# 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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$| \uparrow \downarrow \rangle \pm | \downarrow \uparrow \rangle$, not $| \uparrow \uparrow \rangle$ nor $| \downarrow \downarrow \rangle$ –  user26143 Jun 15 '14 at 20:04

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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What is a nonlinear crystal? –  Nick Oct 21 '14 at 6:11

I will answer the question in the title:

How do particles become entangled?

Entanglement is a shorthand way of saying that "one is dealing with a quantum mechanical system which describes in a probabilistic manner the particles' variables, as solutions of specific quantum mechanical equations with specific boundary conditions."

( aside : Entanglement could be attributed to solutions of classical equations: when one has the solution for a planet and its satellite and the boundary conditions are given, if one knows where the satellite is, one also knows where the planet is, both revolving about their barycenter.)

In the framework of atoms and molecules and photons the quantum mechanical equations describe many body systems, and various quantum mechanical models have been developed to deal with quantum mechanical collective phenomena. Quantum mechanical entanglement means that the probability distributions, ( the square of the state function) for measurable behaviors of the particles are completely determined for the system.

The molecules in a crystal itself are entangled because in principle a state function can be written for the crystal.

In interactions, a new solution has to be used. When a proton hits a proton at the LHC the whole interaction, input particles, output particles and all the correlations and angles of the output particles are entangled . The statistical behavior of the interaction is given by the square of the state function describing the interaction and the experiment can measure the probability of production of the Higgs , for example.

If you want to use an example with light, you have again to go to the quantum mechanical level, the quanta of light, the photons. In lasing action, stimulated emission, a collective interaction, the whole process is entangled, producing a coherent light beam that emerges from the multitude of photons in the process.

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What you've described seems to be one example. In general, interactions between two quantum systems will put the system into some joint state which will generically be entangled. For example, if you have two spins coupled with a spin-spin coupling, then the ground state of that (total) two spin system has some entanglement.

EDIT: BTW, your analogy is not classical but actually quantum. And it's not just an analogy but captures the physics of entanglement accurately. People think of light polarization as classical since they learn it in an electrodynamics course, but essentially it is quantum behaviour -- for "massless" stuff like photons, the classical description has some inherently quantum properties :-)

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