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Search Results (2)

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Keywords = Bures–Helstrom metric tensor

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30 pages, 460 KiB  
Article
Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C∗-Algebras
by Florio M. Ciaglia, Jürgen Jost and Lorenz Schwachhöfer
Entropy 2020, 22(11), 1332; https://doi.org/10.3390/e22111332 - 23 Nov 2020
Cited by 7 | Viewed by 2779
Abstract
A geometrical formulation of estimation theory for finite-dimensional C∗-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with [...] Read more.
A geometrical formulation of estimation theory for finite-dimensional C∗-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented. Full article
(This article belongs to the Special Issue Quantum Statistical Decision and Estimation Theory)
27 pages, 441 KiB  
Article
From the Jordan Product to Riemannian Geometries on Classical and Quantum States
by Florio M. Ciaglia, Jürgen Jost and Lorenz Schwachhöfer
Entropy 2020, 22(6), 637; https://doi.org/10.3390/e22060637 - 8 Jun 2020
Cited by 15 | Viewed by 2370
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 [...] Read more.
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. Full article
(This article belongs to the Section Quantum Information)
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