Generalized Hellinger divergences generated by monotone functions
Người báo cáo: Prof. Hiroyuki Osaka, Ritsumeikan university (Japan)
Time: 9:30 - 10:30, 12 March 2026 (Thursday)
Venue: Room 507, A6, Institute of Mathematics
Abstract: In this talk we discuss quantum Hellinger-type divergences which were studied by Bhatia--Gaubert--Jain (2019), Pitrik--Virosztek (2020), and Dinh--Lie--Osaka--Phan (2025).
In particular, when $g:[0,\infty)\to[0,\infty)$ is a convex function of the form
\[
g(t)=\alpha t^{s}, \qquad \alpha>0,\; s\in[1,2],
\]
and $f:[0,\infty)\to[0,\infty)$ is an operator monotone function satisfying $f'(1)=\lambda\in[0,1]$, we introduce the quantum quantity
\[
\Phi_{g,\sigma}(A,B)
=
\operatorname{Tr}\bigl(g(A\nabla_{\lambda}B - A\sigma_f B)\bigr)
\]
for positive definite matrices $A$ and $B$.
We show that $\Phi_{g,\sigma}$ is a quantum divergence in the sense of Bhatia--Gaubert--Jain. Moreover, it is jointly convex and satisfies the data processing property for any trace-preserving positive unital map $\Phi$, that is,
\[
\Phi_{g,\sigma}(A,B)
\ge
\Phi_{g,\sigma}(\Phi(A),\Phi(B)).
\]
