In the research field of Many-body Localization (MBL), people are always talking about the eigenstate thermalization hypothesis (ETH). ETH asserts that for a isolated quantum system, all many-body eigenstates of the Hamiltonian are thermal, which means all sub-systems can involve to thermalzation in the end. ETH is not always true and violation to it means MBL for an interacting quantum many-body system. Well, my puzzle is as follows:

For a isolated quantum system $A$ and a space-specified sub-system $B\in{A}$. It is assumed the initial state of $A$ is one of the eigenstates $|\psi(t=0)\rangle_{A}$ of its Hamiltonian $H$. Of course it is a pure state. Note the the initial state $|\psi(t=0)\rangle_{B}$ of $B$ is not a pure state unless $|\psi(t=0)\rangle_{A}$ is the direct product state of $|\psi(t=0)\rangle_{B}$ and the state of $A/B$, which means there $B$ is disentangled with the rest part $A/B$. Since $B$ is chosen arbitrarily, mixed initial state of $B$ is the most general case and its state cannot be described by a single state but a density matrix $\rho_{B}(t=0)$.

Now let the system $A$ evolve along time. There are two ways to check $\rho_{B}$ at arbitrary time $t$.

1. I can partially trace $\rho_{A}$ by $\rho_{B}=\text{tr}_{A/B}\rho_{A}$. While $\rho_{A}=|\psi|\rangle_{A}\langle\psi|_{A}$ will not change because $|\psi\rangle_{A}$ is the eigenstate and it will not evolve under the time evolution operator thus $\rho_{B}$ will not change forever.

2. The mixed state $\rho_{B}(t=0)$ evolves along time and it may thermalize to Gibbs density matrix $\tilde{\rho}_{B}=\frac{1}{Z}e^{-\beta{H}}$ where $Z$ is its statistical partition function. This is indeed the statement of ETH.

What's wrong for the paradoxical results viewed from two different perspectives for the same thing?