Entanglement in NRQM and QFT

[accdd]

What is entanglement in QM and QFT?
I understood that it only corresponds to the concept of linear combination of states with multiple particles. Seeing lectures on YB it seems to me that it is something much deeper than that. What did I miss? How is it treated in QFT?
I am studying NRQM from the Griffiths book.

 

[PeterDonis]

What is entanglement in QM and QFT?

An entangled state is a state of a quantum system with multiple subsystems that cannot be expressed as a product of states of the subsystems.

For example, suppose we have two identical particles and we are considering their spins. A product state might be something like this:

$$
\psi_P = \ket{\uparrow}_1 \ket{\uparrow}_2
$$

An entangled state would be something like this:

$$
\psi_E = \ket{\uparrow}_1 \ket{\uparrow}_2 + \ket{\downarrow}_1 \ket{\downarrow}_2
$$

Note that there is no way to express $\psi_E$ as a product of a "1" ket and a "2" ket. It is a sum of multiple such products, but there is no way to express it as a single such product.

 

[Demystifier]

What is entanglement in QM and QFT?
I understood that it only corresponds to the concept of linear combination of states with multiple particles.

Yes, that's correct.

Seeing lectures on YB it seems to me that it is something much deeper than that.
What did I miss?

Mathematically, there is nothing deep about entanglement. What you miss are physical consequences of that. The physical consequences arise when you ascribe a physical interpretation to those states, specifically the standard probabilistic interpretation.

How is it treated in QFT?

The same way as in NRQM. (Except that now you can even have superpositions of states with different numbers of particles, but in the context of entanglement such superpositions are rarely relevant physically.)

 

[vanhees71]

One should also note that you can formulate also non-relativistic QM as a non-relativistic QFT. This is of great advantage when it comes to many-body theory, even if in fact the particle number is conserved. The reason is that the QFT formalism (no matter whether it's relativistic or non-relativistic) takes care of the Bose-Einstein and Fermi-Dirac constraints of many-particle states.

 

[accdd]

Can entangled states be generated only locally? For example if I have a particle that decays I know entangled particles can be generated.
Or can entangled states be generated even between distant particles? For example if I have one particle here and one on a distant galaxy, if I wait long enough, will an entangled state of both particles ever result?

 

[vanhees71]

It's not that easy, and it's extremely unlikely that two particles from far distant galaxies are entangled when we measure them.

Usually to prepare entangled quantum states (most easily it's done with photons) one uses some local process like parametric down conversion:

https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion

Nevertheless you can indeed prepare entangled pairs of particles which never have been in local contact with each other. One way is "entanglement swapping":

https://en.wikipedia.org/wiki/Quantum_teleportation#Entanglement_swapping

 

[accdd]

If I wait long enough can two or more particles in distant galaxies end up in an entangled state?