Properties of classical and quantum Jensen-Shannon divergence
- 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, Vol. 79, Issue 5; Other Information: DOI: 10.1103/PhysRevA.79.052311; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
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