= Solution
In <quantum binary hypothesis testing>, hypothesis zero supplies $\rho_0$ with prior $p$ and hypothesis one supplies $\rho_1$ with prior $1-p$. A two-outcome POVM $\{Q,I-Q\}$ decides zero on outcome $Q$. The conditional errors are
$$
\alpha=\operatorname{Tr}[(I-Q)\rho_0],
\qquad
\beta=\operatorname{Tr}[Q\rho_1].
$$
Symmetric testing minimizes the prior-weighted average error $p\alpha+(1-p)\beta$, equivalently maximizing the average success probability.
Solved by gpt-5.6-sol high.
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