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IEEE International Symposium on Information Theory (ISIT)

Non-Gaussian asymptotic minimizers in entropic uncertainty principles and the dimensional effect

Steeve Zozor, Christophe Vignat

Published Jul 1, 2006• 1 citation
Publisher

Abstract

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle (I. Bialynicki-Birula and J. Mycielski, 1975) and the associated cases of equality. This Shannon entropic version of the well-known Heisenberg inequality 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 cases, equality occurs only for Gaussian random variables. However, we show that in the particular n-dimensional Laplace case, the bound is asymptotically attained as n grows. We also show numerically that this effect exists for Cauchy variables whatever the Renyi entropy considered, extending the results of S. Abe and A.K. Rajagopal (2001), These two cases are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the variable when the dimension increases, so that the effect is rather "dimensional"

DOI
10.1109/isit.2006.261918

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In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle (I. Bialynicki-Birula and J. Mycielski, 1975) and the associated cases of equality. This Shannon entropic version of the well-known Heisenberg inequality 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 cases, equality occurs only for Gaussian random variables. However, we show that in the particular n-dimensional Laplace case, the bound is asymptotically attained as n grows. We also show numerically that this effect exists for Cauchy variables whatever the Renyi entropy considered, extending the results of S. Abe and A.K. Rajagopal (2001), These two cases are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the variable when the dimension increases, so that the effect is rather "dimensional"
In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes 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 indexes 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).
In this paper we study the entropic generalization of the Heisenberg uncertainty principle and its cases of equalities. The Heisenberg uncertainty principle bounds the product of the variances of a random vector and of its conjugated; it has a meaning only if the variances exist. We recall the Renyi entropic generalization to this principle, that permits to bypass the existence of variances. Like for the Heisenberg principle, when the dimension is finite the bound of the inequality is achieved only in the Gaussian context. The goal of this study is to identify the non-Gaussian cases of equality in the context of elliptically distributed vectors and when the dimension tends to the infinity. We show that the equality is asymptotically achieved for numerous cases of Gaussian scale mixtures.
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 un- der 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.
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.
Abstract We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger–Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.
Entropic uncertainty relations H(A)+H(B)⩾γH(A)+H(B)\geqslant γH(A)+H(B)⩾γ give a nonzero lower bound γγγ to the sum of the Shannon entropies HHH of the outcome probabilities of incompatible observables AAA and BBB. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γγγ typically are state independent. Here we provide a state-independent lower bound γsγ_sγs​ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2s\to 2s→2, becomes asymptotically tight for all A,BA,BA,B. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/