= Trine qubit POVM
{title2=$E_j=\tfrac23|\phi_j\rangle\langle\phi_j|$}
Choose three pure <qubit> states whose unit <Bloch vectors> are coplanar, sum to zero and have pairwise angle $120$ degrees. The three effects $E_j=\frac23|\phi_j\rangle\langle\phi_j|$ sum to the identity, giving a <POVM>. If an input is the state orthogonal to $\phi_1$ or to $\phi_2$, outcome one excludes the first input and outcome two excludes the second. Each conclusive outcome has probability one half on its corresponding input; the third outcome is inconclusive with probability one half on either input. This gives <unambiguous quantum state discrimination>, rather than identification on every trial.
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