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We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by BialynickiBirula [1] and Zozor et al. [2]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated Rényi entropies with conjugated indices. We consider here the general case where the entropic indices 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 indices 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).

We show that there exists a very natural, superstatistics-linked extension of the central limit theorem (CLT) to deformed exponentials (also called q-Gaussians): This generalization favorably compares with the one provided by S. Umarov and C. Tsallis [arXiv:cond-mat/0703533], since the latter requires a special "q-independence" condition on the data. On the contrary, our CLT proposal applies exactly in the usual conditions in which the classical CLT is used. Moreover, we show that, asymptotically, the q-independence condition is naturally induced by our version of the CLT. * The authors thank S. Umarov for providing a draft version of [1] † Electronic address: vignat@univ-mlv.fr, plastino@uolsinectis.com.arThe central limit theorems (CLT) can be ranked among the most important theorems in probability theory and statistics and plays an essential role in several basic and applied disciplines, notably in statistical mechanics. Pioneers like A. de Moivre, P.S. de Laplace, S.D. Poisson, and C.F. Gauss have shown that the Gaussian function is the attractor of independent additive contributions with a finite second variance. Distinguished authors like Chebyshev, Markov, Liapounov, Feller, Lindeberg and Lévy have also made essential contributions to the CLT-theory.The random variables to which the classical CLT refers are required to be independent. Subsequent efforts along CLT lines have established corresponding theorems for weakly dependent random variables as well (see some pertinent references in [1,2,3]). However, the CLT does not hold if correlations between far-ranging random variables are not negligible (see [4]).Recent developments in statistical mechanics that have attracted the attention of many researches deal with strongly correlated random variables ([5] and references therein). These correlations do not rapidly decrease with any increasing distance between random variables and are often referred to as global correlations (see [6] for a definition). Is there an attractor that would replace the Gaussians in such a case?The answer is in the affirmative, as shown in [1,2,3], with the deformed or q-Gaussian playing the starring role. It is asserted in [2] that such a theorem cannot be obtained if we rely on classic algebra: it needs a construction based on a special algebra, which is called q-algebra [15]. The goal of this communication is to show that a q-generalization of the central limit theorem becomes indeed possible and in a very simple way without recourse to q-algebra. A. Systems that are q-distributedConsider a system S described by a random vector X with d−components whose covariance matrix readsthe superscript t indicating transposition. We say that X is q−Gaussian (or deformed Gaussian-) distributed if its probability distribution function writes as described by Eqs.(2)-(3) below.