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

Abstract

The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

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]…”

“…According to the work in [41], the gradient vector fields provide an overcomplete basis of the tangent space at each point in every orbit O. Furthermore, on every O we may define a Riemannian metric tensor G given by…”

“…However, G is not invariant under the action of all of G . The metric tensor G turns out to be the C * -algebraic version of some well-known and relevant metric tensors when explicit cases are considered [41]. For instance, if A = C n and O = ∆ + n , then G coincides with the Fisher-Rao metric tensor.…”

“…The geodesic ν v ρ (t) remains inside the space of states S for all t ∈ R, but it also exits and enters the orbit O containing the initial state ρ at multiple times [41].…”

“…The use of -algebras as a theoretical framework to study the geometry of quantum states is not new [ 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 ]. However, the focus was essentially always on the algebra of bounded linear operators on the Hilbert space of the quantum system, and not on a generic -algebra.…”