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

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).

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

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 $$N$$.