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

Abstract

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle 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 Renyi entropies. We recall that in both Shannon and Renyi 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 Renyi entropy considered, extending the results of Abe 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.

References in corpus (3)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• Extensive Renyi Statistics from Non-Extensive Entropy
• The world according to Renyi: Thermodynamics of multifractal systems

Cited by in corpus (31)

• Some extensions of the uncertainty principle
• General entropy-like uncertainty relations in finite dimensions
• Collision entropy and optimal uncertainty
• Majorization approach to entropic uncertainty relations for coarse-grained observables
• Rényi entropy uncertainty relation for successive projective measurements
• Optimal Uncertainty Relations for Extremely Coarse-grained Measurements
• Position-momentum uncertainty relations based on moments of arbitrary order
• Uncertainty relations in terms of Tsallis entropy
• Entropic measures of Rydberg-like harmonic states
• Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
• Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems
• The Shannon-entropy-based uncertainty relation for $D$-dimensional central potentials
• Entropic uncertainty measures for large dimensional hydrogenic systems
• Rényi and Tsallis entropies: three analytic examples
• Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fields
• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• On a generalized entropic uncertainty relation in the case of the qubit
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• On Generalized Stam Inequalities and Fisher-Rényi Complexity Measures
• Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots
• Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
• A geometric formulation of uncertainty principle
• Exact Rényi entropies of $D$-dimensional harmonic systems
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions
• One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field
• Information theoretic measures on quantum droplets in ultracold atomic systems
• Phase space distributions in information theory
• Probing information theoretic measures of nonlinear ultracold quantum gases using phase-space distributions