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I'm coming across some confusion in chapter 10.1.1 of Nielsen and Chaung. In terms of the 'recovery' procedure, how can the result of the syndrome measurement be 0, 2 or 3?

I am assuming that, for example, if the state is $a|010\rangle$ + $b|101\rangle$ (bit flip on the second qubit), then will we not see that $\langle\psi|P_2|\psi\rangle = 1$ and the rest of the projection operators will give a syndrome measurement of 0 (i.e $\langle\psi|P_1|\psi\rangle = 0$)? From this example, and working through the rest in the same manner, I am struggling to see where/how an error syndrome of 2 will be measured.

I hope my current working makes sense!

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    $\begingroup$ Hi and welcome to Quantum Computing SE. Please avoid posting scans of articles and book. Rather cite necessary parts of the source. $\endgroup$ Sep 12, 2022 at 5:27

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The outcome of measuring $P_2$ will not be 2, but rather the outcome of measuring the operator with a spectral decomposition given by the four syndromes and corresponding eigenvalues. Remember that $\langle \psi | P_n | \psi \rangle$ gives the probability of measuring the eigenvalue of $M$ corresponding to $P_n$, not the eigenvalue itself. As such, an outcome of $n$ is not saying that measuring $P_n$ will give $n$, but just refers to the fact that $\langle \psi | P_n | \psi \rangle = 1$, so we are certain that the syndrome is $P_n$.

As for how to perform this syndrome measurement in practice, look at page 430.

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