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
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/129670
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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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http://sedici.unlp.edu.ar/handle/10915/129670 |
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eng |
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