Properties of Classical and Quantum Jensen-Shannon Divergence

TL;DR

This paper explores properties of classical and quantum Jensen-Shannon divergences, proving metric space embeddings for various orders and extending the concepts to quantum states, with implications for information theory and quantum computing.

Contribution

It generalizes Jensen-Shannon divergence to a family of measures and proves their metric properties and embeddings in Hilbert spaces for both classical and quantum cases.

Findings

JD_alpha is the square of a metric for alpha in (0,2].

QJD_alpha^1/2 is a metric space for qubits and pure states when alpha in (0,2].

Bounds are derived relating these divergences to total variation and trace distance.

Abstract

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.