What do the area laws for mutual information and correlations actually state? [closed]

I am trying to understand what are the key points of this paper: Area laws in quantum systems: mutual information and correlations.

If I understand correctly the law states that the information flux between block A and B depends on the boundary between them ($$D$$ is the dimension at which the system is embed in):

I(A : B) ≤ 2|∂A| log D

I don't see why it's surprising that two subcompartment can interact only with their boundary.

closed as unclear what you're asking by Norbert Schuch, Jon Custer, Aaron Stevens, RogerJBarlow, ZeroTheHeroMar 9 at 1:16

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• @DanYand, I agree, but what would be the interpretation of the mutual information $I(A:B)$ in classical system? – 0x90 Feb 17 at 18:57
• What is your question? – Norbert Schuch Feb 17 at 20:54
• Also, shouldn't there be a temperature dependence in the formula? EDIT: Upon a closer look, I realize that you quote a completely random formula from the paper out of context. (This seems to be eq. 4 which talks about PEPS.) How should we know what you are looking for without any context? – Norbert Schuch Feb 17 at 20:55

So, indeed, we get an area law for the mutual information solely from the existence of a length scale $$\xi_M$$, which expresses the common sense explanation of Fig. 1. This area law is also valid for zero temperature and when violated immediately implies an infinite correlation length $$\xi_M$$.
• Thanks for the catch on the dimension - I must not have noticed the distinction between $D$ and $\mathcal{D}$. And yes, reading further, they do generalize the claim more than I thought (I only read up to the equation cited). – Henry Shackleton Feb 17 at 22:37