Publication: Manifolds of classical probability distributions and quantum density operators in infinite dimensions
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2019-12-01
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Springer
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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∗-algebraA which is Abelian for Classical states, and non-Abelian
for Quantum states. In this contribution, the space of states S of a possibly infinitedimensional,
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 .
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Probability distributions, Quantum states, C*-algebras, Banach manifolds, Homogeneous spaces
Bibliographic citation
Ciaglia, F. M., Ibort, A., Jost, J., & Marmo, G. (2019). Manifolds of classical probability distributions and quantum density operators in infinite dimensions. Information Geometry, 2 (2), pp. 231-271.