 Three-way qutrit sign-flip discrimination

For the three qutrit unitary operators $U_j=I-2|j⟩⟨j|$, prepare $|s⟩=(|0⟩+|1⟩+|2⟩)/\sqrt{3}$, query once, apply the known inversion about the mean $D=2|s⟩⟨s|-I$, and use a computational-basis measurement to guess $j$. Since $D^2=I$, the processing is a unitary transformation. Direct multiplication gives $D{U}_j|s⟩=(5|j⟩-{\sum }_{k\ne j}|k⟩)/(3\sqrt{3})$, establishing success probability $25/27$ for each label. This calculation establishes the achieved probability, without requiring an optimality claim.