2008/06/30 by Jop Briët, Peter Harremoës
Computer Science · Mathematics · Physics and Astronomy · #Mathematical Inequalities and Applications #Quantum Information and Cryptography #Statistical Mechanics and Entropy #quant-ph
paper · pdf · doi:10.1103/physreva.79.052311
published as Phys. Rev. A 79, 052311 (2009) · 13 pages, LaTeX, expanded contents, added references and corrected typos
arxiv created 2009/04/14 · openalex publication_date 2009/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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_\ensuremathα for \ensuremathα>0), the Jensen divergences of order \ensuremathα, which generalize JD as JD1=JD. Using a result of Schoenberg, we prove that JD_\ensuremathα is the square of a metric for \ensuremathα∊(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 \ensuremathα (QJD_\ensuremathα). We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, QJD_\ensuremathα1/2 is a metric space which can be isometrically embedded in a real Hilbert space when \ensuremathα∊(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.