(h, Φ)-entropy differential metric

Menéndez, M.; Morales, Domingo; Pardo, Leandro; Salicrú, Miquel

doi:10.1023/a:1022214326758

Cited by 21 publications

(15 citation statements)

References 15 publications

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“…We are interested in quantitative formulations of the uncertainty principle, particularly through the use of information-theoretic quantities. More precisely, as measure of ignorance or of lack of information we employ Salicrú et al (h, φ)-entropies [51,52],…”

Section: Generalized Entropiesmentioning

confidence: 99%

General entropy-like uncertainty relations in finite dimensions

Zozor¹,

Bosyk²,

Portesi³

2014

J. Phys. A: Math. Theor.

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We revisit entropic formulations of the uncertainty principle 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 (c A , c B , c A,B ) with c A (resp. c B ) being the overlap between the elements of the POVM A (resp. B) and c A,B the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] 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 uncertainty principle, 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 c A,B > 1 √ 2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when c A,B ≤ 1 2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

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Section: Generalized Entropiesmentioning

confidence: 99%

General entropy-like uncertainty relations in finite dimensions

Zozor¹,

Bosyk²,

Portesi³

2014

J. Phys. A: Math. Theor.

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“…It is to be noted that Rényi entropy cannot be obtained as a special case of either of the two generalized entropies discussed above. But it can be obtained as a particular case of the (h; /)-entropy defined by Menéndez et al [14]. For different uses of Rényi entropy, one may refer to Nadarajah and Zografos [16].…”

Section: Notations Definitions and Preliminariesmentioning

confidence: 99%

Rényi information measure for a used item

Nanda

Maiti

2007

Information Sciences

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“…In [9], the problem is studied in general populations in which the parameter is estimated by a consistent and asymptotically normal estimator of the unknown parameter. In [10], a metric based on the Hessian of the (h, ¢)-entropy, as well as the geodesic distances induced by the (h, ¢)-entropy for a particular selection of h and ¢, and some probability distributions are studied.…”

Section: I=1mentioning

confidence: 99%