K\"ahler fibrations in quantum information theory

TL;DR

This paper explores the geometric structure of quantum information theory using K"ahler fibrations, revealing how the Fisher information tensor relates to K"ahler geometry on co-adjoint orbits of Lie groups.

Contribution

It introduces a K"ahler geometric framework for quantum information theory based on co-adjoint orbits, extending classical results to the quantum setting.

Findings

Fisher information tensor approximates a K"ahler structure on quantum orbits

Reformulation of quantum information geometry using fiber bundles

Extension of classical co-adjoint orbit results to quantum case

Abstract

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible K\"ahler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a K\"ahler structure to generalise some classical result on co-adjoint orbits.