Workshop

Riemannian metrics on positive definite matrices related to means

  • Fumio Hiai (Tohoku University, Japan)
University n.n. Universität Leipzig (Leipzig)

Abstract

This is a joint work with D.\ Petz and a continuation of the paper in LAA {\bf 430} (2009) with the same title. The n×n Hermitian matrices form the n2-dimensional Euclidean space with respect to Hilbert-Schmidt inner product. The set Pn of all n×n positive definite matrices, being an open subset of Hn, is naturally equipped with a C manifold structure. An smooth kernel function ϕ:(0,)×(0,)(0,) induces a Riemannian metric Kϕ on Pn defined by KDϕ(H,K):=i,jϕ(λi,λj)1TrPiHPjK,DPn, H,KHn, where D=iλiPi is the spectral decomposition. Certain mportant quantities in quantum information geometry, such as statistical metric, quantum Fisher informations, and quantum variances, are Riemannian metrics arising from kernel functions ϕ of the form M(x,y)θ, a θ (R)-power of a symmetric homogenous mean M(x,y) of x,y>0. We discuss the following topics concerning geodesic shortest curves and geodesic distance of Riemannian metrics on Pn of this type.

(1) Since the Riemannian manifold (Pn,Kϕ) with ϕ=M(x,y)θ is complete if and only if θ=2, the existence of geodesic shortest curves in the case θ2 does not seem obvious. When A,B are commuting, we present an explicit formula of a geodesic shortest curve between A,B that is depending on θ but independent of the choice of M. Moreover, we show the existence of a geodesic shortest curve joining A,BPn for the metric Kϕ with ϕ=M(x,y)θ if θ is sufficiently near 2.

(2) We present a necessary and sufficient condition for Riemannian metrics Kϕ and Kψ induced by ϕ=M(x,y)θ and ψ=N(x,y)κ to be isometric under the transformation DPnF(D)Pn given by a smooth function F:(0,)(0,). The condition is explicitly given in terms of M,N,θ, and κ.

(3) From the above (2) we can construct a one-parameter isometric family of Riemannian metrics starting from any Kψ inside the set of Riemannian metrics we are treating. Those isometric families have different features between the cases κ2 and κ=2. We see that each of those families converges to the metric KML2 induced by the square of the logarithmic mean ML(x,y):=(xy)/(logxlogy). Thus KML2 may be regarded as a (unique) attractor in the set of all Riemannian metrics of our discussion. From the fact that the geodesic shortest curve for the metric KML2 is γ(t)=exp((1t)logA+logB) (0t1), this shows the Riemannian geometric interpretation for the limit formulas such as limα0((1t)Aα+tBα)1/α=exp((1t)logA+logB),limα0(Aα#tBα)1/α=exp((1t)logA+logB), where #t is the t-power mean (0t1).

Links

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic