On the geometry of mixed states and the Fisher information tensor

TL;DR

This paper explores the geometric structure of mixed quantum states using co-adjoint orbits, clarifies the Fisher information tensor's properties, and investigates the symplectic form and fibration of these states.

Contribution

It provides a geometric interpretation of the quantum Fisher information tensor and clarifies the symmetric logarithmic derivative within the co-adjoint orbit framework.

Findings

The antisymmetric part of the Fisher information tensor is the pullback of the Kostant-Kirillov-Souriau symplectic form.

The symmetric logarithmic derivative is properly defined as a 1-form and explicitly computed.

The fibration of co-adjoint orbits as spaces of mixed states is analyzed.

Abstract

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.