The real and imaginary parts of the dual GNS Hermitian product, pulled back by a canonical lift, yield the Fisher–Rao, Fubini–Study, and SLD geometries plus a two-form whose closedness is governed by the covariant derivative of the lift.
Stratified manifold of quantum states, actions of the complex special linear group
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Metric tensors and two-forms in information geometry from the GNS construction
The real and imaginary parts of the dual GNS Hermitian product, pulled back by a canonical lift, yield the Fisher–Rao, Fubini–Study, and SLD geometries plus a two-form whose closedness is governed by the covariant derivative of the lift.