{"work":{"id":"29d57c7b-f36c-430e-aece-e013ff1d1015","openalex_id":null,"doi":null,"arxiv_id":"1607.03901","raw_key":null,"title":"The Ryu-Takayanagi Formula from Quantum Error Correction","authors":null,"authors_text":"D","year":2016,"venue":"hep-th","abstract":"I argue that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a \"purely boundary\" interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the \"bit threads\" recently introduced by Freedman and Headrick.","external_url":"https://arxiv.org/abs/1607.03901","cited_by_count":null,"metadata_source":"pith","metadata_fetched_at":"2026-07-01T01:15:12.988324+00:00","pith_arxiv_id":"1607.03901","created_at":"2026-05-11T04:55:59.307481+00:00","updated_at":"2026-07-01T01:15:12.988324+00:00","title_quality_ok":true,"display_title":"The Ryu-Takayanagi Formula from Quantum Error Correction","render_title":"The Ryu-Takayanagi Formula from Quantum Error Correction"},"hub":{"state":{"work_id":"29d57c7b-f36c-430e-aece-e013ff1d1015","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":10,"external_cited_by_count":null,"distinct_field_count":1,"first_pith_cited_at":"2019-05-21T17:27:30+00:00","last_pith_cited_at":"2026-06-10T12:07:41+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-06-27T09:15:56.091837+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":3},{"context_role":"method","n":1}],"polarity_counts":[{"context_polarity":"background","n":3},{"context_polarity":"use_method","n":1}],"runs":{},"summary":{},"graph":{},"authors":[]}}