### authors

- Ciaglia, F. M.
- IBORT LATRE, LUIS ALBERTO
- JOST, J.
- MARMO, GIUSEPPE

- INFORMATION GEOMETRY Journal

- December 2019

- 231

- 271

- 2

- 2

- 2511-2481

- 2511-249X

- 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 A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S of a possibly infinite-dimensional, unital C∗-algebra A is partitioned into the disjoint union of the orbits of an action of the group G of invertible elements of A. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H are smooth, homogeneous Banach manifolds of G=GL(H), and, when 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 G.

- probability distributions; quantum states; c∗-algebras; banach manifolds; homogeneous spaces