Riemannian metrics on positive definite matrices related to means

This is a joint work with D.\ Petz and a continuation of the paper in LAA {\bf 430} (2009) with the same title. The $n\times n$ Hermitian matrices form the $n^2$-dimensional Euclidean space with respect to Hilbert-Schmidt inner product. The set $\mathbb{P}_n$ of all $n\times n$ positive definite matrices, being an open subset of $\mathbb{H}_n$, is naturally equipped with a $C^\infty$ manifold structure. An smooth kernel function $\phi:(0,\infty)\times(0,\infty)\to(0,\infty)$ induces a Riemannian metric $K^\phi$ on $\mathbb{P}_n$ defined by $$ K_D^\phi(H,K):=\sum_{i,j}\phi(\lambda_i,\lambda_j)^{-1}\mathrm{Tr}\, P_iHP_jK, \qquad D\in\mathbb{P}_n,\ H,K\in\mathbb{H}_n, $$ where $D=\sum_i\lambda_iP_i$ 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 $\phi$ of the form $M(x,y)^\theta$, a $\theta$ ($\in\mathbb{R}$)-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 $\mathbb{P}_n$ of this type.

(1) Since the Riemannian manifold $(\mathbb{P}_n,K^\phi)$ with $\phi=M(x,y)^\theta$ is complete if and only if $\theta=2$, the existence of geodesic shortest curves in the case $\theta\ne2$ 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 $\theta$ but independent of the choice of $M$. Moreover, we show the existence of a geodesic shortest curve joining $A,B\in\mathbb{P}_n$ for the metric $K^\phi$ with $\phi=M(x,y)^\theta$ if $\theta$ is sufficiently near $2$.

(2) We present a necessary and sufficient condition for Riemannian metrics $K^\phi$ and $K^\psi$ induced by $\phi=M(x,y)^\theta$ and $\psi=N(x,y)^\kappa$ to be isometric under the transformation $D\in\mathbb{P}_n\mapsto F(D)\in\mathbb{P}_n$ given by a smooth function $F:(0,\infty)\to(0,\infty)$. The condition is explicitly given in terms of $M,N,\theta$, and $\kappa$.

(3) From the above (2) we can construct a one-parameter isometric family of Riemannian metrics starting from any $K^\psi$ inside the set of Riemannian metrics we are treating. Those isometric families have different features between the cases $\kappa\ne2$ and $\kappa=2$. We see that each of those families converges to the metric $K^{M_{\mathrm{L}}^2}$ induced by the square of the logarithmic mean $M_{\mathrm{L}}(x,y):=(x-y)/(\log x-\log y)$. Thus $K^{M_{\mathrm{L}}^2}$ 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 $K^{M_{\mathrm{L}}^2}$ is $\gamma(t)=\exp((1-t)\log A+\log B)$ ($0\le t\le1$), this shows the Riemannian geometric interpretation for the limit formulas such as \begin{align*} \lim_{\alpha\to0}((1-t)A^\alpha+tB^\alpha)^{1/\alpha}&=\exp((1-t)\log A+\log B), \\ \lim_{\alpha\to0}(A^\alpha\,\#_t\,B^\alpha)^{1/\alpha}&=\exp((1-t)\log A+\log B), \end{align*} where $\#_t$ is the $t$-power mean ($0\le t\le1$).