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 (JD(alpha) for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD(1)=JD. Using a result of Schoenberg, we prove that JD(alpha) is the square of a metric for alpha is an element of(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 alpha (QJD(alpha)). We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, QJD(alpha)(1/2) is a metric space which can be isometrically embedded in a real Hilbert space when alpha is an element of(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.