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We give an overview of the role of information theory in statistics, and particularly in biostatistics. We recall the basic quantities in information theory; entropy, cross-entropy, conditional entropy, mutual information and…
Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be…
Entropy and information provide natural measures of correlation among elements in a network. We construct here the information theoretic analog of connected correlation functions: irreducible $N$--point correlation is measured by a decrease…
The use of coarse graining to connect physical and information theoretic entropies has recently been given a precise formulation in terms of ``observational entropy'', describing entropy for observers with respect to a measurement. Here we…
A novel measure, quantumness of correlations is introduced here for bipartite states, by incorporating the required measurement scheme crucial in defining any such quantity. Quantumness coincides with the previously proposed measures in…
Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new…
In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the…
In this article, we discuss the problem of establishing relations between information measures assessed for network structures. Two types of entropy based measures namely, the Shannon entropy and its generalization, the R\'{e}nyi entropy…
Information theory provides a mathematical foundation to measure uncertainty in belief. Belief is represented by a probability distribution that captures our understanding of an outcome's plausibility. Information measures based on…
The Kullback-Leibler (KL) divergence is a foundational measure for comparing probability distributions. Yet in multivariate settings, its single value often obscures the underlying reasons for divergence, conflating mismatches in individual…
In this Thesis, several results in quantum information theory are collected, most of which use entropy as the main mathematical tool. *While a direct generalization of the Shannon entropy to density matrices, the von Neumann entropy behaves…
The universal entropy bound of Bekenstein is considered, at any strength of the gravitational interaction. A proof of it is given, provided the considered general-relativistic spacetimes allow for a meaningful and inequivocal definition of…
In ordinary Boltzmann-Gibbs thermostatistics, the relative entropy expression plays the role of generalized free energy, providing the difference between the off-equilibrium and equilibrium free energy terms associated with Boltzmann-Gibbs…
Let recall that the term 'k-th extreme' was introduced in a limiting sense. That is, if $X_{r:n}$ denote the r-th order statistic then for fix k, as $n\to\infty$, $X_{n-k+1:n}$ is called the k-th extremes or k-th largest order statistics.…
The random matrix ensembles (RME) of quantum statistical Hamiltonians, e.g. Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied in literature to following quantum statistical systems:…
Entropy and information can be considered dual: entropy is a measure of the subspace defined by the information constraining the given ambient space. Negative entropies, arising in na\"ive extensions of the definition of entropy from…
In this paper we provide the asymptotic theory of the general of $\phi$-divergences measures, which include the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence…
Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states…
The quantum Renyi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory.…
We introduce the \textit{Latent Entropy} (L-entropy) as a novel measure to characterize the genuine multipartite entanglement in quantum systems. Our measure leverages the upper bound of reflected entropy and its maximal values attained by…