Kronecker states: a powerful source of multipartite maximally entangled states in quantum information

In quantum information theory, maximally entangled states are essential for well-known protocols like quantum teleportation or quantum key distribution. While many of these protocols focus on bipartite entanglement, other applications, such as quantum error correction or multiparty quantum secret sh...

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Autores:
González Olaya, Walther Leónardo
Tipo de recurso:
Doctoral thesis
Fecha de publicación:
2024
Institución:
Universidad de los Andes
Repositorio:
Séneca: repositorio Uniandes
Idioma:
eng
OAI Identifier:
oai:repositorio.uniandes.edu.co:1992/73974
Acceso en línea:
https://hdl.handle.net/1992/73974
Palabra clave:
Entanglement
Quantum information
Multipartite entanglement
Kronecker states
Física
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openAccess
License
Attribution 4.0 International
id UNIANDES2_aea98cd99777825a5a4d57d75cdc9a67
oai_identifier_str oai:repositorio.uniandes.edu.co:1992/73974
network_acronym_str UNIANDES2
network_name_str Séneca: repositorio Uniandes
repository_id_str
dc.title.eng.fl_str_mv Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
title Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
spellingShingle Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
Entanglement
Quantum information
Multipartite entanglement
Kronecker states
Física
title_short Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
title_full Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
title_fullStr Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
title_full_unstemmed Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
title_sort Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
dc.creator.fl_str_mv González Olaya, Walther Leónardo
dc.contributor.advisor.none.fl_str_mv Botero Mejía, Alonso
dc.contributor.author.none.fl_str_mv González Olaya, Walther Leónardo
dc.contributor.jury.none.fl_str_mv Bagan Capella, Emili
Jiménez Rincón, José Julián
dc.contributor.researchgroup.none.fl_str_mv Facultad de Ciencias::Óptica Cuántica Experimental
dc.subject.keyword.none.fl_str_mv Entanglement
Quantum information
Multipartite entanglement
Kronecker states
topic Entanglement
Quantum information
Multipartite entanglement
Kronecker states
Física
dc.subject.themes.spa.fl_str_mv Física
description In quantum information theory, maximally entangled states are essential for well-known protocols like quantum teleportation or quantum key distribution. While many of these protocols focus on bipartite entanglement, other applications, such as quantum error correction or multiparty quantum secret sharing, are based on multipartite entanglement, precisely, on the so-called locally maximally entangled (LME) multipartite states, where each part is maximally entangled with their complement. Such LME states appear naturally in the invariant subspaces of tensor products of irreducible representations of the symmetric group Sn, which we term Kronecker subspaces, given that their dimensions are the so-called Kronecker coefficients. A Kronecker subspace is a vector space of LME multipartite states that we call Kronecker states, which entangle Hilbert spaces of large dimensions. Although such states can in principle be obtained from the Clebsch-Gordan coefficients of the symmetric group, the known methods to compute these coefficients tend to be inefficient even for small values of n. An alternative quantum-information-based approach is inspired by entanglement concentration protocols, where Kronecker subspaces appear naturally in the isotypic decomposition of tensor products of copies of multipartite entangled states. In this context, closed expressions have been obtained for a limited class of Kronecker states, associated with states in the so-called multiqubit Wclass. Our aim in this thesis is to extend this approach to build bases for Kronecker subspaces associated with any multiqubit system. For developing our method we first propose a graphical construction that we call “W-state Stitching”, where multiqubit entangled states are obtained as tensor networks built from W states. Analyzing the isotypic decomposition of copies of the graph state, an analogous set of graph Kronecker states, made from W-Kronecker states, can be obtained. In particular, the graph states of generic multiqubit states can generate any Kronecker subspace completely. Using this method, we show how to build any Kronecker subspace corresponding to systems of three and four qubits. Independently of the Kronecker state construction, the W-stitching technique has proven to be a powerful method for multiqubit entanglement classification. We hope the results of this work motivate the study of applications of Kronecker states in quantum information, and serve as a starting point for a resource theory of multipartite entanglement, with bipartite states and tripartite W states as building blocks, where the asymptotic analysis is based on Kronecker states.
