Some geometric properties of matrix means with respect to different metrics
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Some geometric properties of matrix means with respect to different metrics Trung Hoa Dinh1 · Raluca Dumitru2 · Jose A. Franco2 Received: 7 October 2019 / Accepted: 2 February 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures–Wasserstein, Hellinger and log-determinant metrics. More precisely, we show that the matrix power means (Kubo–Ando and nonKubo–Ando extensions) satisfy the in-betweenness property in the Hellinger metric. We also show that for two positive definite matrices A and B, the curve of weighted Heron means, the geodesic curve of the arithmetic and the geometric mean lie inside the sphere centered at the geometric mean with the radius equal to half of the logdeterminant distance between A and B. Keywords Function distances · Geometric mean · In-betweenness property · Monotonicity · In-sphere property Mathematics Subject Classification 47A63 · 47A56
1 Introduction Let M n be the algebra of n × n matrices over C and Dn denote cone the positive definite elements of M n . Denote by I the identity matrix of M n . For a real-valued function f and a Hermitian matrix A ∈ M n the matrix f (A) is understood by means of the functional calculus. The space of density matrices or quantum states is as
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Jose A. Franco [email protected] Trung Hoa Dinh [email protected] Raluca Dumitru [email protected]
1
Department of Mathematics, Troy University, Troy, AL 36082, USA
2
Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA
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T. H. Dinh et al.
Dn1 = {ρ ∈ Dn | Tr ρ = 1}. In [2], Audenaert introduced the concept of “in-betweenness” and distance monotonicity for matrix means as follows. A matrix mean σ is said to satisfy the in-betweenness property with respect to the metric d if for any pair of positive definite operators A and B, d(A, Aσ B) ≤ d(A, B). A weighted operator mean σt is said to satisfy the distance monotonicity property with respect to the metric d if for any pair of positive definite operators A and B, the function t → d(A, Aσt B) is monotone on [0, 1]. In the same article, he showed that the in-betweenness property is not stronger than the distance monotonicity for the matrix power means as defined by Bhagwat and Subramaninan [5], μ p (t; A, B) := (t A p + (1 − t)B p )1/ p . Using this comparison he showed that the weighted power means satisfy the inbetweenness property when 1 ≤ p ≤ 2 and 0 ≤ t ≤ 1. He later conjectured that this property should be satisfied for p ≥ 2. However, in [8] we constructed counterexamples for p = 6. Moreover, we showed that the weighted power means satisfy the in-betweenness for p = 1/2 and p = 1/4. Interestingly, in the case of p = 1/2, the property is satisfied with respect to any metric induced from a unitarily invariant norm, i.e., d(A, B) = |||A − B|||. In [10] the first author and co-authors introduced the in-sphere property for matrix means with respect to some distance function
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