Properties of classical and quantum Jensen-Shannon divergence

Journal Article · Fri May 15 04:00:00 EDT 2009 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.79.052311· OSTI ID:21300819

Brieet, Jop; Harremoees, Peter ^[1]

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1. Centrum Wiskunde and Informatica, Science Park 123, 1098 XG Amsterdam (Netherlands)

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{sub {alpha}} for {alpha}>0), the Jensen divergences of order {alpha}, which generalize JD as JD{sub 1}=JD. Using a result of Schoenberg, we prove that JD{sub {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{sub {alpha}}). We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, QJD{sub {alpha}}{sup 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.

• OSTI ID: 21300819
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 5 Vol. 79; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DISTANCE
ENTROPY
HILBERT SPACE
INFORMATION THEORY
METRICS
PROBABILITY
QUBITS