publishDate 2024
dc.date.accessioned.none.fl_str_mv 2024-02-14T13:30:45Z
dc.date.available.none.fl_str_mv 2024-02-14T13:30:45Z
dc.date.issued.none.fl_str_mv 2024-01-31
dc.type.none.fl_str_mv Trabajo de grado - Doctorado
dc.type.driver.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
dc.type.version.none.fl_str_mv info:eu-repo/semantics/acceptedVersion
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dc.type.content.none.fl_str_mv Text
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dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/1992/73974
dc.identifier.doi.none.fl_str_mv 10.57784/1992/73974
dc.identifier.instname.none.fl_str_mv instname:Universidad de los Andes
dc.identifier.reponame.none.fl_str_mv reponame:Repositorio Institucional Séneca
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url https://hdl.handle.net/1992/73974
identifier_str_mv 10.57784/1992/73974
instname:Universidad de los Andes
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dc.language.iso.none.fl_str_mv eng
language eng
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spelling Botero Mejía, Alonsovirtual::420-1González Olaya, Walther LeónardoBagan Capella, EmiliJiménez Rincón, José JuliánFacultad de Ciencias::Óptica Cuántica Experimental2024-02-14T13:30:45Z2024-02-14T13:30:45Z2024-01-31https://hdl.handle.net/1992/7397410.57784/1992/73974instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/In quantum information theory, maximally entangled states are essential for well-known protocols like quantum teleportation or quantum key distribution. While many of these protocols focus on bipartite entanglement, other applications, such as quantum error correction or multiparty quantum secret sharing, are based on multipartite entanglement, precisely, on the so-called locally maximally entangled (LME) multipartite states, where each part is maximally entangled with their complement. Such LME states appear naturally in the invariant subspaces of tensor products of irreducible representations of the symmetric group Sn, which we term Kronecker subspaces, given that their dimensions are the so-called Kronecker coefficients. A Kronecker subspace is a vector space of LME multipartite states that we call Kronecker states, which entangle Hilbert spaces of large dimensions. Although such states can in principle be obtained from the Clebsch-Gordan coefficients of the symmetric group, the known methods to compute these coefficients tend to be inefficient even for small values of n. An alternative quantum-information-based approach is inspired by entanglement concentration protocols, where Kronecker subspaces appear naturally in the isotypic decomposition of tensor products of copies of multipartite entangled states. In this context, closed expressions have been obtained for a limited class of Kronecker states, associated with states in the so-called multiqubit Wclass. Our aim in this thesis is to extend this approach to build bases for Kronecker subspaces associated with any multiqubit system. For developing our method we first propose a graphical construction that we call “W-state Stitching”, where multiqubit entangled states are obtained as tensor networks built from W states. Analyzing the isotypic decomposition of copies of the graph state, an analogous set of graph Kronecker states, made from W-Kronecker states, can be obtained. In particular, the graph states of generic multiqubit states can generate any Kronecker subspace completely. Using this method, we show how to build any Kronecker subspace corresponding to systems of three and four qubits. Independently of the Kronecker state construction, the W-stitching technique has proven to be a powerful method for multiqubit entanglement classification. We hope the results of this work motivate the study of applications of Kronecker states in quantum information, and serve as a starting point for a resource theory of multipartite entanglement, with bipartite states and tripartite W states as building blocks, where the asymptotic analysis is based on Kronecker states.Doctor en Ciencias - FísicaDoctoradoEnredamiento multipartito216 páginasapplication/pdfengUniversidad de los AndesDoctorado en Ciencias - FísicaFacultad de CienciasDepartamento de FísicaAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Kronecker states: a powerful source of multipartite maximally entangled states in quantum informationTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttps://purl.org/redcol/resource_type/TDEntanglementQuantum informationMultipartite entanglementKronecker statesFísica[1] Charles H. 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