arXiv subjects

Laura González-Bravo

Publications and source records attributed to Laura González-Bravo.

5 recordsLinked to original sources

arXiv2026

Fields of covariances on non-commutative probability spaces in finite dimensions

We introduce the notion of a field of covariances, a contravariant functor from non-commutative probability spaces to Hilbert spaces, as a categorical analogue of statistical covariance. In the finite-dimensional setting, we obtain a complete characterization of the fields of covariances satisfying an additional continuity property in terms of a strictly positive constant $α$ and a continuous function $F\colon[0,+\infty)\rightarrow(0,+\infty)$ that is operator monotone on $(0,+\infty)$. In the commutative case, our result recovers a contravariant formulation of Čencov's characterization of the Fisher--Rao metric tensor, and extends the corresponding uniqueness statement to tracial states on possibly noncommutative algebras. On faithful states, a subfamily satisfying a symmetry constraint recovers the regular Morozova--Čencov--Petz monotone metric tensors. More generally, the covariance fields induce positive contravariant symmetric tensors on the homogeneous manifolds of states generated by the action of the invertible elements of an arbitrary finite-dimensional $C^*$-algebra. The geometry internal to the support is determined by the positive spectrum of the modular operator, whereas support-changing directions are governed by $F(0)$. For $F(t)=(1+t)/2$, and with a suitable normalization convention, the resulting tensors recover the inverse of the Riemannian metrics induced by the Jordan product. In particular, on pure states, every $F$ induces a multiple of the inverse Fubini--Study tensor, with coefficient $2F(0)$ in our normalization, in agreement after inversion with the Petz--Sudár radial extension.

math-ph↗

arXiv2026

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

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 $Ω$. 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 $Ω$ 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 $Ω$ 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 $Ω$ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.

math-ph↗

arXiv2025

Towards a category-theoretic foundation of Classical and Quantum Information Geometry

We introduce the category $\mathsf{NCP}$, whose objects are pairs of W$^\ast$-algebras and normal states and whose morphisms are state-preserving unital completely positive (CPU) maps, as a common stage for classical and quantum information geometry, and we formulate two results that will appear in forthcoming works. First, we recast the problem of classifying admissible Riemannian geometries on classical and quantum statistical models in terms of functors $\mathfrak{C}:\mathsf{NCP}\to\mathsf{Hilb}$.These functors provide a generalization of classical statistical covariance, and we call them fields of covariances. A prominent example being the so-called GNS functor arising from the Gelfand-Naimark-Segal (GNS) construction. The classification of fields of covariances on $\mathsf{NCP}$ entails both Čencov's uniqueness of the Fisher-Rao metric tensor and Petz's classification of monotone quantum metric tensors as particular cases. Then, we show how classical and quantum statistical models can be realized as subcategories of $\mathsf{NCP}$ in a way that takes into account symmetries. In this setting, the fields of covariances determine Riemannian metric tensors on the model that reduce to the Fisher-Rao, Fubini-Study, and Bures-Helstrom metric tensor in particular cases.

math-ph↗

arXiv2023

On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality

An extension of Cencov's categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theory in a natural way, while also allowing to formalise the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer-Rao inequality in this setting.

quant-ph↗

arXiv2023

Can Čencov meet Petz?

We discuss how to exploit the recent formulation of classical and quantum information geometry in terms of normal states on $W^{*}$-algebras to formulate a problem that unifies Cencov's theorem and Petz's theorem.

math-ph↗