Date: 12 March  2026 (Thursday)

Venue: Room 507, A6, Institute of Mathematics, 18 Hoàng Quốc Việt

Time: 9:30 - 12:00

9:30 - 10:30: Speaker: Prof. Hiroyuki Osaka,  Ritsumeikan university (Japan)

Tiêu đề: Generalized Hellinger divergences generated by monotone functions 

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

11h00 - 12h00

Speaker: Prof. Dinh Trung Hoa, Troy University (USA)

Tiêu đề:  A Weighted Spectral Quantum Fidelity (this is a joint work with Cong Trinh Le, Minh Toan Ho and The Khoi Vu.)

Tóm tắt: We introduce and study a one-parameter family of fidelity-type quantities based on the weighted spectral geometric mean, which we call the \emph{weighted spectral fidelity}
    \[
    F_{\mathrm{spec}}^{t}(\rho,\sigma)
    :=
    \operatorname{Tr}\!\left[\rho(\rho^{-1}\#\sigma)^{2t}\right],
    \qquad t\in[0,1].
    \]
This family interpolates smoothly between the trivial overlap $(t=0,1)$ and the Uhlmann (root) fidelity at $t=\tfrac12$, and it is distinct from the sandwiched Rényi family except at this midpoint.

We establish core structural features such as unitary invariance, tensor stabilization and multiplicativity, flip symmetry, endpoint behavior, and an orthogonality criterion. We further show explicit violations of the data processing inequality (DPI) for generic $t\neq\tfrac12$.

For concavity in the state variables we obtain concavity in each variable separately. Closed forms are obtained for pure states and for qubits in Bloch coordinates.

We also extend the first Fuchs--van de Graaf inequality to    $F_{\mathrm{spec}}^{t}$ for all $t\in[0,1]$, while the second inequality fails away from the midpoint.