Entanglement density and gravitational thermodynamics

TL;DR

This paper introduces the concept of entanglement density as a quasi-local measure in relativistic quantum theories, linking it to gravitational thermodynamics and energy conditions in holography.

Contribution

It defines entanglement density in relativistic quantum theories and relates its positivity to energy conditions and thermodynamic laws in holographic gravity.

Findings

Entanglement density is positive semi-definite under strong sub-additivity.

Positivity of entanglement density maps to the null energy condition in gravity.

Speculation on a second law-like statement for extremal surfaces.

Abstract

In an attempt to find a quasi-local measure of quantum entanglement, we introduce the concept of entanglement density in relativistic quantum theories. This density is defined in terms of infinitesimal variations of the region whose entanglement we monitor, and in certain cases can be mapped to the variations of the generating points of the associated domain of dependence. We argue that strong sub-additivity constrains the entanglement density to be positive semi-definite. Examining this density in the holographic context, we map its positivity to a statement of integrated null energy condition in the gravity dual. We further speculate that this may be mapped to a statement analogous to the second law of black hole thermodynamics, for the extremal surface.