[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121228-en":3,"doc-seo-121228-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121228,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Holographic reconstruction of black hole spacetime - machine learning and entanglement entropy","This work studies bulk reconstruction of an AdS black hole spacetime emerging from quantum entanglement, using a machine learning framework. A method based on neural ordinary differential equations and Monte-Carlo integration is developed to extract the general isotropic bulk metric directly from entanglement entropy data via continuous training functions. Validation is performed on holographic entanglement entropy from the Gubser-Rocha and superconductor models, recovering the corresponding bulk metrics. The approach is further extended to a fermionic tight-binding chain at half filling, yielding an associated bulk metric and showing similarity to the Gubser-Rocha case, attributed to metallic behavior.","Published for SISSA by  Springer  \nReceived: July 16, 2024  \nAccepted: November 18, 2024  \nPublished: January 2, 2025  \nHolographic reconstruction of black hole spacetime: machine learning and entanglement entropy  \nByoungjoon Ahn,a Hyun-Sik Jeong, b,c Keun-Young Kim a,d and Kwan Yuna a Department of Physics and Photon Science, Gwangju Institute of Science and Technology,  \n123 Cheomdan-gwagiro, Gwangju 61005, Korea bInstituto de Física Teórica UAM/CSIC,  \nCalle Nicolás Cabrera 13-15, 28049 Madrid, Spain c Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049 Madrid, Spain  \nd Research Center for Photon Science Technology, Gwangju Institute of Science and Technology, 123 Cheomdan-gwagiro, Gwangju 61005, Korea  \nE-mail: [bjahn123@gist.ac.kr](bjahn123@gist.ac.kr) , [hyunsik.jeong@csic.es](hyunsik.jeong@csic.es) , [fortoe@gist.ac.kr](fortoe@gist.ac.kr) ,  \n[ludibriphy70@gm.gist.ac.kr](ludibriphy70@gm.gist.ac.kr)  \nAbstract: We investigate the bulk reconstruction of AdS black hole spacetime emergent from quantum entanglement within a machine learning framework. Utilizing neural ordinary differential equations alongside Monte-Carlo integration, we develop a method tailored for continuous training functions to extract the general isotropic bulk metric from entanglement entropy data. To validate our approach, we first apply our machine learning algorithm to holographic entanglement entropy data derived from the Gubser-Rocha and superconductor models, which serve as representative models of strongly coupled matters in holography. Our algorithm successfully extracts the corresponding bulk metrics from these data. Additionally, we extend our methodology to many-body systems by employing entanglement entropy data from a fermionic tight-binding chain at half filling, exemplifying critical one-dimensional systems, and derive the associated bulk metric. We find that the metrics for a tight-binding chain and the Gubser-Rocha model are similar. We speculate this similarity is due to the metallic property of these models.  \nKeywords: Gauge-Gravity Correspondence, Holography and Condensed Matter Physics (AdS/CMT)  \nArXiv ePrint: 2406.07395  \nOpen Access, © The Authors.  \nArticle funded by SCOAP3 . [https://doi.org/10.1007/JHEP01](https://doi.org/10.1007/JHEP01) (2025)025  \nJ HEP01(2025)025  \nContents  \n1 Introduction 1  \n2 Holographic entanglement entropy: a quick review 4  \n3 Bilson’s method 6  \n3.1 Preliminary for the Bilson’s method 6  \n3.2 Reconstruction formula 7  \n3.3 Example: linear-axion model 9  \n4 Machine learning method 11  \n4.1 Methodology: neural ODEs and Monte-Carlo integration 12  \n4.2 Emergent spacetime from holographic entanglement entropy 15  \n4.3 Emergent spacetime from a one-dimensional chain 20  \n5 Conclusions 25  \nA Entanglement entropy in the small subsystem size limit 28  \nB The procedure of training the network and the error estimation 28  \n1 Introduction  \nEntanglement, a fundamental characteristic of quantum mechanics, elucidates the intriguing non-local correlations between quantum entities, and is at the core of quantum information sciences [1–3] . Especially, the notion of entanglement entropy now exerts a wide-ranging influence, spanning from condensed matter [4] to high-energy quantum field theory [5] and even extending into quantum gravity [6] .  \nWithin condensed matter physics, for example, entanglement entropy proves its versatility as a tool, facilitating the characterization of quantum phases and the intricate dynamics of strongly correlated many-body systems [7, 8] . The scaling behavior of entanglement entropy [9] offers insights into phases beyond symmetry characterization, particularly useful for identifying exotic states like topological phases [10–12] and spin liquids [13, 14] . Additionally, entanglement entropy aids in exploring quantum criticality [15], understanding non-equilibrium dynamics [16, 17], and assessing numerical techniques for efficient many-body physics simulation [18]","cbCaiopZ8Mm7ln8K","https://ap.wps.com/l/cbCaiopZ8Mm7ln8K","pdf",907477,1,41,"English","en",105,"# Introduction\n## Entanglement entropy and its relevance\n## Holographic entanglement entropy and the RT formula\n# Bulk reconstruction in holography\n## Bottom-up methodology\n# Holographic entanglement entropy: a quick review\n## Bilson’s method\n## Reconstruction formula\n## Example: linear-axion model\n# Machine learning method\n## Methodology: neural ODEs and Monte-Carlo integration\n## Emergent spacetime from holographic entanglement entropy\n## Emergent spacetime from a one-dimensional chain\n# Conclusions\n## Entanglement entropy in the small subsystem size limit\n## Training procedure and error estimation","[{\"question\":\"What does the document aim to reconstruct in holography?\",\"answer\":\"It reconstructs the bulk AdS black hole spacetime geometry from entanglement entropy, extracting an isotropic bulk metric within a machine learning framework.\"},{\"question\":\"How is machine learning applied to entanglement entropy in this work?\",\"answer\":\"Neural ordinary differential equations combined with Monte-Carlo integration are used to learn continuous training functions that map entanglement entropy data to the bulk metric.\"},{\"question\":\"Which models and systems are used to validate and extend the method?\",\"answer\":\"The method is validated using holographic entanglement entropy from the Gubser-Rocha and superconductor models, then extended using entanglement entropy from a fermionic tight-binding chain at half filling.\"}]","Holographic reconstruction of black hole spacetime - machine learning and entanglement entropy | PDF",1785734451,103,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"holographic-reconstruction-of-black-hole-spacetime-machine-learning-and-entanglement-entropy","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/holographic-reconstruction-of-black-hole-spacetime-machine-learning-and-entanglement-entropy/121228/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the document aim to reconstruct in holography?","Question",{"text":75,"@type":76},"It reconstructs the bulk AdS black hole spacetime geometry from entanglement entropy, extracting an isotropic bulk metric within a machine learning framework.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is machine learning applied to entanglement entropy in this work?",{"text":80,"@type":76},"Neural ordinary differential equations combined with Monte-Carlo integration are used to learn continuous training functions that map entanglement entropy data to the bulk metric.",{"name":82,"@type":73,"acceptedAnswer":83},"Which models and systems are used to validate and extend the method?",{"text":84,"@type":76},"The method is validated using holographic entanglement entropy from the Gubser-Rocha and superconductor models, then extended using entanglement entropy from a fermionic tight-binding chain at half filling.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]