Area laws in quantum systems: mutual information and correlations

TL;DR

The paper investigates how information in a subsystem scales with the boundary rather than the volume across classical and quantum thermal states. It uses mutual information as a rigorous, temperature-aware measure of total correlations and proves area-law bounds for classical Gibbs states and for quantum states, including mixed PEPS representations of commuting Hamiltonians. It further shows that in one-dimensional finitely correlated states the correlation length governs exponential decay of correlations and mutual information, connecting tensor-network structure to area laws. Overall, the work clarifies when and how area laws arise at finite temperature and establishes practical links to tensor-network representations for efficiently describing highly correlated quantum systems.

Abstract

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. This counterintuitive idea which has its roots in the nonextensive nature of black-hole entropy serves as a guiding principle in the search for the fundamental laws of Planck-scale physics. In this paper we show that a similar phenomenon emerges from the established laws 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 microscopic degrees of freedom, it may diverge as the inverse temperature in quantum systems. A rigorous relation between area laws and correlations is established and their explicit behavior is revealed for a large class of quantum many-body states beyond equilibrium systems.

Figures (2)

• 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) is thus 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 which is formed by singlet pairs (indicated by lines joining them) whose length follows a given probability distribution.