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.