[Warning: I'm not a physicist]
Let $A$ and $B$ be microscopic systems, with corresponding Hilbert spaces of state vectors given by $H_A$ and $H_B$ respectively.
Let's say $A$ is in a state $\psi\in H_A$ and $B$ is in a state $\phi\in H_B$, possibly both evolving unitarily in time. When the two systems "interact", a new Hilbert space is formed, for the combined system, which is the (completed) tensor product $H=H_A\otimes H_B$. Likewise, the new combined state becomes $\Psi=\sum_i\psi_i\otimes \phi_i \in H_A\otimes H_B$. Well, or maybe just $\psi\otimes \phi\in H_A\otimes H_B$, but this is a separable state, right? so I don't know why one would say the two systems "get entangled"! In case it's not just $\psi\otimes\phi$, I'll ask:
Q1. Is $\Psi$ determined by the pair $(\psi,\phi)$ or one would need to also know the details of how the two systems interact?
Anyway, it seems to me that the above is just a mathematical idealization: before the "interaction", we have a pair of Hilbert spaces with given state vectors, and just after the "interaction" the Hilbert spaces have magically changed into a tensor product, and likewise the state vectors. This reminds me of the magical collapse of the wave function that is supposed to happen upon measurement according to the Copenhagen interpretation.
But, if quantum mechanics has to hold globally (say for the whole universe, or at least the combined system) and at all times, then the whole "interaction" giving rise to entanglement (or, at least, tensor products) has to take place within a larger Hilbert space $H$ (maybe $H_A\otimes H_B$ is sufficient for this purpose) and according to unitary evolution $\dot{\Psi}(t)=-\frac{\mathrm{i}}{\hbar} \hat{H}\Psi(t)$ where $\Psi(t)\in H$ is a state vector (varying with time). Independently from any interpretation of QM, I would assume the evolution, in $H$, is unitary because no measurement is performed on the combined system "$A+B$" from the outside (or is this wrong cause the two systems somehow "measure each other"?). I imagine the Hamiltonian $\hat{H}$ should depend on the Hamiltonians $\hat{H}_A\in \mathcal{L}(H_A)$ and $\hat{H}_B\in \mathcal{L}(H_B)$ of the two systems, and on how the two systems are supposed to interact.
Q.2 Is there a theory that describes the evolution of $\Psi(t)\in H$ such that $\Psi(0)$ somehow "corresponds" to the datum of noninteracting $\psi$ and $\phi$, and $\Psi(\infty)$ corresponds to $\sum_i\psi_i\otimes \phi_i$ (or $\Psi(T)$ corresponds approximately to $\sum_i\psi_i\otimes\phi_i$ for $T$ some sufficiently large time)? How does the encoding $(\psi,\phi)\mapsto\Psi$ work?
Also, in the case $B$ is a macroscopic measuring device, does the above question have something to do with the Measurement Problem? (If yes, since the Problem is allegedly unsolved, then I guess I'm not expecting a definitive answer but just a justification for the link)
Maybe, the encoding is just $(\psi(t),\phi(t))\mapsto\psi(t)\otimes\phi(t)$ at all times. But then one would need a definition of what it means that somehow, at $t=0$, $\psi(0)\otimes\phi(0)$ describes two separate systems while, at $t=T$, $\psi(T)\otimes\phi(T)$ describes a state of the combined system. Maybe, a "measure of combinedness" $\mu$ such that $\mu(\psi(0)\otimes\phi(0),H_A\otimes H_B)=0$ and it is $>0$ at $t=T$?
Another guess: maybe, the "combinedness" of the systems depends on the choice of an observable $X$, and we have $X(t)=a(t)\otimes 1+1\otimes b(t)+K(t)$ where $X(t)=e^{-\mathrm{i}\hat{H}t/\hbar} X(0)e^{\mathrm{i}\hat{H}t/\hbar }$ etc., and $a$ and $b$ are observables on $A$ and $B$ respectively while $K$ is an observable of the combined system, and $K(0)=0$ (or $||K(0)||\ll 1$).