= Equal-prior discrimination of qubit states
{title2=$p_{\mathrm{succ}}=\tfrac12+\tfrac14|r-s|$}
For equiprobable <qubit> states with <Bloch vectors> $r,s$, their density-matrix difference is $(r-s)\cdot\sigma/2$, with <eigenvalues> $\pm|r-s|/2$. An effect $0\le E\le I$ declaring the first state yields success $1/2+\operatorname{Tr}[E(\rho_r-\rho_s)]/2$. Selecting its positive-eigenvalue eigenspace maximizes this <trace> and proves the displayed result. Measure along $r-s$ and attach the corresponding state labels; if the vectors coincide, no measurement improves on guessing.
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