Entanglement density and gravitational thermodynamics

TL;DR

This paper introduces entanglement density $\hat{n}$ as a quasi-local measure of quantum entanglement in relativistic field theories, defined through second derivatives of the entanglement entropy with respect to infinitesimal deformations of the entangling region. It demonstrates that strong subadditivity enforces $\hat{n} \ge 0$ in $d=2$ and $d=3$ via explicit differential forms, and shows that in holography this positivity translates into an integrated null energy condition along the bulk extremal surface. In the AdS$_3$/CFT$_2$ setup, the authors derive explicit relations connecting $\hat{n}_{\pm}$ to bulk energy via $\hat{n}_{\pm} = \frac{1}{4\,G_N\,|u_{\delta}v\delta|} \int E_{\mu\nu} N_{(\pm)}^{\mu} N_{(\pm)}^{\nu}$ (or equivalently to $T_{\mu\nu}$), which guarantees non-negativity. The work thus strengthens the link between quantum information structure (SSA) and gravitational dynamics, proposing a gravitational second-law–like interpretation for entanglement density and outlining potential nonlinear, higher-dimensional extensions and connections to fundamental theorems in quantum field theory.

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.

Figures (3)

• Figure 1: Illustration of the generic variations $\delta_1 {\cal A}$ and $\delta_2{\cal A}$ which are used to define the entanglement density \ref{['eq:ndef1']}.
• Figure 2: Illustration of the set-up following Casini:2004bw in $d=2$. We choose ${\mathscr C}^+$ to be the origin and the region ${\cal A}$ lies on the time-slice with coordinate $\frac{1}{2}\, t$. We assume $t<0$ and $\epsilon \leq 0$.
• Figure 3: Illustration of the set-up following Casini:2004bw in $d\geq3$ with the same conventions as in Fig. \ref{['fig:2dSSA2']}. The regions $X_i$ in $d=3$ are obtained by considering the future light-cone from points distributed on the (dotted) circle, while their iterated intersections are obtained by considering the future light-cone from points on the (dashed) line-segment.