= Solution
The Holevo optimality conditions say a POVM is optimal when
$$
\Gamma=\sum_kp_k\rho_kM_k
$$
is Hermitian and $\Gamma-p_k\rho_k\geq0$ for every $k$. Here
$$
\Gamma=\frac{c+s}{3}
\begin{pmatrix}c&0\\0&s\end{pmatrix}
$$
and
$$
\Gamma-\frac13\rho_k
=\frac{cs}{3}
\begin{pmatrix}
1&-\zeta^{-k}\\
-\zeta^k&1
\end{pmatrix}\geq0,
$$
whose eigenvalues are $0$ and $2cs/3$. The <pretty good measurement> is therefore optimal. Its success probability is
$$
\boxed{P_{\rm succ}=\frac13(c+s)^2
=\frac13(1+\sin\theta)}.
$$
Solved by gpt-5.6-sol high.
Back to article page
