Continuous-variable entropic uncertainty relations

Hertz, Anaelle; Cerf, Nicolas J.

doi:10.1088/1751-8121/ab03f3

Cited by 63 publications

(58 citation statements)

References 77 publications

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“…differs from its Dirichlet counterpart, Equation (19), by the presence of the zero energy state, E N 1 = 0, whose position wave function is just a constant, as follows from Equation (38a). Expressions for the densities [51] ρ N n x ð Þ = 1=a, n = 1 2 a cos 2 n− 1 ð Þπ a…”

Section: Neumann Wellmentioning

confidence: 99%

“…Let us note also that in the presence of the magnetic fields the orbital that at α = 1/2 saturates the Rényi and Tsallis inequalities (Equations (13) and (16), respectively) is not necessarily the lowest energy level, as was shown for the 2D ring. [10] A study of the entropic uncertainty relations [16][17][18][19][20] is an essential endeavor both from a fundamental point of view with respect to their role in quantum foundations and from their miscellaneous applications, first of all, in information theory.…”

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confidence: 99%

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Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Olendski

2020

Int J of Quantum Chemistry

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A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the Rényi R(α) as well as Tsallis T(α) entropies between these two geometries. It is shown, in particular, that for either BC the dependences of the Rényi position components on the parameter α are the same for all orbitals but the lowest Neumann one, for which the corresponding functional R is not influenced by the variation of α. Lower limit α TH of the semi-infinite range of the dimensionless Rényi/Tsallis coefficient where momentum entropies exist crucially depends on the position BC and is equal to 1/4 for the Dirichlet requirement and 1/2 for the Neumann one. At α approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds α TH of the two BCs causes different behavior of the Rényi uncertainty relations as functions of α. For both configurations, the lowest energy level at α = 1/2 does saturate either type of the entropic inequality, thus confirming an earlier surmise about it. It is also conjectured that the threshold α TH of ½ is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view. K E Y W O R D S Dirichlet boundary condition, Neumann boundary condition, quantum well, Rényi entropy, Tsallis entropy

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Section: Neumann Wellmentioning

confidence: 99%

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confidence: 99%

Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Olendski

2020

Int J of Quantum Chemistry

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“…Entropic uncertainty relations based on the Shannon entropies have been formulated as

S_{t} = S_{r} + S_{p} \geq D ((), 1 + \ln π),

where D is the dimension of the system, in our particular (atomic) case D = 3. S r and S p are the respective Shannon entropies in position and in momentum space, defined as

\begin{matrix} lefttrue \\ S_{r} = - \int ρ (boldr) \ln [ρ (boldr)] d r, \\ S_{p} = - \int π (boldp) \ln [π (boldp)] d p, \end{matrix}

ρ ( r ) and π ( p ) are the position and momentum space densities that are usually normalized to unity.…”

Section: Introductionmentioning

confidence: 99%

“…S r and S p are the respective Shannon entropies in position and in momentum space, defined as

\begin{matrix} lefttrue \\ S_{r} = - \int ρ (boldr) \ln [ρ (boldr)] d r, \\ S_{p} = - \int π (boldp) \ln [π (boldp)] d p, \end{matrix}

ρ ( r ) and π ( p ) are the position and momentum space densities that are usually normalized to unity. There are advantages to the use of Equation over the usual one in terms of standard deviations, to express the uncertainty relation . The physical dimensions of the position and momentum densities and their entropies have also been discussed in detail by others .…”

Section: Introductionmentioning

confidence: 99%

Shannon‐information entropy sum in the confined hydrogenic atom

Salazar

Laguna

Prasad

et al. 2020

Int J of Quantum Chemistry

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The appearance of critical points in the Shannon entropy sum as a function of confinement radius, in ground and excited state confined hydrogenic systems, is discussed. We illustrate that the Coulomb potential in tandem with the hard sphere confinement are responsible for these points. The positions of these points are observed to vary with the intensity of the potential. The effects of the Coulomb potential on the system are further probed, by examining the differences between the densities of the confined atom and those of the particle confined in a spherical box, for the same confinement radius. These differences are quantified by using Kullback-Leibler and cumulative residual Kullback-Leibler distance measures from information theory. These measures detect that the effects of the Coulomb potential are squeezed out of the system as the confinement radius decreases. That is, the confined atom densities resemble the particle in a box ones, for smaller confinement radii. Furthermore, the critical points in the entropy sum lie in the same regions where there are changes in the distance measures, as the atom behaves more particle in a spherical box-like. The analysis is further complemented by examination of the derivative of the entropy sum with respect to confinement radius. This study illustrates the inhomogeneity in the magnitudes of the derivatives of the entropy sum components and their dependence on the Coulomb potential. A link between the derivative and the entropic force is also illustrated and discussed. Similar behaviors are observed when the virial ratio is compared to the entropic power one, as a function of confinement radius. K E Y W O R D S confined hydrogen atom, Kullback-Leibler distance measures, quantum uncertainties, Shannon entropy sum

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“…There is no general consensus concerning a proper formulation of the uncertainty principle [6]. Entropic functions provide a powerful tool to characterize uncertainties in quantum measurements [7][8][9][10]. Other approaches to express uncertainties in quantum measurements are currently the subject of interest.…”

Section: Introductionmentioning

confidence: 99%

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

Rastegin

2019

Annalen der Physik

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Reformulation of Heisenberg's uncertainty principle in application to energy and time is a powerful heuristic principle. In a qualitative form, this statement plays the important role in foundations of quantum theory and statistical physics. A typical meaning of energy-time uncertainties is as follows. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is also well known that the case of energy and time principally differs from more familiar examples of two non-commuting observables. Since quantum theory was originating, many approaches to energy-time uncertainties have been proposed. Entropic approach to formulating the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, we obtain uncertainty relations of the "energy-time" type in terms of the Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state-dependent and state-independent formulations are of interest. Some of the derived stateindependent bounds are similar to the results obtained within a more general approach on the base of sandwiched relative entropies. In this regard, our relations provide an alternative viewpoint on such bounds. The developed approach also allows us to address the case of detection inefficiencies. It is worth for several reasons including possible information-processing applications.

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Continuous-variable entropic uncertainty relations

Cited by 63 publications

References 77 publications

Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Shannon‐information entropy sum in the confined hydrogenic atom

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

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