I am trying to understand how Boltzmann's statistical entropy relates to Von Neumann's quantum entropy. For example, the canonical ensemble describes a system in contact with a thermal bath at fixed temperature $T$, where the probability to be in a state $n$ is
\begin{equation} p_n=e^{-E_n /{k_b T}} \end{equation}
we can compute the entropy of this system as
\begin{equation} S=-\sum_n p_n \ln p_n \end{equation}
Conceptually, what's happening is that our actual system is rather complicated in its interactions with the thermal bath, so we instead make $N$ copies of the system with different definite energies such that the probability distribution follows $p_n$ and then make predictions about the ensemble.
The important thing is that even if we don't know what's happening with the system, our physical real system is very much in some state. However, at the quantum level, if we think of the system as trully being in one state, then the Von Neumann entropy automatically gives as zero entropy. I think this would be solved if we notice that the system is in contact with a thermal bath and thus is entangled with the enviroment. Local measurements on the system would then look as if the system is in a mixed state, thus giving entropy.
My question is: Is classical statistical entropy caused at the quantum level by tracing out the degrees of freedom of the thermal bath? If not, how can we explain that a macroscopical classical system in a definite state has some entropy but if we describe it quantum mechanically it does not?
Related to this, if entanglement with the enviroment is what creates statistical entropy by tracing out the degrees of freedom in the thermal bath, then how can we explain entropy in the microcanonical ensemble?