Similar to fuzzy set on a possibility space, uncertain set is a set-valued function on an uncertainty space, and attempts to model unsharp concepts. Entropy provides a quantitative measurement of the uncertainty associated with an uncertain set. This paper presents a formula for calculating the entropy of an uncertain set via its inverse membership function. Based on the formula, the entropy op...

In this paper, at first we derive a family of maximum Tsallis entropy distributions under optional side conditions on the mean income and the Gini index. Furthermore, corresponding with these distributions a family of Lorenz curves compatible with the optional side conditions is generated. Meanwhile, we show that our results reduce to Shannon entropy as $beta$ tends to one. Finally, by using ac...

Discerning possible candidates for measuring distances between quantum states is a subject of perennial interest. Many of these measures were first defined as distances between probability distributions and subsequently employed as distance-measures in Hilbert space. Let H be the Hilbert space associated with a quantum system and let S be the set of all states, i.e. the set of self-adjoint, (se...

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...

Relative entropy is an essential tool in quantum information theory. There are so many problems which are related to relative entropy. In this article, the optimal values which are defined by max U∈U(Xd) S(UρU∗ ‖ σ) and min U∈U(Xd) S(UρU∗ ‖ σ) for two positive definite operators ρ, σ ∈ Pd(X ) are obtained. And the set of S(UρU∗ ‖ σ) for every unitary operator U is full of the interval [ min U∈U...

Abstract— The direct part of Stein’s lemma in quantum hypothesis testing is revisited based on a key operator inequality between a density operator and its pinching. The operator inequality is used to show a simple proof of the direct part of Stein’s lemma without using Hiai-Petz’s theorem, along with an operator monotone function, and in addition it is also used to show a new proof of Hiai-Pet...