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 (${\text{JD}}_{\alpha }$ for $\alpha >0$), the Jensen divergences of order $\alpha$, which generalize JD as ${\text{JD}}_1=\text{JD}$. Using a result of Schoenberg, we prove that ${\text{JD}}_{\alpha }$ is the square of a metric for $\alpha ∊\left(0,2\right]$, 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 $\alpha$ $\left({\text{QJD}}_{\alpha }\right)$. We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, ${\text{QJD}}_{\alpha }^{1/2}$ is a metric space which can be isometrically embedded in a real Hilbert space when $\alpha ∊\left(0,2\right]$. 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.

• Received 20 March 2009

DOI:https://doi.org/10.1103/PhysRevA.79.052311