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arXiv:0804.3662 · International Journal of Quantum Information

JENSEN–SHANNON DIVERGENCE AS A MEASURE OF THE DEGREE OF ENTANGLEMENT

A. P. Majtey, Antoni Borràs, M. Casas, Pedro W. Lamberti, Angel Ricardo Plastino

Published Jul 1, 2008• 20 citations
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Abstract

The notion of distance in Hilbert space is relevant in many scenarios. In particular, "distances" between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.

DOI
10.1142/s0219749908004006

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The notion of distance in Hilbert space is relevant in many scenarios. In particular, "distances" between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.
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.
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 JD1=JD{\text{JD}}_{1}=\text{JD}JD1​=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.
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.
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.
The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states.