2009-05-01

# Properties of classical and quantum Jensen-Shannon divergence

## Publication

### Publication

*Physical Review A: Atomic, Molecular and Optical Physics , Volume 79 - Issue 5*

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_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (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_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (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.

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Keywords | Quantum Jensen-Shannon divergence, Kullback divergence, Schoenberg's Theorem |

MSC | Applications in statistical physics (msc 46N55) |

Publisher | American Physical Society |

Journal | Physical Review A: Atomic, Molecular and Optical Physics |

Citation |
Briët, J, & Harremoës, P. (2009). Properties of classical and quantum Jensen-Shannon divergence.
Physical Review A: Atomic, Molecular and Optical Physics, 79(5). |