# Canonical divergence for flat $\alpha$-connections: Classical and Quantum

## TL;DR

This paper explores a canonical divergence on manifolds with dualistic structures, showing it coincides with classical and quantum $ extalpha$-divergences in respective settings, bridging classical and quantum information geometry.

## Contribution

It demonstrates that the recent canonical divergence aligns with classical and quantum $ extalpha$-divergences on their respective manifolds, unifying these concepts.

## Key findings

- Canonical divergence matches classical $ extalpha$-divergence on positive measures.
- Canonical divergence coincides with quantum $ extalpha$-divergence on positive definite Hermitian operators.
- Provides a unified geometric framework for classical and quantum information divergences.

## Abstract

A recent canonical divergence, which is introduced on a smooth manifold $\mathrm{M}$ endowed with a general dualistic structure $(\mathrm{g},\nabla,\nabla^*)$, is considered for flat $\alpha$-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical $\alpha$-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum $\alpha$-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum $\alpha$-divergence.