Many problems of statistical and quantum mechanics can be established in terms of a distance between probability distributions. The present work is devoted to review some of the most relevant applications of the notion of distance in the context of statistical mechanics, quantum mechanics and information theory. Although we make a general overview of the most frequently used distances between probability distributions, we will center our presentation in a distance known as the Jensen-Shannon divergence both in their classical and quantum versions. For the classical one we present its main properties and we discuss its relevance as a segmentation tool for symbolic sequences. In the quantum case we show that the quantum Jensen-Shannon divergence is an adequate measure of entanglement.
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 (${\text{JD}}_{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}>0$), the Jensen divergences of order $\ensuremath{\alpha}$, which generalize JD as
JD1=JD. Using a result of Schoenberg, we prove that ${\text{JD}}_{\ensuremath{\alpha}}$ is the square of a metric for $\ensuremath{\alpha}∊(0,2]$, 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 $\ensuremath{\alpha}$ $({\text{QJD}}_{\ensuremath{\alpha}})$. We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, ${\text{QJD}}_{\ensuremath{\alpha}}^{1/2}$ is a metric space which can be isometrically embedded in a real Hilbert space when $\ensuremath{\alpha}∊(0,2]$. 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.