Adding the algebraic Ryu-Takayanagi formula to the algebraic reconstruction theorem

TL;DR

This paper proves that the algebraic Ryu-Takayanagi formula can be included within the algebraic reconstruction theorem for type I/II factors, advancing the algebraic understanding of holographic entanglement.

Contribution

It demonstrates the inclusion of the algebraic Ryu-Takayanagi formula in the algebraic reconstruction theorem for type I/II factors, clarifying their role in holography.

Findings

Inclusion holds for type I/II factors

Supports the algebraic approach to holographic entanglement

Advances the algebraic reconstruction theorem

Abstract

A huge progress in studying holographic theories is that holography can be interpreted via the quantum error correction, which makes equal the entanglement wedge reconstruction, the Jafferis-Lewkowycz-Maldacena-Suh formula, the radial commutativity and the Ryu-Takayanagi formula. We call the equivalence the reconstruction theorem, whose infinite-dimensional generalization via algebraic language was believed to exclude the algebraic version of the Ryu-Takayanagi formula. However, recent developments regarding gravitational algebras have shown that the inclusion of the algebraic Ryu-Takayanagi formula is plausible. In this letter, we prove that such inclusion holds for the cases of type I/II factors, which are expected to describe holographic theories.