A holographic proof of the strong subadditivity of entanglement entropy

TL;DR

This paper provides a general geometric proof that strong subadditivity of entanglement entropy holds in holographic theories, using the Ryu-Takayanagi prescription. It shows that, for static bulk geometries, minimal-surface rearrangements into $m_{A\cup B}$ and $m_{A\cap B}$ preserve total area, yielding $S(A)+S(B) \ge S(A\cup B)+S(A\cap B)$, with the homology constraint ensuring validity in nontrivial topologies. The result reinforces the consistency of holography with fundamental quantum information inequalities and clarifies geometric intuition behind SSA, including implications for Araki-Lieb saturation and mutual information. It also notes a link to coplanar Wilson loop concavity and outlines directions for covariant extensions.

Abstract

When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as "strong subadditivity". For a field theory this inequality can be stated as follows: given any two regions of space $A$ and $B$, $S(A) + S(B) \ge S(A \cup B) + S(A \cap B)$. Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. In this note we give a simple geometrical proof of strong subadditivity employing this holographic prescription.