Properties of Classical and Quantum Jensen Shannon Divergence
J. Briet, Peter Harremoes and F. Topsoe
Communication in Information and Systems 2008.

Abstract

The Jensen-Shannon divergence (JSD) is a symmetrized and smoothed version of the all important divergence measure of informaton theory, the Kullback-Leibler divergence. It defines a true metric – precisely, it is the square of a metric. We prove a stronger result for a new family of divergence measures based on the Tsallis entropy, that includes the JSD. Furthermore we elaborate on details of geometric properties of the JSD. Analogously, the quantum Jensen-Shannon divergence (QJSD) is a symmetrized version of the quantum relative entropy that has recently been considered as a distance measure for quantum states. We prove for a new family of distance measures for states, including the QJSD, that each member is the square of a metric for all qubits, strengthening recent results by Lamberti et al. We also discuss geometric properties of the QJSD. In analogy to Lin’s generalization of the JSD, we also define the general QJSD for a weighting of any number of states and discuss interpretations of both quantities.

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