Projective state exclusion ($⟨b_j|{\psi }_j⟩=0$)

For $d$ labelled pure states in dimension $d$, an orthonormal basis $\{|b_j⟩{\}}_{j=0}^{d-1}$ supplies error-free projective state exclusion when $⟨b_j|{\psi }_j⟩=0$ for every $j$. The Born rule gives zero probability to reporting $j$ on input ${\psi }_j$; orthonormal-basis completeness ensures that an outcome is always obtained. Conversely, zero error for this labelled basic measurement forces precisely those zero overlaps. The matrix of transition probabilities is therefore a matrix of column probability distributions with zero diagonal.