On Informational Divergences for General Statistical Theories

Qiuping Alexandre Wang

William Demopoulos

Sumiyoshi Abe

The Kullback-Leibler divergence offers an information-theoretic basis for measuring the difference between two given distributions. Its quantum analog, however, fails to play a corresponding role for comparing two density matrices, if the reference states are pure states. Here it is shown that nonadditive quantum information theory inspired by nonextensive statistical mechanics is free from such a difficulty and the associated quantity, termed the quantum q-divergence, can in fact be a good information-theoretic measure of the degree of state purification. The correspondence relation between the ordinary divergence and the q-divergence is violated for the pure reference states, in general.

Many problems of statistical and quantum mechanics can be established in terms of a distance between probability distributions. The present work is devoted to review some of the most relevant applications of the notion of distance in the context of statistical mechanics, quantum mechanics and information theory. Although we make a general overview of the most frequently used distances between probability distributions, we will center our presentation in a distance known as the Jensen-Shannon divergence both in their classical and quantum versions. For the classical one we present its main properties and we discuss its relevance as a segmentation tool for symbolic sequences. In the quantum case we show that the quantum Jensen-Shannon divergence is an adequate measure of entanglement.

Roger Balian

Zhaoqi Wu, Chuanxi Zhu, Shunlong Luo +1 more