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

Abstract

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $C^{\ast }$-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 $C^{\ast }$-algebra $\mathcal{A}$ which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $\mathcal{S}$ of a possibly infinite-dimensional, unital $C^{\ast }$-algebra $\mathcal{A}$ is partitioned into the disjoint union of the orbits of an action of the group $\mathcal{G}$ of invertible elements of $\mathcal{A}$. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $\mathcal{H}$ are smooth, homogeneous Banach manifolds of $\mathcal{G}=\mathcal{G}\mathcal{L}(\mathcal{H})$, and, when $\mathcal{A}$ 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 $\mathcal{G}$.

35 pages. Revised version in which some imprecise statements have been amended. Comments are welcome!

References in corpus (4)

• Dynamical Vector Fields on the Manifold of Quantum States
• Quantum Statistical Manifolds
• Stratified Manifold of Quantum States, actions of the complex special linear group
• Geometry of quantum dynamics in infinite dimension

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• Group actions and monotone metric tensors: The qubit case