= Solution
For any linear operator, the <trace norm> is $\|X\|_1=\operatorname{Tr}\sqrt{X^\dagger X}$, the sum of its <singular values>. For a <Hermitian operator>,
$$
\boxed{\|X\|_1=\operatorname{Tr}|X|=\sum_i|\lambda_i|.}
$$
The <Holevo–Helstrom theorem> concerns minimum-error discrimination of two <density operators> $\rho,\sigma$. If their prior probabilities are $p$ and $1-p$, the maximum success probability over all <quantum measurements> is
$$
\boxed{P_{\mathrm{succ}}^*=\frac12\left(1+\|p\rho-(1-p)\sigma\|_1\right).}
$$
An optimal binary <POVM> declares $\rho$ on the positive spectral subspace of $p\rho-(1-p)\sigma$ and declares $\sigma$ on its negative spectral subspace; zero-eigenvalue vectors can be assigned either way. For equal priors the formula becomes $P_{\mathrm{succ}}^*=\tfrac12+\tfrac14\|\rho-\sigma\|_1$.
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