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Lobachevskii Journal of Mathematics

A Generalized Matrix Power Mean and a New Quantum Hellinger Divergence

Trung Hoa Dinh, Anh Vu Le, Thi Thu Hoai Nguyen, Ngoc Yen Phan

Published Feb 1, 2024• 0 citations
Publisher

Abstract

In this paper, we study a matrix equation involving the matrix geometric mean $$A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},$$ $$t\in(1,2)$$ . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean $$A\natural_{t}B$$ to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.

DOI
10.1134/s1995080224600304

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In this paper, we study a matrix equation involving the matrix geometric mean A♮tB=A1/2(A−1/2BA−1/2)tA1/2,A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},A♮t​B=A1/2(A−1/2BA−1/2)tA1/2, t∈(1,2)t\in(1,2)t∈(1,2) . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean A♮tBA\natural_{t}BA♮t​B to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.
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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 this paper, we introduce a new quantum divergence$$\Phi (X,Y) = \Tr \left[\left(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right)X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$where 0<α<10< \alpha <10<α<1.We study the least square problem with respect to this divergence. We also show that the new quantum divergence satisfies the Data Processing Inequality in quantum information theory. In addition, we show that the matrix ppp-power mean μp(t,A,B)=((1−t)Ap+tBp)1/p\mu_p(t, A, B) = ((1-t)A^p + tB^p)^{1/p}μp​(t,A,B)=((1−t)Ap+tBp)1/p satisfies the in-betweenness property with respect to the new divergence.
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].
Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.
Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.