quantum mechanics - How are entangled states created?

I understand that when I have two separate states that their combination state increases the Hilbert Space to $|\psi_1\rangle \otimes |\psi_2\rangle$

For example, looking at a simple example where we are considering two possible states, this can be expanded to: $(a|H_1\rangle+b|V_1\rangle)\otimes(c|H_2\rangle+d|V_2\rangle)$.

This can be then be written as $(ac|H_1\rangle |H_2\rangle + ad|H_1\rangle |V_2\rangle + bc|V_1\rangle |H_2\rangle + bd|V_1\rangle |V_2\rangle)\frac{1}{2}$

Now entanglement is defined as when we get something different than this. We have entanglement when the state can not be written as simply a Kroniker product of any superposition state of its component states ($\psi \neq |\psi_1\rangle \otimes |\psi_2\rangle$)

There are a number of different procedures for checking if a given state is entangeled, but how are entanglement states created in the first place?

I'm looking for examples of entanglement in which the mechanism that creates the entanglement is explicit.

The only example I can think of is Hong-Ou-Mendel interference creating NOON states like, $|2,0\rangle + |0, 2\rangle$. I get that generally identical possible outcomes can sometimes destructively interfere, but I'm looking for something a little bit more clear generally. In particular I'd like to build some intuition so that when I see am looking at given physical system I'll have an idea if entanglement could be generated.

Answer

Any procedure in a quantum system can be described by a unitary operator $U$ (quantum evolution) and/or a projection operator $P$ (measurement). If you want to bring two isolated subsystems in a state $|\psi_1\rangle\otimes|\psi_2\rangle$ into an entangled state $|\psi\rangle$ you need to ask what type of unitary operator $U$ and/or projection operator $P$ you should use such that: $$ P\left(U\left(|\psi_1\rangle\otimes|\psi_2\rangle\right)\right)=|\psi\rangle $$ As an example, imagine two $1/2-$spin systems in an initial state $|\uparrow\rangle \otimes|\uparrow\rangle$, doing the following procedures:

1. A measurement of $\vec{S}_1\cdot \vec{S}_2=\frac{1}{2}\left[(\vec{S}_1+\vec{S}_2)^2 - \vec{S}_1^{2}-\vec{S}_2^{2}\right]=\frac{1}{2}(\vec{S}_1+\vec{S}_2)^2-\frac{3}{4}$.


2. Or a evolution by an hamiltonian $H\propto\vec{S}_1\cdot \vec{S}_2$ by $\Delta t\neq T$, where $T$ is the period of precession.

you are going to get an entangled state.

More generally, any measurement of a global observable like $\vec{S}_1\cdot \vec{S}_2$ produces an entangled state.

For the $U$ operators, any hamiltonian that cannot be writen as $H\neq H_1\otimes1 + 1\otimes H_2$ will produce entangled states for times different than the period of oscillation, if there is any. This means that is sufficient to have an interaction between this two subsystems and avoid intervals of time $\Delta t=T$, where $T$ is some period of the system.