= Solution
\b[Yes: distinct density operators have a projective measurement with different outcome probabilities.] Let $\rho,\sigma$ be the two <density operators> and let $D=\rho-\sigma$. It is a nonzero <Hermitian operator> with trace zero. Its nonzero eigenvalues cannot all have the same sign, since their sum is zero. Let $P_+$ be the <orthogonal projection> onto its positive-eigenvalue subspace. Then the two-outcome <projective measurement> $\{P_+,I-P_+\}$ satisfies
$$
P_\rho(+)-P_\sigma(+)=\operatorname{Tr}[P_+(\rho-\sigma)]
=\sum_{\lambda_j(D)>0}\lambda_j(D)>0.
$$
The outcome therefore carries statistical information about which <quantum ensemble> was used.
For example, with equal prior probabilities, guess $\rho$ after $+$ and $\sigma$ after the other outcome. The success probability is
$$
P_{\rm succ}=\frac12\operatorname{Tr}(P_+\rho)+\frac12\operatorname{Tr}[(I-P_+)\sigma]
=\frac12+\frac12\operatorname{Tr}(P_+D).
$$
Since the positive and negative eigenvalue sums of $D$ have equal absolute value, $\operatorname{Tr}(P_+D)=\|D\|_1/2$. Consequently
$$
\boxed{P_{\rm succ}=\frac12+\frac14\|\rho-\sigma\|_1>\frac12.}
$$
This is the equal-prior measurement in the <Holevo–Helstrom theorem>, expressed using the <trace distance>. The conclusion is some information, not necessarily perfect one-shot discrimination: nonorthogonal density operators generally cannot be distinguished with certainty.
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