= Quantum state tomography
{wiki}
<Quantum state tomography> estimates a <density operator> from measurement frequencies on many independently and identically prepared systems. One fixed <projective measurement> reveals only its basis populations, so further measurement settings are generally needed. For a <qubit>, write
$$
\rho=\tfrac12(I+r_xX+r_yY+r_zZ),\qquad
r_j=\operatorname{Tr}(\rho\sigma_j)=p_{j,+}-p_{j,-}.
$$
The three <Pauli matrices> thus turn measured <expectation values> into the <Bloch vector>. This reconstruction uses the <Bloch ball> representation of arbitrary <qubit> <density operators>. https://www.preskill.caltech.edu/ph219/chap2_15.pdf[Preskill, Quantum Information, §2.3.2]{ref}.
Divide the prepared ensemble into groups measured in the $X$, $Y$ and $Z$ bases. For example, positive-outcome frequencies $0.8,0.5,0.9$ give the estimate $\mathbf r=(0.6,0,0.8)$ and
$$
\rho=\begin{pmatrix}0.9&0.3\\0.3&0.1\end{pmatrix}.
$$
Here $|\mathbf r|=1$, so the reconstructed state is the <pure state> $(3|0\rangle+|1\rangle)/\sqrt{10}$. Its $Z$ frequencies alone also fit the diagonal <mixed state> with populations $0.9,0.1$; the $X$ data distinguish them. For the angular <pure state> in <measurement in quantum mechanics>, $r_x=\sin\theta\cos\phi$, $r_y=\sin\theta\sin\phi$ and $r_z=\cos\theta$, so $Y$ resolves the sign of the phase that $X$ alone cannot.
Finite samples introduce statistical uncertainty: with $N$ independent binary trials in one basis, the frequency has <variance> $p(1-p)/N$. At $p=1/2$ and $N=100$, its <standard deviation> is $0.05$. Fluctuations can place a naive <Bloch vector> estimate outside the <Bloch ball>; a physical reconstruction must enforce positivity of the <density operator>. Repeated measurements on one unprepared specimen sample its changing conditional state, rather than independent copies of its original state. Multiple bases are needed in this projective protocol; a single fixed <informationally complete POVM>, such as the <tetrahedral qubit POVM>, is an alternative. It still requires an ensemble of trials.
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