= Solution
A <basic measurement> of a finite-dimensional <quantum system> is specified by an <orthonormal basis> $\{|b_j\rangle\}_j$ of its <Hilbert space>. For a normalized input <pure state> $|\chi\rangle$, the <Born rule> gives
$$
\boxed{p_j=|\langle b_j|\chi\rangle|^2,\qquad
|\chi_j\rangle=\frac{|b_j\rangle\langle b_j|\chi\rangle}{\sqrt{p_j}}
\quad(p_j>0).}
$$
The <post-measurement state> is therefore $|b_j\rangle$ up to a <global phase>. An outcome of zero <probability> has no conditional <post-measurement state>. For a general input <density matrix> $\rho$, the same <basic measurement> has $p_j=\langle b_j|\rho|b_j\rangle$ and conditional <density matrix> $|b_j\rangle\langle b_j|$ whenever $p_j>0$.
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