ A=\sum_i\lambda_iP_i, $$ where the $\lambda_i$ are the distinct (real) eigenvalues, and the $P_i$ are projectors onto the different eigenspaces, and satisfy the completeness relation $\sum_iP_i=I$.

So, basically, we're using $A$ to define the set $\{P_i\}$ which are the projectors in the formalism of projective measurement, i.e. we measure in this basis and get answer $i$ with probability $p_i=\langle\psi|P_i|\psi\rangle$. After the measurement, the system is in the state $P_i|\psi\rangle/\sqrt{p_i}$.

In your case, since $A^2=I$, you know the eigenvalues are $\pm 1$, and hence you can write $$ P_{\pm}=\frac12(I\pm A). $$

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