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_ for >0), the Jensen divergences of order, which generalize JD as JD₁=JD. Using a result of Schoenberg, we prove that JD_ is the square of a metric for ∊ (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 (QJD_). We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, QJD_^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when ∊ (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.