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BENTHAM SCIENCE PUBLISHERS eBooks

Distances Measures for Probability Distributions

Published Dec 11, 2013• 1 citation
Publisher

Abstract

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.

DOI
10.2174/9781608057603113010011

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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.
Many problems of statistical and quantum mechanics can be established in terms of a distance; in the first case the distance is usually defined between probability distributions; in the second one, between quantum states. The present work is devoted to review the main properties of a distance known as the Jensen‐Shannon divergence (JSD) in its classical and quantum version. We present two examples of application of this distance: in the first one we use it as a quantifiers of the stochastic resonance phenomenon in ion channels; in the second one we use the JSD to propose a geometrical view of entanglement for two qubits states.
We discuss an alternative to relative entropy as a measure of distance between mixed quantum states. The proposed quantity is an extension to the realm of quantum theory of the Jensen-Shannon divergence (JSD) between probability distributions. The JSD has several interesting properties. It arises in information theory and, unlike the Kullback-Leibler divergence, it is symmetric, always well-defined, and bounded. We show that the quantum JSD shares with the relative entropy most of the physically relevant properties, in particular those required for a ``good'' quantum distinguishability measure. We relate it to other known quantum distances and we suggest possible applications in the field of the quantum information theory.
Chapter 5 delves into divergences and distance measures, which are crucial for comparing quantum states. It begins with classical divergences such as the Kullback–Leibler and Jensen–Shannon, then advances to their quantum counterparts, discussing their optimal characteristics. Influenced by quantum resource theories, these quantum extensions provide foundational insights into the robust tools of resource theories. The chapter concentrates on particular divergences that serve as true metrics, including the trace distance and a variant of the fidelity, and explores the concept of distance between subnormalized states, which is essential in the context of quantum measurements. It emphasizes the purified distance, a useful tool for understanding the entanglement cost of quantum systems, setting the stage for further exploration in later chapters. The chapter offers a mathematically approachable survey of these measures, underscoring their practical importance in quantum information theory.
In a recent paper, the generalization of the Jensen-Shannon divergence in the context of quantum theory has been studied [Majtey et al., Phys. Rev. A 72, 052310 (2005)]. This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of a numerical procedure for mixed states, that its square root verifies the triangle inequality.