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Electronic Journal of Linear Algebra

Barycenter of the arithmetic-harmonic quantum divergence

Miran Jeong, Sejong Kim

Published Apr 3, 2025• 1 citation
Publisher

Abstract

A notion of divergence is a very important and useful tool to measure the difference between probability distributions or between data (information). We consider a quantum divergence constructed by the difference of two-variable weighted arithmetic and harmonic means on the open convex cone of positive definite Hermitian matrices, called the arithmetic-harmonic quantum divergence. We see its invariance properties and study the barycenter minimizing the weighted sum of arithmetic-harmonic quantum divergences to given variables. We provide the lower bound for the barycenter of the arithmetic-harmonic quantum divergence in terms of Loewner order and its upper bound in terms of operator norm.

DOI
10.13001/ela.2025.8883

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A notion of divergence is a very important and useful tool to measure the difference between probability distributions or between data (information). We consider a quantum divergence constructed by the difference of two-variable weighted arithmetic and harmonic means on the open convex cone of positive definite Hermitian matrices, called the arithmetic-harmonic quantum divergence. We see its invariance properties and study the barycenter minimizing the weighted sum of arithmetic-harmonic quantum divergences to given variables. We provide the lower bound for the barycenter of the arithmetic-harmonic quantum divergence in terms of Loewner order and its upper bound in terms of operator norm.
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.
Let Pn\mathbb{P}_nPn​ be the cone of n×nn\times nn×n complex positive definite matrices. For 0<θ<10<θ<10<θ<1 and r≥0r\ge0r≥0, let Mr(θ)M_r^{(θ)}Mr(θ)​ denote the weighted matrix power mean, with the logarithmic endpoint at r=0r=0r=0, defined by Mr(θ)(A,B)={((1−θ)Ar+θBr)1/r,r>0,exp⁡((1−θ)log⁡A+θlog⁡B),r=0. M_r^{(θ)}(A,B) = \begin{cases} \big((1-θ)A^r+θB^r\big)^{1/r}, & r>0,\\[2mm] \exp\big((1-θ)\log A+θ\log B\big), & r=0. \end{cases} Mr(θ)​(A,B)=⎩⎨⎧​((1−θ)Ar+θBr)1/r,exp((1−θ)logA+θlogB),​r>0,r=0.​ For p>q≥0p>q\ge0p>q≥0, we introduce Φp,q(θ)(A,B)=Tr⁡[Mp(θ)(A,B)−Mq(θ)(A,B)]. Φ_{p,q}^{(θ)}(A,B) = \operatorname{Tr}\big[M_p^{(θ)}(A,B)-M_q^{(θ)}(A,B)\big]. Φp,q(θ)​(A,B)=Tr[Mp(θ)​(A,B)−Mq(θ)​(A,B)]. We prove that Φp,q(θ)Φ_{p,q}^{(θ)}Φp,q(θ)​ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for 0≤q≤1≤p≤2,p>q, 0\le q\le1\le p\le2,\qquad p>q, 0≤q≤1≤p≤2,p>q, it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.
In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (also called ”sandwiched Rényi relative entropy”) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.
Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.
In this paper we investigate quantum Hellinger type divergences which were studied by Bhatia-Gaubert-Jain (2019), Pitrik-Virosztek (2020), and Dinh-Lie-Osaka-Phan (2025). In particular, when [Formula: see text] is a convex function defined by the form [Formula: see text] ([Formula: see text]) and [Formula: see text] is an operator monotone function with [Formula: see text], we introduce the quantum quantitative measure [Formula: see text] for positive definite matrices [Formula: see text] and [Formula: see text], and show that [Formula: see text] is a quantum divergence in the sense of Bhatia-Gaubert-Jain and also show that it is jointly convex and satisfies the data processing property by a trace preserving positive unital map [Formula: see text], that is, [Formula: see text].