= Projective measurement
{title2=$P_iP_j=\delta_{ij}P_i,\quad\sum_iP_i=I$}
A <projective measurement> has self-adjoint <orthogonal projections> $P_i$ satisfying $P_iP_j=\delta_{ij}P_i$ and $\sum_iP_i=I$. Measuring an <observable> with discrete <spectral decomposition> $A=\sum_i a_iP_i$ reports <eigenvalue> $a_i$ with <Born rule> <probability> $p_i=\operatorname{Tr}(P_i\rho)$. A projector can have <rank> greater than one: the outcome then identifies an <eigenspace>, rather than a particular <eigenvector> inside it. Projectors specify the outcome statistics; the usual ideal conditional update is the additional <Lüders rule>.
In the <qubit> example from <measurement in quantum mechanics>, $Z=|0\rangle\langle0|-|1\rangle\langle1|$. At $\theta=\pi/3$, the zero and one <probabilities> are $3/4$ and $1/4$, regardless of the relative <quantum phase> $\phi$. Under the <Lüders rule>, obtaining zero prepares $|0\rangle$; obtaining one prepares $|1\rangle$, up to <global phase>. Repeating the same measurement returns the same label with certainty, provided no intervening evolution or operation changes the measured sector. Repeatability concerns the conditional output, and does not mean the first outcome was predetermined.
Change to $X$ and its <Pauli X eigenstates> $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$. Adding the two <probability amplitudes> before squaring gives
$$
p_\pm=|\langle\pm|\psi\rangle|^2
=\frac12\left|\cos(\theta/2)\pm e^{i\phi}\sin(\theta/2)\right|^2
=\frac{1\pm\sin\theta\cos\phi}{2}.
$$
At $\theta=\pi/3$ and $\phi=0$, $p_+=(2+\sqrt3)/4\simeq0.9330$ and $p_-\simeq0.0670$. At $\phi=\pi/2$, both are $1/2$; at $\phi=\pi$, the first pair is exchanged. The $Z$ statistics remain $3/4,1/4$ throughout. This is the same addition-and-cancellation principle as <wave interference>: relative <quantum phase> changes how amplitudes reinforce or cancel in another basis.
The beauty of this basis dependence is that information invisible to one measurement becomes accessible to another. A $Z$ measurement of an ensemble estimates its <quantum state populations>, while $X$ probes one component of its <quantum coherence>. These are experiments on separate identically prepared copies. Sequentially measuring $Z$ and then $X$ on one copy instead probes the disturbed <post-measurement state>. <Quantum state tomography> combines suitable ensemble statistics to reconstruct the original state.
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