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"?


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 $$ P_{\pm}=\frac12(I\pm A). $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.