Solution
= Solution
For priors $p_1,p_2$,
$$
p_{\rm succ}
=p_2+\operatorname{Tr}[E(p_1\rho_1-p_2\rho_2)].
$$
If $\Delta=p_1\rho_1-p_2\rho_2$, the maximum is $p_2+\operatorname{Tr}\Delta_+$. Since
$$
\operatorname{Tr}\Delta_+
=\frac12(\|\Delta\|_1+\operatorname{Tr}\Delta)
=\frac12(\|\Delta\|_1+p_1-p_2),
$$
one obtains
$$
\boxed{p_{\rm succ}^{\max}
=\frac12\left(1+\|p_1\rho_1-p_2\rho_2\|_1\right)}.
$$