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

TL;DR

This paper introduces a geometric framework for quantum relative entropies, specifically the $q$-$z$ family, deriving coordinate-independent metric tensors that unify and generalize well-known quantum metrics.

Contribution

It develops an intrinsic, coordinate-free geometric formalism for quantum metrics derived from the $q$-$z$ relative entropies, extending to arbitrary finite-dimensional quantum systems.

Findings

Explicit coordinate-free expressions for quantum metrics like von Neumann-Umegaki, Bures, and Wigner-Yanase.

Unified geometric approach applicable to all $n$-level quantum systems.

Recovery of known quantum metrics as special cases within the $q$-$z$ family.

Abstract

The so-called $q$-z-\textit{R\'enyi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant $(0,2)$ symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic $n$-level system. By suitably varying the parameters $q$ and $z$, we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann-Umegaki metric, for the Bures metric and for the Wigner-Yanase metric in the arbitrary $n$-level case.