Speaker
Description
Abstract:
Many of the standard Riemannian structures in information geometry arise from a single source: the Hilbert space provided by the Gelfand-Naimark-Segal (GNS) construction for states on a C*-algebra. I will explain how the Fisher-Rao metric (classical models), the Fubini-Study/quantum geometric tensor (pure quantum states), and the Bures-Helstrom metric (faithful quantum states in finite dimensions) can all be obtained as pullbacks of the same GNS-induced geometry along a statistical model. The same viewpoint naturally yields an associated 2-form that vanishes in the commutative case, reduces to the standard Berry curvature/symplectic form for pure quantum states, and measures the commutativity of symmetric logarithmic derivatives for faithful quantum states in finite dimensions.