# Eigenvalues of Pauli Gate and connection to measurement

Suppose I measure a qubit in the $$Z$$ basis. If I measure and obtain the outcome $$+1$$, I get the post-measurement state $$\vert 0\rangle\langle 0\vert$$ and if I measure and obtain the outcome $$-1$$, I get the post-measurement state $$\vert 1\rangle\langle 1\vert$$.

These eigenvalue choices seem arbitrary. But the $$Z$$ operator is given in the standard basis by $$Z = \vert 0\rangle\langle 0\vert - \vert 1\rangle\langle 1\vert$$. Is $$\pm 1$$ in the measurement outcomes related to the eigenvalues of the $$Z$$ operator?

To "measure an observable" $$O$$ means in practice to

1. perform a projective measurement in the eigenbasis of $$O$$;
2. assign to each observed outcome the corresponding eigenvalue (ie the eigenvalue corresponding to the eigenvector corresponding to the observed outcome);
3. compute the average of such values.

So in this sense yes, you might say that the eigenvalues are "arbitrary", in the sense that they are classical values "arbitrarily" assigned in post-processing to the measurement outcomes. You can see them as ways to select which features you want to extract from the data.

For example, if you considered the observable $$|0\rangle\!\langle 0|$$ instead of $$Z$$ for a single qubit, the performed measurement (as in, the PVM used) would be identical, both are computational basis measurements, but in the former case you only count the number of times you get the outcome $$|0\rangle$$, while in the latter case you compute the difference between the number of times you see $$|0\rangle$$ and the number of times you see $$|1\rangle$$.

• If the label you assign to the measurement is arbitrary, can the Pauli Z gate also be redefined to have different eigenvalues from $\pm 1$? Commented Jul 24 at 14:30
• @Jorge well, if you change the eigenvalues, it won't be a Pauli Z gate anymore. But you will get an observable which describes the same physical measurement, yes (though asking different features in post-processing)
– glS
Commented Jul 24 at 16:37