Generalized Hellinger divergences generated by monotone functions 

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,
        \[
        \Phi_{g,\sigma}(A,B)
        \ge
        \Phi_{g,\sigma}(\Phi(A),\Phi(B)).
        \]