Metric tensors and two-forms in information geometry from the GNS construction

Abstract

We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over $C^*$-algebras. Since the state space of a $C^*$-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor $K$ on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor $G$ and a smooth two-form $\Omega$. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher--Rao metric in the commutative dominated case, the Fubini--Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form $\Omega$ is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that $\Omega$ need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of $\Omega$ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.