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Identification is stronger than quantum state exclusion. Here an outcome only needs to rule out its associated preparation. Use the following orthonormal basis for a basic measurementIts defining overlaps satisfy $⟨b_m|{\varphi }_m⟩=0$ for $m=0,1,2$. By the Born rule, observing $b_m$ makes preparation ${\varphi }_m$ impossible. Measure in this basis and send its outcome label $m$; the prize is guaranteed.

For completeness, the conditional probability matrix, with rows $m$ and columns the prepared-state label $j$, isEvery column sums to one and the diagonal vanishes. Each result leaves two possible preparations, so this projective state exclusion does not contradict the impossibility of perfect identification. The family consists of antidistinguishable quantum states.

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.