Given that a qubit in equal superposition of $|0\rangle$ and $|1\rangle$ is represented by following wave function
\begin{equation} \Psi = \frac{1}{\sqrt 2}(|0\rangle + |1\rangle) \end{equation}
and associated density matrix
\begin{equation} \rho_1 = |\Psi\rangle \langle\Psi| = \frac{1}{2} \begin{pmatrix} 1 && 1 \\ 1 && 1 \end{pmatrix} \end{equation}
Further a qubit in this state is considered to be in a pure state (even though its in a superposition) since the density matrix can be cleanly factored into a product.
Fact $F$ : When measured this qubit will turn out to be 0 or 1 with 50% probability each.
But there exists another density matrix consistent with the same fact $F$ as follows:
\begin{equation} \rho_2 = \frac{1}{2} (|0\rangle \langle0|) + \frac{1}{2} (|1\rangle \langle1|) = \frac{1}{2} \begin{pmatrix} 1 && 0 \\ 0 && 1 \end{pmatrix} \end{equation}
and this matrix corresponds to a mixed state. Clearly $\rho_1 \neq \rho_2$ but they have same physical characteristics which is the fact $F$.
My Question: What is a physical fact or property that can be used to distinguish $\rho_1$ from $\rho_2$?