[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-45402-en":3,"doc-seo-45402-105":29,"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":20,"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":13,"seo_description":14,"update_tm":27,"read_time":28},45402,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","ER=EPR, Entanglement Topology and Tensor Networks","This paper examines the ER = EPR hypothesis by constructing a new topological space that fuses a given background space with a quantum tensor network. The method ties quantum entanglement directly to topological connectivity, so the resulting augmented space inherits connectivity properties determined by entanglement structure. The discussion interprets wormhole-like connections through a topological analogy, connecting event-horizon roles with augmentation points and network neighborhoods.","arXiv :2203 .09797v2 [ quant-ph] 23 Mar 2022  \nER=EPR, Entanglement Topology and Tensor Networks  \nLouis H. Kau􀀋mana  \na Department of Mathematics, Statistics and Computer Science (m/c 249), 851 South Morgan  \nStreet, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA  \nABSTRACT  \nThis paper discusses ER = EPR. Given a background space and a quantum tensor network, we describe how to construct a new topological space, that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.  \nKeywords: quantum entanglement,topological space, topological connectivity, linking, augmented space, tensor network, wormhole, Heyting algebra  \n1. INTRODUCTION  \nWe discuss the relationship of space, spacetime and quantum entanglement in the context of the hypothesis of Susskind and Malcedena.15–17 Their ER = EPR hypothesis is based on the suggestion that connectivity in spacetime is equivalent to quantum entanglement. Susskind asserts that quantum entanglement of distant black holes is equivalent to the existence of an Einstein-Rosen bridge connecting them. If this hypothesis is true, then there is indeed a topological underpinning for quantum entanglement. Here we make foundational comments on the ER = EPR hypothesis. In the discussion below we examine entanglement and teleportation in relation to the construction of a space that is augmented by quantum states. Since an entangled state such as |􀀎i = 2 (|01i+|10i) can be formulated without any background space, we point out that it is possible graphically to form a new space from the given space or spactime S of the physics by attaching a corresponding quantum network to S. The new space S′ has connectivity related to the entanglement. This construction can then be considered as a precursor to the spacetime with an Einstein-Rosen bridge connecting the sites of the entangled particles. This analogy is illustrated in Figure 1 where we show on the left the bare bones of a line space augmented by the tensor network for an entangled pair and on the right a schema for a wormhole connection of two entangled black holes. The event horizon of the wormhole plays the same topological role as the extra point E in the augmentation. Any neighborhood of E must contain neighborhoods of the ends of the network. Any neighborhood of the event horizon is a connection of the two black holes. We see that underlying the properties of an event horizon are the simplest possiblities for e􀀋ecting a topological connection.  \nIn the paper we review the idea of tensor networks and their relationship with topology in Section1 . We consider network topologies and the augmentation referred to above in Section 2 . We discuss topological entanglement and quantum entanglement in Section 3 . We show how Heyting algebra structures are deeply related to topological connectivity and thus to ER = EPR in Section 5 . Section 6 is a summary of the ideas in the paper. Ideas in this paper are discussed from other viewpoints in our other papers,10, 11 and the present paper is intended as a source for further work.  \nFurther author information: L.H.K. E-mail: [loukau@gmail.com](loukau@gmail.com)  \nFigure 1 . Augmented Space and ER-Bridge  \n2. TENSOR NETWORKS  \nA tensor network is a graph G with tensors or matrices associated with each of its nodes and and index set I that can be used to label the edges of the graph. A contraction of the tensor net G is obtained by assigning 􀀌xed indices to all external edges of G and then summing over all possible assignments of indices to internal edges the products of the corresponding matrix entries for the nodes of the graph. See Figure 2 for illustrations of abstract tensor networks. It is often useful to choose a form for the nodes of the graph that is mnemonic for particular uses. For example, in Figure 3 will illustrate a tensor network that is associated with a knot diag","cbCailTFD1sUtAyM","https://ap.wps.com/l/cbCailTFD1sUtAyM","pdf",598780,1,14,"English","en",105,"# Introduction\n## ER = EPR and augmented space construction\n# Tensor Networks\n## Definition and contraction of tensor networks\n## Graphs, categories, and functors\n## Joining tensor networks with topological spaces\n# Topological entanglement and quantum entanglement\n## Heyting algebra and topological connectivity\n# Summary","[{\"question\":\"What does the paper aim to demonstrate about ER = EPR?\",\"answer\":\"It describes how to construct an augmented topological space by welding a quantum tensor network to a background space, embodying the connection between quantum entanglement and topological connectivity.\"},{\"question\":\"How is the “augmented space” formed in the construction?\",\"answer\":\"Starting from a background space and a quantum tensor network, the paper constructs a new topological space that welds the network and the background together so the space’s connectivity reflects entanglement.\"},{\"question\":\"What role do tensor networks play in relating topology to quantum phenomena?\",\"answer\":\"Tensor networks are used as a graphical/computational structure whose connectivity is tied to entanglement; the paper discusses how joining tensor networks with topological spaces can model a quantum world through topology.\"}]",1783459016,35,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"erepr-entanglement-topology-and-tensor-networks","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/erepr-entanglement-topology-and-tensor-networks/45402/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-10","2026-07-07",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the paper aim to demonstrate about ER = EPR?","Question",{"text":75,"@type":76},"It describes how to construct an augmented topological space by welding a quantum tensor network to a background space, embodying the connection between quantum entanglement and topological connectivity.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the “augmented space” formed in the construction?",{"text":80,"@type":76},"Starting from a background space and a quantum tensor network, the paper constructs a new topological space that welds the network and the background together so the space’s connectivity reflects entanglement.",{"name":82,"@type":73,"acceptedAnswer":83},"What role do tensor networks play in relating topology to quantum phenomena?",{"text":84,"@type":76},"Tensor networks are used as a graphical/computational structure whose connectivity is tied to entanglement; 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