The holographic map as a conditional expectation Article Swipe
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.
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Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2008.04810
- https://arxiv.org/pdf/2008.04810
- OA Status
- green
- Cited By
- 21
- References
- 56
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- 10
- OpenAlex ID
- https://openalex.org/W3048584138
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W3048584138Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2008.04810Digital Object Identifier
- Title
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The holographic map as a conditional expectationWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2020Year of publication
- Publication date
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2020-08-11Full publication date if available
- Authors
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Thomas FaulknerList of authors in order
- Landing page
-
https://arxiv.org/abs/2008.04810Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2008.04810Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2008.04810Direct OA link when available
- Concepts
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Holography, Computer science, Artificial intelligence, Optics, PhysicsTop concepts (fields/topics) attached by OpenAlex
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21Total citation count in OpenAlex
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2024: 7, 2023: 7, 2022: 2, 2021: 4, 2020: 1Per-year citation counts (last 5 years)
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56Number of works referenced by this work
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.superselection | 144 |
| abstract_inverted_index.dual-additivity, | 172 |
| abstract_inverted_index.reconstructable. | 205 |
| abstract_inverted_index.boundary/physical | 37 |
| cited_by_percentile_year | |
| corresponding_author_ids | https://openalex.org/A5109902567 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 1 |
| corresponding_institution_ids | https://openalex.org/I157725225 |
| sustainable_development_goals[0].id | https://metadata.un.org/sdg/16 |
| sustainable_development_goals[0].score | 0.5400000214576721 |
| sustainable_development_goals[0].display_name | Peace, Justice and strong institutions |
| citation_normalized_percentile |