Year: 2009
Type: article
Abstract: 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 (${\text{JD}}_{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}>0$), the Jensen divergences of order $\ensuremath{\alpha}$, which generalize JD as ${\text{JD}}_{1}=\text{JD}$. Using a result of Schoenberg, we prove that ${\text{JD}}_{\ensuremath{\alpha}}$ is the square of a metric for $\ensuremath{\alpha}∊(0,2]$, 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 $\ensuremath{\alpha}$ $({\text{QJD}}_{\ensuremath{\alpha}})$. We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, ${\text{QJD}}_{\ensuremath{\alpha}}^{1/2}$ is a metric space which can be isometrically embedded in a real Hilbert space when $\ensuremath{\alpha}∊(0,2]$. 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.
Source: Physical Review A
Authors: Jop Briët, Peter Harremoës
Institution: Centrum Wiskunde & Informatica
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Award: G5946206379
OpenAlex ID: W2034306513