The f↔f˜ Correspondence and Its Applications in Quantum Information Geometry

Abstract

Due to the classifying theorems by Petz and Kubo–Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive half line; this last class is usually denoted as Fop. This class of operator monotone function has a significant structure, which is worthy of study; indeed, any step in understanding Fop, besides being interesting per se, immediately translates into a property of the classes of operator means and therefore of Quantum Fisher Information(s). In recent years, the f↔f correspondence has been introduced, which associates a non-regular element of Fop to any regular element of the same set. In terms of operator means, this amounts to associating a mean with multiplicative character to a mean that has an additive character. In this paper, we survey a number of different settings where this technique has proven useful in Quantum Information Geometry. In Sections 1–4, all the needed background is provided. In Sections 5–14, we describe the main applications of the f↔f˜ correspondence.