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

Florio Maria Ciaglia, Alberto Ibort, Jürgen Jost, and Giuseppe Marmo

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Submission date: 01. Jul. 2019
Pages: 37
published in: Information geometry, 2 (2019) 2, p. 231-271 
DOI number (of the published article): 10.1007/s41884-019-00022-1
Bibtex
Keywords and phrases: Probability distributions, Quantum states, C*-algebras, Banach manifolds, homogeneous spaces
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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 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 τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ is a smooth, homogeneous Banach manifold for 𝒢 .