On estimating some distances involving operator entropies via Riemannian metric

Document Type : Original Article

Authors

Mohamed Chergui ¹

Abdenbi El Hilali ²

Bouazza El Wahbi ¹

¹ Department of Mathematics, CRMEF Rabat-Sale-Kenitra, EREAM Team, LaREAMI-Lab, Kenitra 14000, Morocco

² Department of Mathematics, Faculty of Sciences, Ibn Tofail University, LAGA-Lab Kenitra, Morocco.

10.22034/kjm.2021.260901.2082

Abstract

In this paper, we focus on geometric properties for relative operator entropy and its extensions for positive definite matrices by considering Riemannian metric. In particular, we prove that the Tsallis relative entropy $T_p(A|B)$ lies inside the sphere centered at the geometric mean of $A$ and $B$ with the radius equal to the half of the Riemannian distance between $A$ and $B$. Some numerical examples are given in the aim to verify the validity of the reverse of some results.

Keywords

Relative operator entropy

Parametric relative operator entropy

Tsallis relative operator entropy

Operator inequalities

---