Parameter-free description of the manifold of non-degenerate density matrices

TL;DR

This paper develops a parameter-free framework for non-commutative Information Geometry on the manifold of non-degenerate density matrices, introducing exponential arcs, dual connections, and affine coordinates.

Contribution

It introduces a parameter-free approach to non-commutative Information Geometry, including exponential arcs, dual charts, and affine coordinates, enhancing the geometric understanding of density matrices.

Findings

Defined exponential arcs in the density matrix manifold

Established duality of m- and e-connections under the Bogoliubov metric

Introduced convex potentials and affine coordinates

Abstract

The paper gives a definition of exponential arcs in the manifold of non-degenerate density matrices and uses it as a starting point to develop a parameter-free version of non-commutative Information Geometry in the finite-dimensional case. Given the Bogoliubov metric the m- and e-connections are each other dual. Convex potentials are introduced. They allow to introduce dual charts. Affine coordinates are introduced at the end to make the connection with the more usual approach.