Peter Harremoës

Properties of Classical and Quantum Jensen-Shannon Divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JDα for α > 0), the Jensen divergences of order α, which generalize JD as JD1 = JD. Using a result of Schoenberg, we prove that JDα is the square of a metric for α ∈ (0, 2] , and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order α (QJDα). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD α is a metric space which can be isometrically embedded in a real Hilbert space when α ∈ (0, 2] . In analogy with Burbea and Rao’s generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

Evolutionary Entropy: A Predictor of Body Size, Metabolic Rate and Maximal Life Span

Body size of organisms spans 24 orders of magnitude, and metabolic rate and life span present comparable differences across species. This article shows that this variation can be explained in terms of evolutionary entropy, a statistical parameter which characterizes the robustness of a population, and describes the uncertainty in the age of the mother of a randomly chosen newborn. We show that entropy also has a macroscopic description: It is linearly related to the logarithm of the variables body size, metabolic rate, and life span. Furthermore, entropy characterizes Darwinian fitness, the efficiency with which a population acquires and converts resources into viable offspring. Accordingly, entropy predicts the outcome of natural selection in populations subject to different classes of ecological constraints. This predictive property, when integrated with the macroscopic representation of entropy, is the basis for enormous differences in morphometric and life-history parameters across species.

Information Topologies with Applications

Topologies related to information divergence are introduced. The conditional limit theorem is taken as motivating example, and simplified proofs of the relevant theorems are given. Continuity properties of entropy and information divergence are discussed.

Entropy 2008, 10, 240-247: Ferri et al. Deformed Generalization of the Semiclassical Entropy

It has come to our attention that names of authors have been misspelled in a paper recently published in Entropy [1]. The correct names are: Gustavo Ferri, Felipe Olivares, Flavia Pennini, Angelo Plastino, Angel R. Plastino and Montserrat Casas. We apologize for this mistake and any inconvenience caused. [...]