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Florio M. Ciaglia

Publications and source records attributed to Florio M. Ciaglia.

At least 19 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

A two-point approach to the inverse problem in information geometry

We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affine structure, realized through a local parallelism obtained from parallel transport. This yields a contrast bi-form inducing the original metric-affine manifold. The inverse problems for statistical manifolds admitting torsion and for statistical manifolds are then recovered by homotopical reduction. In this way, suitable pre-contrast and contrast functions are obtained, including several established constructions for statistical manifolds and SMATs. We apply the general theory to reductive homogeneous pseudo-Riemannian manifolds endowed with invariant affine connections. Particular attention is devoted to semisimple Lie groups with Cartan-Schouten connections and to odd-dimensional spheres equipped with Berger metrics.

math.DG↗
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↗
arXiv2026

Potential functions in information geometry via bi-forms

In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

math.DG↗
arXiv2025

Bi-forms Approach to Potential Functions in Information Geometry

Contrast functions play a fundamental role in information geometry, providing a means for generating the geometric structures of a statistical manifold: a pseudo-Riemannian metric and a pair of torsion-free conjugate affine connections. Conventional contrast-based approaches become indeed insufficient within settings where torsion is naturally present, such as quantum information geometry. This paper introduces contrast bi-forms, a generalisation of contrast functions that systematically encode metric and connection data, allowing for arbitrary affine connections regardless of torsion. It will be shown that they provide a unified framework for statistical potentials, offering new insights into the inverse problem in information geometry. As an example, we consider teleparallel manifolds, where torsion is intrinsic to the geometry, and show how bi-forms naturally accommodate these structures.

math.DG↗
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

The Geometry of the solution space of first order Hamiltonian field theories III: Palatini's formulation of General Relativity

We complete the program started in two companion papers of defining a Poisson bracket structure on the space of solutions of the equations of motion of first order Hamiltonian field theories. The case of General Relativity is addressed by looking at it as a particular non-Abelian gauge theory in a suitable low-energy limit and via a technique related to the coisotropic embedding theorem.

math-ph↗
arXiv2023

The Geometry of the solution space of first order Hamiltonian field theories I: from particle dynamics to free Electrodynamics

We analyse the problem of defining a Poisson bracket structure on the space of solutions of the equations of motions of first order Hamiltonian field theories. The cases of Hamiltonian mechanical point systems (as a (0 + 1)-dimensional field) and more general field theories without gauge symmetries are addressed by showing the existence of a symplectic (and, thus, a Poisson) structure on the space of solutions. Also the easiest case of gauge theory, namely free electrodynamics, is considered: within this problem, a pre-symplectic tensor on the space of solutions is introduced, and a Poisson structure is induced in terms of a flat connection on a suitable bundle associated to the theory.

math-ph↗
arXiv2023

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

math.DG↗
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

Monotone metric tensors in Quantum Information Geometry

We review some geometrical aspects pertaining to the world of monotone quantum metrics in finite dimensions. Particular emphasis is given to an unfolded perspective for quantum states that is built out of the spectral theorem and is naturally suited to investigate the comparison with the classical case of probability distributions.

quant-ph↗
arXiv2023

G-dual teleparallel connections in Information Geometry

Given a real, finite-dimensional, smooth parallelizable Riemannian manifold $(\mathcal{N},G)$ endowed with a teleparallel connection $\nabla$ determined by a choice of a global basis of vector fields on $\mathcal{N}$, we show that the $G$-dual connection $\nabla^{*}$ of $\nabla$ 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 $\nabla$. We call any such pair $(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair. Then, after defining a covariant $(0,3)$ tensor $T$ uniquely determined by $(\mathcal{N},G,\nabla,\nabla^{*})$, we show that $T$ being symmetric in the first two entries is equivalent to $\nabla$ being torsion-free, that $T$ being symmetric in the first and third entry is equivalent to $\nabla^{*}$ being torsion free, and that $T$ being symmetric in the second and third entries is equivalent to the basis vectors determining $\nabla$ ($\nabla^{*}$) being parallel-transported by $\nabla^{*}$ ($\nabla$). 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.

math-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↗
arXiv2022

Parametric models and information geometry on W*-algebras

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W*-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on parametric models satsfying suitable regularity conditions. This Riemannian metric tensor reduces to the Fisher-Rao metric tensor, or to the Fubini-Study metric tensor, or to the Bures-Helstrom metric tensor when suitable choices for the W*-algebra and the models are made.

math-ph↗
arXiv2022

Group actions and Monotone Quantum Metric Tensors

The interplay between actions of Lie groups and monotone quantum metric tensors on the space of faithful quantum states of a finite-level system observed recently in DOI: 10.1140/epjp/s13360-020-00537-y and DOI: 10.1007/978-3-030-80209-7_17 is here further developed.

quant-ph↗
arXiv2022

Dynamical maps and symmetroids

Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.

math-ph↗