How are entangled states created?

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.

One of the most popular methods to create entangled states in quantum optics is with the aid of spontaneous parametric down-conversion (SPDC). It is a nonlinear optical process where one photon (the pump photon) is converted into two photons (signal and idler). Conservation of momentum and energy implies that the two photons are entangled.

The nature of the entanglement is determined by the type of phase matching that is used. For type I phase matching one gets entanglement in the spatial degrees of freedom. With type II phase matching one also gets entanglement in the polarization degrees of freedom.

One way to look at the entanglement of a pure state is to express it as a Schmidt expansion $$ |\psi\rangle = \sum_n \lambda_n |\phi_n\rangle_s |\phi_n\rangle_i . $$ where $\lambda_n$ denotes the Schmidth coefficients and $|\phi_n\rangle_s$ and $|\phi_n\rangle_i$ are the Schmidt bases in the signal and idler systems, respectively. If the state is entangled, then there would be more than one nonzero $\lambda_n$. For collinear SPDC, the Schmidt bases are orbital angular momentum (OAM) eigenstates. This is a property that is often exploited in quantum optics, because it allows one to prepare entangled states in an infinite dimensional Hilbert space.