We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller–Lennert et al. Recently, Müller–Lennert et al. [12] and Wilde et al. [15] modified the traditional notion of relative Rényi entropy and showed that their new definition has several desirable properties of a relative entropy. One of th...

Entanglement Rényi-α entropy is an entanglement measure. It reduces to the standard entanglement of formation when α tends to 1. We derive analytical lower and upper bounds for the entanglement Rényi-α entropy of arbitrary dimensional bipartite quantum systems. We also demonstrate the application our bound for some concrete examples. Moreover, we establish the relation between entanglement Rény...

We associate to the p-th Rényi entropy a definition of entropy power, which is the natural extension of Shannon’s entropy power and exhibits a nice behaviour along solutions to the p-nonlinear heat equation in Rn. We show that the Rényi entropy power of general probability densities solving such equations is always a concave function of time, whereas it has a linear behaviour in correspondence ...

Entropy is well known to be Schur concave on finite alphabets. Recently, the authors have strengthened the result by showing that for any pair of probability distributions P and Q with Q majorized by P , the entropy of Q is larger than the entropy of P by the amount of relative entropy D(P ||Q). This result applies to P and Q defined on countable alphabets. This paper shows the counterpart of t...

Estimation of the entropy rate of a stochastic process with unknown statistics, from a single sample path is a classical problem in information theory. While universal estimators for general families of processes exist, the estimates have not been accompanied by guarantees for fixed-length sample paths. We provide finite sample bounds on the convergence of a plug-in type estimator for the entro...

Recently, a variety of new measures of quantum Rényi mutual information and quantum Rényi conditional entropy have been proposed, and some of their mathematical properties explored. Here, we show that the Rényi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternate hypothesis consists o...

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller–Lennert et al. Recently, Müller–Lennert et al. [12] and Wilde et al. [15] modified the traditional notion of relative Rényi entropy and showed that their new definition has several desirable properties of a relative entropy. One of th...

The conventional channel resolvability problem refers to the determination of the minimum rate needed for an input process to approximate the output distribution of a channel in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in [0,2]) to measure the leve...

For a large class of stationary probability measures on AN, where A is a finite alphabet, we compute the specific Rényi entropy of order α and the specific guesswork moments of order β > −1. We show that the specific guesswork moment of order β equals the specific Rényi entropy of order α = 1/(1+ β) multiplied by β. The method is based on energy–entropy estimates suggested by statistical physic...

The notion of smooth entropy allows a unifying, generalized formulation of privacy ampli-cation and entropy smoothing. Smooth entropy is a measure for the number of almost uniform random bits that can be extracted from a random source by probabilistic algorithms. It is known that the R enyi entropy of order at least 2 of a random variable is a lower bound for its smooth entropy. On the other ha...