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The work analyzes local and pairwise entanglement for a four-site spin chain at electron counts N=4 and N=6 under different potential settings. The study is extended to L=6 and L=8, comparing N=L and N=L+2, showing sharper entanglement transitions and smaller finite-size effects with increasing size. Results connect local potential shifts and Coulomb interaction strength to electron redistribution and complex phase-diagram entanglement dynamics.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/exploring-entanglement-spectra-and-phase-diagrams-in-multi-electron-quantum-dot-chains/657083/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/exploring-entanglement-spectra-and-phase-diagrams-in-multi-electron-quantum-dot-chains/657083.png","ImageObject",300,407,{"name":92,"@type":93},"SundayMiles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What model is used to describe the quantum dot chains in this study?","Question",{"text":112,"@type":113},"The study uses the extended Hubbard model (EHM), including multiple energy levels within each quantum dot, to model semiconductor quantum dot spin chains.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which entanglement measures are analyzed?",{"text":117,"@type":113},"It analyzes local and pairwise entanglement by focusing on the entanglement spectrum derived from one-site and two-site reduced density matrices.",{"name":119,"@type":110,"acceptedAnswer":120},"How do system size and electron number affect the entanglement behavior?",{"text":121,"@type":113},"For larger systems (L=6 and L=8), comparing N=L and N=L+2 shows sharper entanglement transitions and reduced finite-size effects as the system size increases.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657083,1792973496,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Article  \nExploring Entanglement Spectra and Phase Diagrams in MultiElectron Quantum Dot Chains  \nGuanjie He and Xin Wang  \n[https://doi.org/10.3390/e27050479](https://doi.org/10.3390/e27050479)  \nArticle  \nExploring Entanglement Spectra and Phase Diagrams in Multi-Electron Quantum Dot Chains  \nGuanjie He 1 and Xin Wang 1,2,3, *  \n1 Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China; [guanjiehe2-c@my.cityu.edu.hk](guanjiehe2-c@my.cityu.edu.hk)  \n2 Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China  \n3 Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area, Shenzhen 518045, China  \n* Correspondence: [x.wang@cityu.edu.hk](x.wang@cityu.edu.hk)  \nAbstract: We investigate the entanglement properties in semiconductor quantum dot systems modeled by the extended Hubbard model, focusing on the impacts of potential energy variations and electron interactions within a four-site quantum dot spin chain. Our study explores local and pairwise entanglement across con􀀂gurations with electron counts N = 4 and N = 6, under different potential energy settings. By adjusting the potential energy in speci􀀂c dots and examining the entanglement across various interaction regimes, we identify signi􀀂cant variations in the ground states of quantum dots. We extend this analysis to larger systems with L = 6 and L = 8, comparing electron counts N = L and N = L + 2, revealing sharper entanglement transitions and reduced 􀀂nite-size effects as the system size increases. Our results show that local potential shifts and the Coulomb interaction strength lead to notable redistributions of the electron con􀀂gurationsin the quantum dot spin chain, signi􀀂cantly affecting the entanglement properties. These changes are depicted in phase diagrams that highlight entanglements’ dependencies on the interaction strengths and potential energy adjustments, illustrating complex entanglement dynamics shifts triggered by interdot interactions.  \nKeywords: Hubbard model; entanglement; quantum dot  \n1. Introduction  \nAcademic Editor: Rosario Lo Franco  \nReceived: 28 February 2025  \nRevised: 8 April 2025  \nAccepted: 23 April 2025  \nPublished: 29 April 2025  \nCitation: He, G.; Wang, X. Exploring Entanglement Spectra and Phase Diagrams in Multi-Electron Quantum Dot Chains. Entropy 2025, 27, 479 . [https://doi.org/10.3390/e27050479](https://doi.org/10.3390/e27050479)  \n[Copyright:](Copyright:) © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://creativecommons.org/](https://creativecommons.org/)[ ](https://creativecommons.org/)[licenses/by/4.0/](licenses/by/4.0/)) .  \nQuantum entanglement plays a crucial role in various 􀀂elds of quantum physics, including quantum communication and quantum information processing [1,2] . In condensed matter physics, especially in many-body quantum systems, quantum entanglement serves as a fundamental criterion for quantum phase transitions and many-body localization [3–7] . Among various systems, semiconductor quantum dots have emerged as scalable, implementable, and precisely controllable [8–16] platforms for the simulation of many-body systems of interest, particularly the Fermi–Hubbard physics [17–23] . The Fermi–Hubbard model provides a common framework describing quantum dot systems in the regime of low temperatures and strong Coulomb interactions, 􀀂nding extensive application in the physical realization of quantum information processing and entanglement creation [24–32] . Consequently, a comprehensive understanding of quantum dots from the perspective of Fermi–Hubbard physics becomes imperative.  \nHigh-􀀂delity qubit gate operations [33,34] and noise suppression schemes [35] commonly applied to conventional quantum dot systems, where each dot accommodates at most two electrons, traditionally rely on the monoto","cbCaine2WTcDNoxW","https://ap.wps.com/l/cbCaine2WTcDNoxW","pdf",4242656,26,"English","# Introduction\n## Quantum entanglement in many-body systems\n## Quantum dots and the Hubbard-model framework","[{\"question\":\"What model is used to describe the quantum dot chains in this study?\",\"answer\":\"The study uses the extended Hubbard model (EHM), including multiple energy levels within each quantum dot, to model semiconductor quantum dot spin chains.\"},{\"question\":\"Which entanglement measures are analyzed?\",\"answer\":\"It analyzes local and pairwise entanglement by focusing on the entanglement spectrum derived from one-site and two-site reduced density matrices.\"},{\"question\":\"How do system size and electron number affect the entanglement behavior?\",\"answer\":\"For larger systems (L=6 and L=8), comparing N=L and N=L+2 shows sharper entanglement transitions and reduced finite-size effects as the system size increases.\"}]","Exploring Entanglement Spectra and Phase Diagrams in Multi-Electron Quantum Dot Chains | PDF",66]