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Metrics Induced by Quantum Jensen-Shannon-Renyí and Related Divergences

Suvrit Sra

Published Nov 6, 2019• 0 citations
OpenAlex

Abstract

We study symmetric divergences on Hermitian positive definite matrices generated by functions closely related to Pick-Nevanlinna functions. Our main result proves that the square root of these divergences is a distance metric. As a corollary we obtain the metric property for Quantum Jensen-Shannon-Tsallis divergences (parameterized by $\alpha\in [0,2]$), which in turn (for $\alpha=1$) yields a proof of the metric property conjectured by Briet and Harremoes a decade ago (\emph{Properties of classical and quantum Jensen-Shannon divergence}, Phy. Rev. A, \textbf{79}, 052311 (2009)). A somewhat more intricate argument also establishes metric properties of Jensen-Renyi divergences (for $\alpha \in (0,1)$), and outlines a techniques for proving this property for symmetric divergences involving completely monotonic functions.

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We study symmetric divergences on Hermitian positive definite matrices generated by functions closely related to Pick-Nevanlinna functions. Our main result proves that the square root of these divergences is a distance metric. As a corollary we obtain the metric property for Quantum Jensen-Shannon-Tsallis divergences (parameterized by α∈[0,2]\alpha\in [0,2]α∈[0,2]), which in turn (for α=1\alpha=1α=1) yields a proof of the metric property conjectured by Briet and Harremoes a decade ago (\emph{Properties of classical and quantum Jensen-Shannon divergence}, Phy. Rev. A, \textbf{79}, 052311 (2009)). A somewhat more intricate argument also establishes metric properties of Jensen-Renyi divergences (for α∈(0,1)\alpha \in (0,1)α∈(0,1)), and outlines a techniques for proving this property for symmetric divergences involving completely monotonic functions.
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.
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.
One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.
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.
There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold P of positive definite matrices. The geodesic connecting two points on P is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on P. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮α-Rényi divergence of real order α∈R, including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α–z-Rényi divergence and ♮α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.