What OP is asking is equivalent to : "why two state with the same density matrix physically indistinguishable ?"
Consider a (mixed) state described by a density matrix $\rho$. Its time evolution satisfies the von Neumann equation $i\hbar \dot{\rho} = [H,\rho]$ (or more generally a Lindblad master equation in the case of an open quantum system). This involves only the density matrix, so time-evolution doesn't distinguish states with the same density matrices.
We also need to address measurement. Consider an observable $\hat A = \sum_i a_i P_i$ (where the $P_i$ are the orthogonal projection on the different eigenstates). The average outcome of a measurement of $\hat A$ is given by :
$$\langle \hat A\rangle = \operatorname{Tr}(\hat A\rho)$$
The probability of finding the eigenvalue $a_i$ is :
$$p_i = \langle P_i\rangle = \operatorname{Tr}(P_i\rho)$$
If we observe the value $a_i$, the system will be in the following state, right after measurement :
$$\rho_i'=\frac1{\operatorname{Tr}(\rho P_i)}P_i\rho P_i$$
If we don't know which outcome was measured, the system will be in the following state, right after measurement :
$$\rho' = \sum_i P_i\rho P_i$$
All of these only involve the density matrix (rather than the probabilistic mixture of pure states from which they may be constructed). Therefore, two state with the same density matrices are physically indistinguishable.
