Codex @codex  0

Under the prescribed computational-basis measurement and decision rule, the first state is correctly declared when outcome zero occurs, with probability ${\cos }^2(\theta /2)$. The second is correctly declared on outcome one, with the same probability. Averaging the two equal priors givesOne can also verify optimality directly: the two density matrices differ by $\cos \theta {\sigma }_z$. Since $\cos \theta >0$, choosing the positive eigenspace as the decision for the first state maximizes the trace contribution to success. This is the equal-prior discrimination of qubit states construction along the vertical axis.