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The so-called $q$-z-\textit{R\'enyi Relative Entropies} provide a hugetwo-parameter family of relative entropies which includes almost all well-knownexamples of quantum relative entropies for suitable values of the parameters.In this paper we consider a log-regularized version of this family and use itas a family of potential functions to generate covariant $(0,2)$ symmetrictensors on the space of invertible quantum states in finite dimensions. Thegeometric formalism developed here allows us to obtain the explicit expressionsof such tensor fields in terms of a basis of globally defined differentialforms on a suitable unfolding space without the need to introduce a specificset of coordinates. To make the reader acquainted with the intrinsic formalismintroduced, we first perform the computation for the qubit case, and then, weextend the computation of the metric-like tensors to a generic $n$-levelsystem. By suitably varying the parameters $q$ and $z$, we are able to recoverwell-known examples of quantum metric tensors that, in our treatment, appearwritten in terms of globally defined geometrical objects that do not depend onthe coordinates system used. In particular, we obtain a coordinate-freeexpression for the von Neumann-Umegaki metric, for the Bures metric and for theWigner-Yanase metric in the arbitrary $n$-level case.

A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the $q$-$z$ Family