The Ryu-Takayanagi Formula from Quantum Error Correction: An Algebraic Treatment of the Boundary CFT

In recent years, an interpretation of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence in the language of quantum error correction has been developed. This language shines light on several puzzling features of the correspondence and has therefore played a crucial role in advancing our understanding of AdS/CFT. In particular, in a recent work by Daniel Harlow, it is shown that sub-algebra quantum erasure-correcting codes with complementary recovery naturally give rise to a version of quantum-corrected Ryu-Takayanagi formula that captures the physics of AdS/CFT. In his interpretation, Harlow considers a Von Neumann algebra on the bulk, but assumes a simple tensor product structure on the boundary Hilbert space. In this work, we developed the mathematical framework for extending Harlow's results to the more physical case where a Von Neumann algebra is also given on the boundary CFT. We showed that the resulting code more accurately captures the properties of AdS/CFT.