An area law for the entropy of low-energy states

TL;DR

This paper proves that low-energy states in quantum systems with local interactions obey an area law for entropy, with a possible logarithmic correction, under certain decay and eigenstate conditions.

Contribution

It establishes conditions under which low-energy states satisfy an area law for entropy, extending understanding beyond ground states.

Findings

Low-energy states obey an area law with a logarithmic correction.

Conditions include decay of correlations and limited eigenstates with zero energy-density.

The approach involves measuring energy fluctuations via exterior and shell measurements.

Abstract

It is often observed in the ground state of spatially-extended quantum systems with local interactions that the entropy of a large region is proportional to its surface area. In some cases, this area law is corrected with a logarithmic factor. This contrasts with the fact that in almost all states of the Hilbert space, the entropy of a region is proportional to its volume. This paper shows that low-energy states have (at most) an area law with the logarithmic correction, provided two conditions hold: (i) the state has sufficient decay of correlations, (ii) the number of eigenstates with vanishing energy-density is not exponential in the volume. These two conditions are satisfied by many relevant systems. The central idea of the argument is that energy fluctuations inside a region can be observed by measuring the exterior and a superficial shell of the region.