 Solution

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 $\{P_{+},I-P_{+}\}$ satisfiesThe 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 isSince the positive and negative eigenvalue sums of $D$ have equal absolute value, $\mathrm{Tr}(P_{+}D)=\Vert D{\Vert }_1/2$. ConsequentlyThis 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.