= Solution
\b[No measurement on the supplied system alone distinguishes ensembles with the same density operator.] A completely general <quantum measurement> has <POVM> effects $E_r\geq0$ with $\sum_r E_r=I$. The outcome probability for a <quantum ensemble> is
$$
P(r)=\sum_i p_i\langle\psi_i|E_r|\psi_i\rangle
=\operatorname{Tr}(E_r\rho).
$$
If both ensembles have the same $\rho$, every outcome probability is identical. This proves the claim for all <positive operator-valued measures>, and includes <projective measurements> as a special case. An <quantum ancilla> in a fixed independent state does not help: the input density operators remain identical after tensoring with that ancilla, and the same argument applies to any joint measurement.
The premise concerns access only to the prepared state. An accessible preparation label or a correlated <purification of a density operator> is additional information outside that premise and can distinguish global preparations. Without such side information, different ensemble decompositions do not define distinguishable states.
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