General entropy-like uncertainty relations in finite dimensions

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 meas...

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Detalles Bibliográficos
Autores principales: Zozor, Steeve, Bosyk, Gustavo Martín, Portesi, Mariela Adelina
Formato: Articulo
Lenguaje:Inglés
Publicado: 2014
Materias:
Física
Entropic uncertainty relation
Pure and mixed qudit states
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/129670
Aporte de:
SEDICI (UNLP) de Universidad Nacional de La Plata
id I19-R120-10915-129670
record_format dspace
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Física
Entropic uncertainty relation
Pure and mixed qudit states
spellingShingle Física
Entropic uncertainty relation
Pure and mixed qudit states
Zozor, Steeve
Bosyk, Gustavo Martín
Portesi, Mariela Adelina
General entropy-like uncertainty relations in finite dimensions
topic_facet Física
Entropic uncertainty relation
Pure and mixed qudit states
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.
format Articulo
Articulo
author Zozor, Steeve
Bosyk, Gustavo Martín
Portesi, Mariela Adelina
author_facet Zozor, Steeve
Bosyk, Gustavo Martín
Portesi, Mariela Adelina
author_sort Zozor, Steeve
title General entropy-like uncertainty relations in finite dimensions
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
publishDate 2014
url http://sedici.unlp.edu.ar/handle/10915/129670
work_keys_str_mv AT zozorsteeve generalentropylikeuncertaintyrelationsinfinitedimensions
AT bosykgustavomartin generalentropylikeuncertaintyrelationsinfinitedimensions
AT portesimarielaadelina generalentropylikeuncertaintyrelationsinfinitedimensions
bdutipo_str Repositorios
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