On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Zozor, Steeve; Vignat, Christophe

doi:10.1016/j.physa.2006.09.019

Cited by 50 publications

(69 citation statements)

References 42 publications

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“…[1,2] in the continuous-continuous, discrete-continuous (periodic) and discrete-discrete cases. In terms of the Rényi λ-entropy power, these uncertainty relations read …”

Section: Previous Results On Entropic Uncertainty Relationsmentioning

confidence: 99%

Some extensions of the uncertainty principle

Zozor

Portesi

Vignat

2008

Physica A: Statistical Mechanics and its Applications

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We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by BialynickiBirula [1] and Zozor et al. [2]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated Rényi entropies with conjugated indices. We consider here the general case where the entropic indices are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indices α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).

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“…[1,2] in the continuous-continuous, discrete-continuous (periodic) and discrete-discrete cases. In terms of the Rényi λ-entropy power, these uncertainty relations read …”

Section: Previous Results On Entropic Uncertainty Relationsmentioning

confidence: 99%

Some extensions of the uncertainty principle

Zozor

Portesi

Vignat

2008

Physica A: Statistical Mechanics and its Applications

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“…Bounds to the product of logarithmic uncertainties [12] and the sum of Shannon [13] and Rényi [14] entropies are also known. Although these relationships are usually applied in three-dimensional systems (i.e., with vectors of three components r and p), all of them are valid for arbitrary dimensionality [15,16]. Such uncertainty relations are physically relevant because of their importance not only in a theoretical quantum-mechanical framework [17][18][19] but also in the development of quantum information and computation [20,21].…”

Section: Introductionmentioning

confidence: 99%

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Angulo

2011

Phys. Rev. A

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Rigorous and universal relationships among radial expectation values of any D-dimensional quantummechanical system are obtained, using Rényi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products r a 1/a p b 1/b , widely considered in the literature, including the famous Heisenberg relationship r 2 p 2 D 2 /4. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b > 0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.

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“…It is worth noting that on the "conjugated curve" β = α * = α/(2α − 1) in the (α, β ) plane, the bound is sharp and attainable if (and only if) ρ is Gaussian [5,6]. We also showed that for β > α * , no uncertainty principle exists [6], in the sense that it is possible to find states for which the positive product of Rényi's entropy powers can be made arbitrarily small.…”

Section: Entropic Uncertainty Relations: a Brief Reviewmentioning

confidence: 99%

On moments-based Heisenberg inequalities

Zozor¹,

Portesi²,

Sánchez-Moreno³

et al. 2011

AIP Conference Proceedings

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In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: x 2 u 2 ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy under statistical moments' constraints ( x a and u b ). Our result improves a relation proposed very recently by Dehesa et al.[1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref.[1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.

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On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Cited by 50 publications

References 42 publications

Some extensions of the uncertainty principle

Some extensions of the uncertainty principle

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

On moments-based Heisenberg inequalities

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