Adding the algebraic Ryu-Takayanagi formula to the algebraic reconstruction theorem
Mingshuai Xu, Haocheng Zhong
TL;DR
This work extends the holographic reconstruction program to include the algebraic RT formula within the algebraic quantum field theory (AQFT) framework by focusing on type I/II factors. It develops and employs Araki's relative entropy and the algebraic von Neumann entropy to define and compare bulk and boundary entropies in a setting where type III obstructions are avoided. The main result is an algebraic reconstruction theorem for type I/II factors, establishing equivalences between bulk operator reconstruction, preserved relative entropy, and an algebraic RT relation S(Ψ;A_phys) = S(˜Ψ;A_code) plus the boundary-area-like contribution, under modular-splitting assumptions. The discussion highlights how the area term may emerge as a regulated object and addresses extensions to more general factors and holographic contexts, including large-N considerations and potential algebraic regularization schemes for RT. Overall, the paper provides a rigorous algebraic foundation for bulk reconstruction in holography beyond finite dimensions and outlines a path to a fully algebraic RT formula in the holographic correspondence.
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.