Related papers: Inequalities Revisited
Probability theory is fundamental for modeling uncertainty, with traditional probabilities being real and non-negative. Complex probability extends this concept by allowing complex-valued probabilities, opening new avenues for analysis in…
For noncomposite systems in classical and quantum domains, we obtain new inequalities such as the subadditivity and strong subadditivity conditions for Shannon entropies and information determined by the probability distributions and for…
Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a…
In recent years, various quantum inequalities have been established on quantum symmetries in the framework of quantum Fourier analysis. We provide a detailed introduction to quantum inequalities including Hausdorff-Young inequality, Young's…
The R{\'e}nyi entropy is one of the important information measures that generalizes Shannon's entropy. The quantum R{\'e}nyi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital…
We discuss linear programming techniques that help to deduce corollaries of non classic inequalities for Shannon's entropy. We focus on direct applications of the copy lemma. These applications involve implicitly some (known or unknown)…
Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we…
While most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information, up to now Shannon's entropy power inequality (EPI) is an exception: Existing information theoretic proofs of the…
We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The…
We investigate quantum R\'enyi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed R\'enyi generalisations of the quantum relative entropy. We may define R\'enyi…
The concept of entropy, firstly introduced in information theory, rapidly became popular in many applied sciences via Shannon's formula to measure the degree of heterogeneity among observations. A rather recent research field aims at…
We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…
We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below…
In this paper we remark that Shannon entropy can be expressed as a function of the self-information (i.e. the logarithm) and the inverse of the Lambert $W$ function. It means that we consider that Shannon entropy has the trace form: $-k…
The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy and are…
The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or…
We compare two different techniques for proving non-Shannon-type information inequalities. The first one is the original Zhang-Yeung's method, commonly referred to as the copy/pasting lemma/trick. The copy lemma was used to derive the first…
This paper addresses the correspondence between linear inequalities of Shannon entropy and differential entropy for sums of independent group-valued random variables. We show that any balanced (with the sum of coefficients being zero)…
Gagliardo--Nirenberg inequalities are interpolation inequalities which were proved independently by Gagliardo and Nirenberg in the late fifties. In recent years, their connections with theoretic aspects of information theory and nonlinear…
We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate the basic notions of both theories:…