# Manifolds of classical probability distributions and quantum density operators in infinite dimensions

## TL;DR

This paper explores the geometric structure of classical and quantum states in infinite-dimensional settings using $C^{*}$-algebras, showing that their state spaces form smooth Banach manifolds under certain group actions.

## Contribution

It characterizes the manifold structure of state spaces of $C^{*}$-algebras, including classical and quantum states, as smooth Banach manifolds under the action of invertible elements.

## Key findings

- Orbits of density operators form smooth, homogeneous Banach manifolds.
- The orbit through a faithful tracial state is a smooth, homogeneous Banach manifold.
- The structure applies to infinite-dimensional, separable Hilbert spaces.

## Abstract

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $C^{*}$-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $C^{*}$-algebra $\mathscr{A}$ which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $\mathscr{S}$ of a possibly infinite-dimensional, unital $C^{*}$-algebra $\mathscr{A}$ is partitioned into the disjoint union of the orbits of an action of the group $\mathscr{G}$ of invertible elements of $\mathscr{A}$. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $\mathcal{H}$ are smooth, homogeneous Banach manifolds of $\mathscr{G}=\mathcal{GL}(\mathcal{H})$, and, when $\mathscr{A}$ admits a faithful tracial state $\tau$ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $\tau$ is a smooth, homogeneous Banach manifold for $\mathscr{G}$.