General properties of holographic entanglement entropy

TL;DR

The paper analyzes general properties of holographic entanglement entropy in the static RT framework, focusing on how EE is constrained by geometry and how a canonical bulk region map r(A) encodes boundary information. It clarifies continuity properties, introduces a precise map from boundary to bulk regions, and surveys key entropic inequalities (SSA, Araki-Lieb, MMI) along with saturation conditions and a novel reflection inequality, linking geometric configurations to density-matrix structure. The work highlights the kinematic nature of these results, their interpretation in terms of the holographic reduced density matrix ρ_A, and the implications for understanding how bulk geometry constrains boundary entanglement, with potential extensions to time-dependent cases. Overall, it provides a unified geometric framework to probe how spacetime information is organized in holographic theories.

Abstract

The Ryu-Takayanagi formula implies many general properties of entanglement entropies in holographic theories. We review the known properties, such as continuity, strong subadditivity, and monogamy of mutual information, and fill in gaps in some of the previously-published proofs. We also add a few new properties, including: properties of the map from boundary regions to bulk regions implied by the RT formula, such as monotonicity; conditions under which subadditivity-type inequalities are saturated; and an inequality concerning reflection-symmetric states. We attempt to draw lessons from these properties about the structure of the reduced density matrix in holographic theories.

Figures (8)

• Figure 1: Illustration of the various regions and boundaries defined in section \ref{['sec:setup']}. $\Sigma$ is a constant-time slice of the bulk spacetime $M$. $\dot\Sigma$ is its conformal boundary, which is a constant-time slice of the boundary spacetime $\dot M$. $H$ is the horizon of black hole in the bulk. The bulk is also bounded at the bottom by a wall of some kind. $r$ (in green) is a region of the bulk. The part of its boundary along $\dot\Sigma$ is denoted $\dot r$. The rest of its boundary, including the part along $H$ but not including the part along the wall, is denoted $\partial r$.
• Figure 2: Left: Situation excluded by Property \ref{['r(dotSigma)']}, in which there exists a minimal surface homologous to $\dot \Sigma$ with smaller area than $H$, that would therefore be $m(\dot \Sigma)$. (The proof actually excludes the existence of even a local minimal surface homologous to $\dot \Sigma$ other than $H$.) Right: Counterexample to Property \ref{['r(dotSigma)']}: The walls on the left and right bend in to create a minimal surface homologous to $\dot \Sigma$, with smaller area than $H$. Presumably such behavior for walls is unphysical.
• Figure 3: Left: Illustration of the situation excluded by Property \ref{['monotonicity']}. $A$ and $m(A)$ are in blue, $B$ is in green, and $m(AB)$ is in red. Right: Surfaces $m'(A):=\partial r'(A)$ (blue) and $m'(AB):=\partial r'(AB)$ (red) used in the proof of Property \ref{['monotonicity']}, for the surfaces shown on the left.
• Figure 4: Left: Illustration of the situation excluded by Property \ref{['nonoverlapping']}. $A$ and $m(A)$ are in blue, while $B$ and $m(B)$ are in green. Right: Surfaces $m'(A):=\partial r'(A)$ (blue) and $m'(B):=\partial r'(B)$ (green) used in the proof of Property \ref{['nonoverlapping']}, for the surfaces shown on the left.
• Figure 5: Left: Illustration of a case where the subadditivity inequality is saturated. Right: Example of two thermodynamic systems coupled via a macroscopic variable: Two species of gas in a box separated by a movable piston. $A,B$ represent the states of the two gases respectively. As discussed in the main text, such a system is closely analogous to the state of regions in a holographic field theory such that subadditivity is saturated, as on the left side.