= Three-way qutrit sign-flip discrimination
{title2=$p_{\mathrm{success}}=25/27$}
For the three <qutrit> <unitary operators> $U_j=I-2|j\rangle\langle j|$, prepare $|s\rangle=(|0\rangle+|1\rangle+|2\rangle)/\sqrt3$, query once, apply the known <inversion about the mean> $D=2|s\rangle\langle s|-I$, and use a <computational-basis measurement> to guess $j$. Since $D^2=I$, the processing is a <unitary transformation>. Direct multiplication gives $DU_j|s\rangle=(5|j\rangle-\sum_{k\ne j}|k\rangle)/(3\sqrt3)$, establishing success <probability> $25/27$ for each label. This calculation establishes the achieved <probability>, without requiring an optimality claim.
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