Why does the entanglement of quantum fields depend on their distance?

Question

When watching Seans Carrol's "A Brief History of Quantum Mechanics", he mentioned around the 50th minute (the video I linked to starts at that point) that

[about quantum fields in vacuum] ... and guess what! The closer they are to each other, the more entangled they are.

Why is it so? I was under the impression that entanglement is not dependent on the distance (two entangled particles getting further from each other are not less entangled).

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If this is at all possible I would be grateful for an answer understandable by an arts major - just kidding a bit, I simply would like to avoid an answer which starts with

^{courtesy of Redorbit}

Answer 1

I expect that Carrol is referring to cluster decomposition, a principle satisfied by many quantum field theories. This principle says that if two quantities are located in spacelike-separated regions very far from one another, then they are going to be uncorrelated. That is, if operators $A$ and $B$ are localized in two spacelike-separated regions a distance $\ell$ apart, we have $\langle A B \rangle \to \langle A \rangle\langle B \rangle$ as $\ell \to \infty$. We can usually give a more detailed statement of how quickly the correlation dies off. For example, in a theory of a field with mass $m$ the bound on the difference $\lvert\langle A B \rangle - \langle A \rangle\langle B \rangle\rvert$ goes like $e^{-m\ell}$, and in a massless theory the difference goes to zero faster than $\ell^{-N}$ for any $N$.

Answer 2

I don't think you should pay any attention to this. Entanglement refers to quantum states, not to quantum fields.