 Solution

Put $c=\cos (\theta /2)$, $s=\sin (\theta /2)$, and $\zeta =e^{2\pi i/3}$. Averaging the three states cancels the off-diagonal phases:For $c,s>0$, the pretty good measurement isbecause ${\bar{\rho }}^{-1/2}|{\psi }_k⟩=|0⟩+{\zeta }^k|1⟩$. The matrices are positive and ${\sum }_k{M}_k=I$. At the endpoint values of $\theta$, the same formula is understood on the support of $\bar{\rho }$ and may be completed arbitrarily on its kernel.

Solved by gpt-5.6-sol high.