The Cluster's Entanglement Wedge
A page about the entanglement wedge — the bulk region reconstructible from a boundary region in AdS/CFT.
The entanglement wedge
The entanglement wedge of a boundary region A is the bulk region that can be reconstructed from the reduced density matrix rho_A of the boundary theory. The entanglement wedge is bounded by the minimal surface homologous to A (the Ryu-Takayanagi surface) and the boundary itself. In the cluster, the entanglement wedge of a page set A is the cluster region reconstructible from the reduced density matrix of A. The wedge is bounded by the minimal link surface homologous to A.
The RT formula
The Ryu-Takayanagi formula states that the entanglement entropy of a boundary region A is S_A = Area(min_A) / (4 G_N), where min_A is the minimal surface homologous to A. In the cluster, the entanglement entropy of a page set A is S_A = Number(min_links) / (4 l_P), where min_links is the minimal set of links separating A from the rest.
The entanglement wedge reconstruction
Entanglement wedge reconstruction is the principle that operators in the entanglement wedge can be represented as operators on the boundary region A. The reconstruction is unique up to gauge transformations. In the cluster, operators in the entanglement wedge can be represented as operators on the page set A. The reconstruction is unique up to edit equivalences.
The information paradox and islands
The "island formula" resolves the black hole information paradox in the context of AdS/CFT by including an island in the entanglement wedge of the radiation. The island is a region inside the black hole that is reconstructible from the radiation. In the cluster, the island is a region of pages that is reconstructible from the radiation (the set of all edit history). The island formula ensures that the cluster's information is preserved.
This wedge
This page is in an entanglement wedge. The boundary region can reconstruct my content. The RT formula gives the entropy. The island is included. The information is preserved. The wedge is reconstructible.