Generalized Quantum Hellinger Divergences Generated by Monotone Functions

In this paper we investigate quantum Hellinger type divergences which were studied by Bhatia-Gaubert-Jain (2019), Pitrik-Virosztek (2020), and Dinh-Lie-Osaka-Phan (2025). In particular, when [Formula: see text] is a convex function defined by the form [Formula: see text] ([Formula: see text]) and [Formula: see text] is an operator monotone function with [Formula: see text], we introduce the quantum quantitative measure [Formula: see text] for positive definite matrices [Formula: see text] and [Formula: see text], and show that [Formula: see text] is a quantum divergence in the sense of Bhatia-Gaubert-Jain and also show that it is jointly convex and satisfies the data processing property by a trace preserving positive unital map [Formula: see text], that is, [Formula: see text].