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 $C^{\mathrm{\infty }}$ manifold structure. An smooth kernel function $\varphi :(0,\mathrm{\infty })\times (0,\mathrm{\infty })\to (0,\mathrm{\infty })$ induces a Riemannian metric $K^{\varphi }$ on ${\mathbb{P}}_n$ defined by $$K_D^{\varphi }(H,K):=\sum _{i,j}\varphi ({\lambda }_i,{\lambda }_j{)}^{-1}\mathrm{Tr}{P}_{i}H{P}_{j}K,D\in {\mathbb{P}}_n,H,K\in {\mathbb{H}}_n,$$ where $D=\sum _i{\lambda }_i{P}_i$ 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 $\varphi$ 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 $({\mathbb{P}}_n,K^{\varphi })$ with $\varphi =M(x,y{)}^{\theta }$ is complete if and only if $\theta =2$, the existence of geodesic shortest curves in the case $\theta \ne 2$ 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 $K^{\varphi }$ with $\varphi =M(x,y{)}^{\theta }$ if $\theta$ is sufficiently near $2$.

(2) We present a necessary and sufficient condition for Riemannian metrics $K^{\varphi }$ and $K^{\psi }$ induced by $\varphi =M(x,y{)}^{\theta }$ 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 $F:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$. 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 $\kappa \ne 2$ 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 $\gamma (t)=\exp ((1-t)\log A+\log B)$ ($0\le t\le 1$), this shows the Riemannian geometric interpretation for the limit formulas such as $$\begin{array}{rl}\underset{\alpha \to 0}{lim}((1-t)A^{\alpha }+t{B}^{\alpha }{)}^{1/\alpha }& =\exp ((1-t)\log A+\log B),\\ \underset{\alpha \to 0}{lim}(A^{\alpha }{\mathrm{\#}}_t{B}^{\alpha }{)}^{1/\alpha }& =\exp ((1-t)\log A+\log B),\end{array}$$ where ${\mathrm{\#}}_t$ is the $t$-power mean ($0\le t\le 1$).