# 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.

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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

## 1 Answer

I agree that it doesn't sound too surprising. Intuitively, if we have a system with local interactions, it's reasonable to suppose that the only relevant interactions between two subsystems only occurs at the boundaries, and therefore the mutual information scales with the area (this is also the intuition that the paper uses in Figure 1). However, maybe I can motivate the importance of the result:

• The specific numerical factor in the inequality is notable. Not only is the mutual information of two subsystems bounded by some value proportional to the surface area, but it is the same value for a wide variety of cases.
• By verifying this bound rigorously, one can also better understand the necessary assumptions that go into it. A notable quote from the paper is

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 $$\xi_M$$.

In other words, the area law can be violated in certain cases, and now we better understand why that is.

• "Additionally, what's up with the logarithmic dependence on the dimension?" -- This is the Hilbert space dimension of each particle, not the spatial dimension. It is most natural that it appears. (EDIT: Or the bond dimension. In any case, not the spatial dimension.) – Norbert Schuch Feb 17 at 20:55
• "Note that the paper you referenced only verifies the bound for a certain class of states." -- It is verified for all thermal states at non-zero temperature. This is rather general! – Norbert Schuch Feb 17 at 21:04
• 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