Tracing $\rho (t)$ with respect to the Bath when system and bath are coupled in an open quantum system

Consider a system S that is coupled to a bath B. Let {$$|s_i\rangle 's$$} and {$$|b_j\rangle 's$$} be the eigen states of the system and bath hamiltonians respectively (i.e)

\begin{align} \hat{H}_{S}|s_i\rangle = E_{s_i} |s_i\rangle\\ \hat{H}_{B}|b_j\rangle = E_{b_j} |b_j\rangle\\ \end{align} and the total Hamiltonian of the system is given by: \begin{align} \hat{H}_T = \hat{H}_{S}+\hat{H}_{B}+\hat{H}_{int} \end{align} Then the direct product space $$S\otimes B$$ is spanned by {$$|s_i b_j\rangle\langle s_i b_j|$$} and it is not the set of eigen states due to the interaction term $$\hat{H_{int}}$$. But, it could be used as a basis to find the trace of $$\rho(t) \hat{A}(q_s)$$. Where $$\hat{A}(q_s)$$ is any operator in S and only depends on the system degrees of freedom. The expectation value of this system operator is given by: \begin{align} \langle A(q_{s_i}) \rangle = \textbf{Tr} \ \rho(t) A(q_s) \end{align}

In construction of the reduced density matrix $$\sigma_{s_i,s_k}(t)$$, we get the expression to be: \begin{align} \sigma_{s_i,s_k}(t) = \sum_{b} \langle s_i b_j| \rho(t)|s_k b_j\rangle = \textbf{Tr}_\textbf{B} \langle s_i| \rho(t)| s_j \rangle \end{align}

Questions

1. How do we arrive at that final expression considering that systems and bath states are interacting and are entangled to each other?

2. What is the physical significance of constructing the reduced density matrix? Could point to some calculation that could help me the reader appreciate the need for constructing the reduced density matrix.

3. What are the subtleties that are involved in constructing the reduced density matrix in the Many body physics or molecular physics?

• For question #2, this simple perspective may help: Using a reduced density matrix is equivalent to ignoring all observables except those that are associated exclusively with the sub-system $S$. The reduced density matrix is just the minimum amount of information we need to retain about the state in order to calculate expectation values of that restricted set of observables. So there is really no "need" to use a reduced density matrix, unless we just don't want to carry around the extra information that we never plan to use. Commented May 21, 2019 at 4:15
• as far as I understand your notation, the fact that system and bath are interacting doesn't really have much to do with how you write the reduced density matrix. What will happen is that the reduced density matrix will not evolve unitarily if there is an interaction with the bath. The reason you are interested in the reduced density matrix is often that you do not have access to the full state of system+environment.
– glS
Commented May 22, 2019 at 11:04
• What will happen to loss of information on coherences when we trace out? Commented May 22, 2019 at 11:24