The holographic map as a conditional expectation

TL;DR

This paper models the holographic map in AdS/CFT as a quantum error correcting code with exact complementary recovery, revealing how local conditional expectations determine the map and relate to entanglement entropy and phase transitions.

Contribution

It introduces a framework connecting holographic maps to quantum error correction, using conditional expectations to explain entanglement and recovery properties, and discusses phase transitions and limitations.

Findings

Holographic map determined by local conditional expectations.

Black Hole area law arises from a sum of entropies on the relative commutant.

Dual-additivity condition leads to phase transitions between entanglement wedges.

Abstract

We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expectation can also be found via a minimization procedure, similar to the minimization involved in the RT formula. For a local net of algebras associated to connected boundary regions, we show the complementary recovery condition is equivalent to the existence of a standard net of inclusions -- an abstraction of the mathematical structure governing QFT superselection sectors given by Longo and Rehren. For a code consisting of algebras associated to two disjoint regions of the boundary theory we impose an extra condition, dubbed dual-additivity, that gives rise to phase transitions between different entanglement wedges. Dual-additive codes naturally give rise to a new split code subspace, and an entropy bound controls which subspace and associated algebra is reconstructable. We also discuss known shortcomings of exact complementary recovery as a model of holography. For example, these codes are not able to accommodate holographic violations of additive for overlapping regions. We comment on how approximate codes can fix these issues.