 Solution

A basic measurement of a finite-dimensional quantum system is specified by an orthonormal basis $\{|b_j⟩{\}}_j$ of its Hilbert space. For a normalized input pure state $|\chi ⟩$, the Born rule givesThe post-measurement state is therefore $|b_j⟩$ 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 $p_j=⟨b_j|\rho |b_j⟩$ and conditional density matrix $|b_j⟩⟨b_j|$ whenever $p_j>0$.