# How do particles become entangled?

A person asked me this and I'm just a lowly physical chemist.

I used a classical analogy (how good or bad is this and how to fix?) Basically, light has a net angular momentum of zero, insofar as it is not polarized into its left and right plane polarized forms until it hits a crystalline structure.

However, once it does hit such a structure, we have left and right plane polarized light--that is left and right photon beams.

Since the original light was not polarized, this polarization (left right) be inherent in the light. The original light is a superposition of left and right polarized light, each with a total angular momentum of -1 and 1, so that they result in the total zero polarized incident light.

Thuse they are entangled to lower their spin (quantum angular momentum). Once we measure one of the particles in the superposition, we know the other by conservation of angular momentum.

Is this close?

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The production of entangled photons is often accomplished in the process of spontaneous parametric downconversion (wikipedia page), in which a beam of light is sent through a nonlinear crystal. Through interactions with the quantum structure of the crystal, there is a non-zero amplitude for a single photon to transition to a superposition of multiparticle states (i.e. the photon number operators do not commute with the effective Hamiltonian). With very high probability, a resultant two-particle state will be entangled, in the sense that it cannot be factored into a single tensor product of two single-particle states.

Note that the likelihood of obtaining an entangled state is "increased" by the fact that photons are indistinguishable: if such a factorization were possible, then the factor states would need to be identical. In general, the nature of the resulting entangled states depends on the conserved quantities associated with the interaction with the crystal.

It is also possible to produce an entangled state through a global transformation on a system in a way that does not involve interactions between the constituent particles. For example, since the $|++\rangle$ and $(1/N)(|+-\rangle +|-+\rangle)$ two-spin states are both in the triplet representation of $\mathrm{SO}(3)$, it is possible to find a group element of $\mathrm{SO}(3)$ that maps the tensor product state to the entangled state, thereby "producing" an entangled state (albeit in a very contrived way: basically, there's a way of looking at the reducible $|++\rangle$ state such that it appears to be entangled).

The theory of entanglement and quantum mechanics in general is very closely related to the mathematical subject of representation theory, which is very useful in theoretical chemistry.

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