An entropy-based uncertainty principle for a locally compact abelian group

Özaydın, Murad; Przebinda, Tomasz

doi:10.1016/j.jfa.2003.11.008

Cited by 23 publications

(18 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Conversely, for p = 2, since only Gaussian waves functions achieve equality in (4) as it is proved in [23], the Gaussian waves are the only wave packets which achieve equality in (6). The Shannon case p = 2 is more subtle, but it has been recently proved that equality is reached in inequality (2) only in the Gaussian case [24].…”

Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 97%

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

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle [1] and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both Shannon and Rényi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Rényi entropy considered, extending the results of Abe [2] and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases.

show abstract

Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 97%

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

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

show abstract

“…So the minimizer of the latter one has to be that of the former one. Theorem 6.4 is a noncommutative version of Theorem 1.5 in [24] when A is a finite abelian group. As showed in [24], the minimizer of the classical uncertainty principle is a nonzero scalar multiple of a translation and a modulation of the indicator function of a subgroup of A.…”

Section: Minimizers For Noncommutative Uncertainty Principlesmentioning

confidence: 99%

“…Theorem 6.4 is a noncommutative version of Theorem 1.5 in [24] when A is a finite abelian group. As showed in [24], the minimizer of the classical uncertainty principle is a nonzero scalar multiple of a translation and a modulation of the indicator function of a subgroup of A. Their techniques to describe the extremal bi-partial isometries do not work in subfactor planar algebras, since we do not have the translation or the modulation to shift an extremal bi-partial isometry to a biprojection.…”

Section: Minimizers For Noncommutative Uncertainty Principlesmentioning

confidence: 99%

“…We will first show that the two uncertainty principles have the same minimizers which are extremal bi-partial isometries. The proof of the following theorem benefits a lot from [24], and we also need Hopf's maximum principle. For readers' convenience, we state it here: Proposition 6.3 (Hopf's maximum principle, [15]).…”

Section: Minimizers For Noncommutative Uncertainty Principlesmentioning

confidence: 99%

“…In 1990, K. Smith [27] generalized the Donoho-Stark uncertainty principle to finite abelian groups. In 2004,Özaydm and Przebinda [24] generalized the Hischman-Beckner uncertainty principle and the Donoho-Stark uncertainty principle to abelian groups. They characterized all minimizers of the uncertainty principles and noticed that the minimizers of two uncertainty principles coincide.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Noncommutative uncertainty principles

Jiang

Liu

2016

Journal of Functional Analysis

View full text Add to dashboard Cite

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's λ-lattices, modular tensor categories etc.

show abstract

The Quantum Condition Space

Kais

2022

Adv Quantum Tech

View full text Add to dashboard Cite

The fundamental properties of quantum physics are exploited to evaluate event probabilities with projection measurements. Next, to study what events can be specified by quantum methods, the concept of the condition space is introduced, which is found to be the dual space of the classical outcome space of bit strings. Like the classical outcome space generates the quantum state space, the condition space generates the quantum condition space being the central idea of this work. The quantum condition space permits the existence of entangled conditions having no classical equivalent. In addition, the quantum condition space is related to the quantum state space by a Fourier transform guaranteed by the Pontryagin duality, and therefore an entropic uncertainty principle can be defined. The quantum condition space offers a novel perspective of understanding quantum states with the duality picture. Furthermore, the quantum conditions have physical meanings and realizations of their own and thus may be studied for purposes beyond the original motivation of characterizing events for probability evaluation. Finally, the relation between the condition space and quantum circuits provides insights into how quantum states are collectively modified by quantum gates, which may lead to deeper understanding of the complexity of quantum circuits.

show abstract

An entropy-based uncertainty principle for a locally compact abelian group

Cited by 23 publications

References 12 publications

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

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

Noncommutative uncertainty principles

The Quantum Condition Space

Contact Info

Product

Resources

About