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 * -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 .

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

“…The space is not a smooth manifold in the usual sense of differential geometry (see [ 59 , 60 ] for the appropriate definition of smooth manifold). However, there is a linear left action of the Lie group of invertible elements in on the space given by (see [ 29 , 30 , 31 ]) This action preserves , and every orbit of is a smooth submanifold of . That is, even though itself does not have a smooth structure, it is stratified by the orbits of this action all of which are homogeneous spaces and hence smooth manifolds; in fact, the manifold structure of these homogeneous spaces coincides with that induced by the inclusion into as the action of is defined on all of .…”

“…Now, we define the vector fields by means of Quite interestingly, a direct computation shows that which means that we also have Comparing Equation ( 26 ) with Equation ( 16 ), we conclude that the vector fields provide a representation of the Lie algebra of which is tangent to . Furthermore, if we define the map given by (see also [ 29 , 30 , 31 ]) it is not hard to show that it is a left action of on which is transitive (essentially because is transitive on ). In particular, the space of faithful states is an orbit of .…”

“…In [ 29 , 30 , 31 ], it has been shown that the relevant manifolds of quantum states may be described as submanifolds of homogeneous spaces of the group of invertible operators on . Specifically, if denotes a self-adjoint operator on , the group acts on the space of self-adjoint operators according to the map , where † denotes the adjoint operator on .…”

Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C∗-Algebras

“…as well as for its restrictions to the various submanifolds of V we will introduce below (with an evident abuse of notation). There is a group action of G on S given by [41,68]…”

“…In the infinite-dimensional case, the technical difficulties would often obscure the conceptual aspects, and this unavoidably leads to be less communicative. Moreover, it is even not yet clear what are the geometrical players on the fields when infinite dimensions are considered because there is no general consensus on which are the most appropriate manifolds of states to consider in this case (see the works in [ 4 , 23 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 ] for some examples).…”