On the Rényi divergence and the joint range of relative entropies
نویسنده
چکیده
This paper starts with a study of the minimum of the Rényi divergence subject to a fixed (or minimal) value of the total variation distance. Relying on the solution of this minimization problem, we determine the exact region of the points ( D(Q||P1), D(Q||P2) ) where P1 and P2 are any probability distributions whose total variation distance is not below a fixed value, and the probability distribution Q is arbitrary (none of these three distributions is assumed to be fixed). It is further shown that all the points of this convex region are attained by a triple of 2-element probability distributions. As a byproduct of this characterization, we provide a geometric interpretation of the minimal Chernoff information subject to a minimal total variation distance.
منابع مشابه
A Preferred Definition of Conditional Rényi Entropy
The Rényi entropy is a generalization of Shannon entropy to a one-parameter family of entropies. Tsallis entropy too is a generalization of Shannon entropy. The measure for Tsallis entropy is non-logarithmic. After the introduction of Shannon entropy , the conditional Shannon entropy was derived and its properties became known. Also, for Tsallis entropy, the conditional entropy was introduced a...
متن کاملα-z-Rényi relative entropies
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...
متن کاملOptimized quantum f-divergences and data processing
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properti...
متن کاملMinimization Problems Based on a Parametric Family of Relative Entropies I: Forward Projection
Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative α-entropies (denoted Iα), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual r...
متن کاملA duality relation connecting different quantum generalizations of the conditional Rényi entropy
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. This generalizes the well-known duality relation...
متن کامل