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

publication date

  • December 2019

start page

  • 231

end page

  • 271

issue

  • 2

volume

  • 2

International Standard Serial Number (ISSN)

  • 2511-2481

Electronic International Standard Serial Number (EISSN)

  • 2511-249X

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 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.

keywords

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