Is the establishment of entanglement inevitable with the passage of time?

Question

Derivation of the Lindblad master equation starts with the assumption that at an initial time $t=0$, the total density matrix is the product of the density matrices of the system $\rho_S$ and that of the environment $\rho_E$ i.e. $$\rho_{\rm tot}(0)=\rho_S(0)\otimes\rho_E(0)\tag{1}$$

Question If we start with such a factorizable density matrix (of the form Eq.$(1)$), is it inevitable that entanglement will be established between system and environment at $t>0$ irrespective of the interaction Hamiltonian that couples the system and environment? Can we think of a situation where the interaction Hamiltonian such that no entanglement is ever produced under time development?

Answer 1

Yes, it is inevitable that entanglement will emerge at $t>0$ if you want your system to evolve in a non-trivial way. This is due to the fact that, if the action of the bath on the system is not trivial, it means that the system-environment interaction is non-local and generates entanglement. Although for Markovian systems the amount of correlations built up at each time-step are neglected during the computation of the master equation, they still exist, e.g. have a look at the nice discussion in this reference: if the bath is large but finite, the emergence of correlations is clearly visible.

Still, it may happen that the system-environment interaction has no effect on a certain subspace of the Hilbert space of the system. This is the case of the decoherence-free subspaces. If the initial state of the system belongs to a decoherence-free subspace, then its dynamics will be decoupled from the environment one and no entanglement will arise.