Area Laws in Quantum Systems: Mutual Information and Correlations

The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. In this Letter we show that this phenomenon not only emerges in the search for new Planck-scale laws but also in lattice models of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of degrees of freedom, it may diverge as the inverse temperature in quantum systems. It is shown that an area law is generally implied by a finite correlation length when measured in terms of the mutual information.

Figure 1

We are interested in the mutual information (or entanglement) between the two regions $A$ and $B$. Heuristically, if there is a correlation length $\xi$, then sites in $A$ and $B$ that are separated by more than $\xi$ (the shaded stripe) should not contribute to the information or entanglement between $A$ and $B$. The mutual information (or entanglement) should thus be bounded by the number of sites at the boundary.

Figure 2

Left: We consider regions $A$ and $B$ separated by a spherical shell of thickness $L\ll R$; Right: Simple 1D model for a state that is formed by singlet pairs (indicated by lines joining them) whose length follows a given probability distribution.