Uncertainty Relations for Coarse–Grained Measurements: An Overview

Toscano, Fabricio; Tasca, D. S.; Rudnicki, Łukasz; Walborn, S. P.

doi:10.3390/e20060454

Cited by 23 publications

(22 citation statements)

References 199 publications

(436 reference statements)

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“…where x = (x 1 , · · · ,x n ) and p = (p 1 , · · · ,p n ) are two vectors of pairwise canonicallyconjugate quadratures and h(·) is the differential entropy defined in Eq. (27). The covariance matrix γ is defined as…”

Section: Tight Entropic Uncertainty Relation For Canonically Conjugatmentioning

confidence: 99%

Continuous-variable entropic uncertainty relations

Hertz

Cerf

2019

J. Phys. A: Math. Theor.

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Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p. This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x-p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p, but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.

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Section: Tight Entropic Uncertainty Relation For Canonically Conjugatmentioning

confidence: 99%

Continuous-variable entropic uncertainty relations

Hertz

Cerf

2019

J. Phys. A: Math. Theor.

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“…In the same limit (ie, at β → 1/2), the Neumann sum

{(([], R_{ρ_{n}}^{N} ((), α) + R_{γ_{n}}^{N} ((), β))|}_{α = \infty}

diverges due to the similar behavior of the momentum item. Knowledge of the different behavior of the entropic uncertainty relations [ 14–19 ] will help in the correct choice of the substances in the design of devices for data compression, quantum cryptography, entanglement witnessing, quantum metrology, and other tasks employing correlations between the position and momentum components of the information measures. [ 16,19 ] Equation ) is a result of logarithmization of the Sobolev inequality of the Fourier transform [ 20 ]

{(\frac{α}{π})}^{d / ((), 4 α)} {[\int_{D_{ρ}^{d}} ρ_{n}^{α} (boldr) d r]}^{1 / ((), 2 α)} \geq {(\frac{β}{π})}^{d / ((), 4 β)} {[\int_{D_{γ}^{d}} γ_{n}^{β} (boldk) d k]}^{1 / ((), 2 β)},

for which, in addition to the requirement from Equation ), an extra constraint

\frac{1}{2} \leq α \leq 1

is imposed, which directly leads to the Tsallis uncertainty relation [ 21 ] :

{(\frac{α}{π})}^{d / ((), 4 α)} {[1 + (1 - α) T_{ρ_{n}} (α)]}^{1 / ((), 2 α)} \geq (β...)

…”

Section: Introductionmentioning

confidence: 99%

Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Olendski

2020

Int J of Quantum Chemistry

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d-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) allows analytic solution of the Schrödinger equation in position space and the Fourier transform of the corresponding wave function leads to the analytic form of its momentum counterpart too. This paves the way to an efficient computation in either space of Shannon, Rényi and Tsallis entropies, Onicescu energies and Fisher informations; for example, for the latter measure, some particular orbitals exhibit simple expressions in either space at any BC type. A comparative study of the influence of the edge requirement on the quantum information measures proves that the lower threshold of the semi-infinite range of the dimensionless Rényi/Tsallis coefficient where one-parameter momentum entropies exist is equal to d/(d + 3) for the Dirichlet hyperball and d/(d + 1) for the Neumann one what means that at the unrestricted growth of the dimensionality both measures have their Shannon fellow as the lower verge. Simultaneously, this imposes the restriction on the upper value of the interval [1/2, α R ) inside which the Rényi uncertainty relation for the sum of the position R ρ (α) and wave vector R γ α 2α−1 components is defined: α R is equal to d/(d − 3) for the Dirichlet geometry and to d/(d − 1) for the Neumann BC. Some other properties are discussed from mathematical and physical points of view. Parallels are drawn to the corresponding properties of the hydrogen atom and similarities and differences are explained based on the analysis of the associated wave functions.

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“…Another important advantage to using the information-theoretic approach is that the entanglement effect can be incorporated into the uncertainty paradigm by introducing the concept of quantum memory [19,20,21]. Those EURs form crucial key elements in detecting entanglement and proving the security of quantum cryptography, as extensively reviewed in [22,23,24]. More recently, it has been discovered that the EURs with quantum memory allow for trade-offs between the concepts of quantum uncertainty and reality for quantum observables [25].…”

Section: Introductionmentioning

confidence: 99%

Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

Baek

Nha

Son

2019

Entropy

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We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K.Życzkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [ L. Rudnicki, Z. Pucha la and K.Życzkowski, Phys. Rev. A 89, 052115 (2014)]. We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen-Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.

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Uncertainty Relations for Coarse–Grained Measurements: An Overview

Cited by 23 publications

References 199 publications

Continuous-variable entropic uncertainty relations

Continuous-variable entropic uncertainty relations

Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

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