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

Ciaglia, Florio M.; Jost, Jürgen; Schwachhöfer, Lorenz

doi:10.3390/e22111332

Cited by 14 publications

(18 citation statements)

References 113 publications

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“…[8,20,23,39,57,60]) ubiquitous in quantum estimation theory (cf. [19,25,45,49,54,55,56]), while when κ = 1 4 , the monotone quantum metric tensor G f coincides with the Wigner-Yanase metric tensor [28,29,37,42]. In all these cases, the commutator between gradient vector fields is a fundamental vector field of α and, in general, it does not vanish so that the connection ∇ f presents torsion.…”

Section: G-dual Teleparallel Pairs In Quantum Information Geometrymentioning

confidence: 99%

G-dual teleparallel connections in Information Geometry

Ciaglia¹,

Cosmo²,

Ibort³

et al. 2022

Preprint

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Given a real, finite-dimensional, smooth parallelizable Riemannian manifold (N , G) endowed with a teleparallel connection ∇ determined by a choice of a global basis of vector fields on N , we show that the G-dual connection ∇ * of ∇ in the sense of Information Geometry must be the teleparallel connection determined by the basis of G-gradient vector fields associated with a basis of differential one-forms which is (almost) dual to the basis of vector fields determining ∇. We call any such pair (∇, ∇ * ) a G-dual teleparallel pair. Then, after defining a covariant (0, 3) tensor T uniquely determined by (N , G, ∇, ∇ * ), we show that T being symmetric in the first two entries is equivalent to ∇ being torsion-free, that T being symmetric in the first and third entry is equivalent to ∇ * being torsion free, and that T being symmetric in the second and third entries is equivalent to the basis vectors determining ∇ (∇ * ) being parallel-transported by ∇ * (∇). Therefore, G-dual teleparallel pairs provide a generalization of the notion of Statistical Manifolds usually employed in Information Geometry, and we present explicit examples of G-dual teleparallel pairs arising both in the context of both Classical and Quantum Information Geometry.

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Section: G-dual Teleparallel Pairs In Quantum Information Geometrymentioning

confidence: 99%

G-dual teleparallel connections in Information Geometry

Ciaglia¹,

Cosmo²,

Ibort³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Bures-Helstrom metric tensor is particularly relevant for all the information-theoretical tasks related with the quantum formulation of estimation theory [21,62,72,83,84,87] because it allows to give the lowest quantum version of the classical Cramer-Rao bound for unbiased estimators [36]. From a more geometrical point of view, G H BH is naturally connected with the concept of purification for quantum states [30,89,90], and with the Jordan product (anticommutator) among self-adjoint operators [22].…”

Section: The Bures-helstrom Metric Tensormentioning

confidence: 99%