The Renyi entropy coprises a group of data estimates that sums up the well-known Shannon entropy, acquiring considerable lot its properties. It appears as unqualified and restrictive relative or common data, has found numerous applications in information theory. Renyi's argument, area law black hole plays significant role. However, total can be modified by some quantum effects, motivated random...

We investigate the Rényi entropy of sums independent integer-valued random variables through Fourier theoretic means, and give sharp comparisons between variance for Bernoulli variables. As applications, we prove that a discrete “min-entropy power” is superadditive with respect to convolution modulo universal constant, new bounds on an entropic generalization Littlewood-Offord problem are in “P...

Distributions derived from the maximization of Rényi-Tsallis entropy are often called Tsallis’ distributions. We first indicate that these distributions can arise as mixtures, and can be interpreted as the solution of a standard maximum entropy problem with fluctuating constraints. Considering that Tsallis’ distributions appear for systems with displaced or fluctuating equilibriums, we show tha...

We consider the maximum entropy problems associated with Rényi Q-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are der...

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on R under constrained Rényi-αentropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In th...

We consider a two-parameter family of Rényi relative entropies Dα,z(ρ||σ) that are quantum generalisations of the classical Rényi divergence Dα(p||q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum general...

This paper is twofold. In the first part, we derive an improvement of the Rényi Entropy Power Inequality (EPI) recently obtained by Bobkov and Marsiglietti [10]. The proof largely follows Lieb’s [22] approach of employing Young’s inequality. In the second part, we prove a reverse Rényi EPI, that verifies a conjecture proposed in [4, 23] in two cases. Connections with various p-th mean bodies in...