Toy model for entangling interaction? Most of the time entanglement is described using some instantaneous classical measurement from an apparatus, but it is possible to describe it between two quantum systems, with two systems $O$ (for the observer) and $S$ undergoing the following process 
$$\vert \psi^O \rangle \otimes\vert \psi^S \rangle \to \sum_\alpha a_\alpha  \vert \psi^O_\alpha \rangle \otimes \vert \phi_\alpha \rangle$$
for a bunch of measurement with value $\alpha$ and probability $a_\alpha^2$.
But is there a toy model to show this process actually happening, where two particles, originally just a tensor product of the free states, evolve into entangled states, ideally without using some instantaneous interaction like a quantum gate? Also does this conflict with the fact that for long range interactions, particles are always interacting with each other?
 A: Assume that $\left| \psi^O \right. \rangle$, $\left| \psi^S \right. \rangle$ are just two spin 1/2 particles. Then a Heisenberg interaction:
$$\vec{S^o} \cdot \vec{S^s} = \hat{S_x^o} \otimes \hat{S_x^s} + \hat{S_y^o} \otimes \hat{S_y^s} + \hat{S_z^o} \otimes \hat{S_z^s}$$
will evolve a product state into an entangled singlet state:
$$\frac{1}{\sqrt{2}} \left( \left| \uparrow^O \right. \rangle \left| \downarrow^S \right. \rangle - \left| \downarrow^O \right. \rangle \left| \uparrow^S \right. \rangle \right)$$
It is possible to generalize this to a more realistic "observer" and "system". A simple way of playing around with this is to numerically generate a random hermitian $(NM \times NM$) matrix, where $N = dim(system)$, $M=dim(observer)$. Diagonalize the matrix and find the lowest energy eigenvector. Reshape this $NM$-vector into a $(N,M)$ matrix and do a singular value decomposition of it. The entanglement entropy can then be found from the singular values $\Lambda_i$ as
$$S = \sum_i -\Lambda^2_i \log(\Lambda^2_i)$$
You will find that most Hamiltonians have an entangled ground state. This would mean that your initial product state (with sufficient overlap) would also evolve to such an entangled state.
More realistic would be to include the effect of an environment by using the theory of open quantum systems. In "most cases" the environment destroys the entanglement, which is an explanation why quantum entanglement is so fragile. See https://arxiv.org/pdf/quant-ph/0306072.pdf for more on the subject.
Maybe this at least answers the first part of your question.
