General entropy-like uncertainty relations in finite dimensions

Autores
Zozor, Steeve; Bosyk, Gustavo Martín; Portesi, Mariela Adelina
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h,ϕ ) -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (cA, cB, cA,B) with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and cA B, the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given cA B, > 1/2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when cA B, ⩽ 1/2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
Instituto de Física La Plata
Materia
Física
Entropic uncertainty relation
Pure and mixed qudit states
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/129670

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spelling General entropy-like uncertainty relations in finite dimensionsZozor, SteeveBosyk, Gustavo MartínPortesi, Mariela AdelinaFísicaEntropic uncertainty relationPure and mixed qudit statesWe revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h,ϕ ) -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (cA, cB, cA,B) with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and cA B, the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given cA B, > 1/2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when cA B, ⩽ 1/2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.Instituto de Física La Plata2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://sedici.unlp.edu.ar/handle/10915/129670enginfo:eu-repo/semantics/altIdentifier/issn/1751-8113info:eu-repo/semantics/altIdentifier/issn/1751-8121info:eu-repo/semantics/altIdentifier/arxiv/1311.5602info:eu-repo/semantics/altIdentifier/doi/10.1088/1751-8113/47/49/495302info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-09-29T11:31:17Zoai:sedici.unlp.edu.ar:10915/129670Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-09-29 11:31:17.816SEDICI (UNLP) - Universidad Nacional de La Platafalse
dc.title.none.fl_str_mv General entropy-like uncertainty relations in finite dimensions
title General entropy-like uncertainty relations in finite dimensions
spellingShingle General entropy-like uncertainty relations in finite dimensions
Zozor, Steeve
Física
Entropic uncertainty relation
Pure and mixed qudit states
title_short General entropy-like uncertainty relations in finite dimensions
title_full General entropy-like uncertainty relations in finite dimensions
title_fullStr General entropy-like uncertainty relations in finite dimensions
title_full_unstemmed General entropy-like uncertainty relations in finite dimensions
title_sort General entropy-like uncertainty relations in finite dimensions
dc.creator.none.fl_str_mv Zozor, Steeve
Bosyk, Gustavo Martín
Portesi, Mariela Adelina
author Zozor, Steeve
author_facet Zozor, Steeve
Bosyk, Gustavo Martín
Portesi, Mariela Adelina
author_role author
author2 Bosyk, Gustavo Martín
Portesi, Mariela Adelina
author2_role author
author
dc.subject.none.fl_str_mv Física
Entropic uncertainty relation
Pure and mixed qudit states
topic Física
Entropic uncertainty relation
Pure and mixed qudit states
dc.description.none.fl_txt_mv We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h,ϕ ) -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (cA, cB, cA,B) with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and cA B, the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given cA B, > 1/2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when cA B, ⩽ 1/2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
Instituto de Física La Plata
description We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h,ϕ ) -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (cA, cB, cA,B) with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and cA B, the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given cA B, > 1/2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when cA B, ⩽ 1/2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
publishDate 2014
dc.date.none.fl_str_mv 2014
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info:eu-repo/semantics/publishedVersion
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dc.identifier.none.fl_str_mv http://sedici.unlp.edu.ar/handle/10915/129670
url http://sedici.unlp.edu.ar/handle/10915/129670
dc.language.none.fl_str_mv eng
language eng
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info:eu-repo/semantics/altIdentifier/issn/1751-8121
info:eu-repo/semantics/altIdentifier/arxiv/1311.5602
info:eu-repo/semantics/altIdentifier/doi/10.1088/1751-8113/47/49/495302
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
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rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:SEDICI (UNLP)
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