Properties of Classical and Quantum Jensen-Shannon Divergence

By J. Briet, Peter Harremoës. 2008

145 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Artificial Intelligence, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3172efd0a0d2d2d2428652846e4f4ac70d9af3e1

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The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ingemar Bengtsson
2. Karol Zyczkowski
3. M. A. Porter

In Artificial Intelligence

1. Alex Arenas
2. Ingemar Bengtsson
3. Karol Zyczkowski
4. M. A. Porter
5. Manlio De Domenico
6. Santo Fortunato

In Statistical and Nonlinear Physics

1. M. A. Porter
2. Manlio De Domenico
3. Alex Arenas
4. Ingemar Bengtsson
5. Karol Zyczkowski
6. Santo Fortunato
7. Olaf Sporns
8. Mark Newman

Citations in the corpus

• Linear Algebra and its Applications (2010, unknown): “So we require The number of iterations k i , i = 1,2, is now chosen as follows.”
• Multilayer Analysis and Visualization of Networks (2014, background): “By definition, the quantum Jensen–Shannon divergence is symmetric and its square root, which is usually called the Jensen–Shannon distance, satisfies the properties of a metric [49].”
• Geometry of Quantum States: Index (2007, background): “2 This was shown in 2003 by Endres and Schindelin [295], and the result is expected to generalise to the quantum case [175, 570].”