Statistical models play a central role whenever one wishes to describe, compare, and infer from incomplete information. In practice, observations are finite, noisy, and often indirect, and one works with structured families of admissible states depending on a finite number of parameters. Such models make it possible to perform estimation, prediction, uncertainty quantification, hypothesis testing, and experimental design. In physics they encode families of accessible states of a system; in statistics they organize the passage from data to inference; in information theory they provide a setting in which one can quantify how distinguishable nearby states are. For this reason, understanding the geometry naturally carried by statistical models is not merely a formal matter, but one that is closely tied to the operational content of inference and estimation.

A particularly fruitful way to study statistical models is through differential geometry. Once a smooth manifold whose points parametrize the relevant states is selected, classical and quantum information geometry provide systematic ways to endow the parameter manifold with a Riemannian metric [1, 2, 3, 4, 5], obtaining a local notion of statistical distance, typically related to the distinguishability of nearby points on the model. At the same time, the geometric structures arising in these two settings seem strikingly different, and understanding both their differences and their possible points of contact is one of the central themes behind this work.

The classical case is governed by the geometry of the Fisher–Rao metric tensor [6, 7, 5]. On smooth manifolds of strictly positive probability measures on a discrete and finite outcome space, the Fisher–Rao metric tensor is not merely one possible choice among many. In his seminal work, Čencov introduced a categorical framework in which he showed that this metric is uniquely characterized by its invariance under congruent embeddings induced by Markov morphisms [5]. This celebrated theorem shows that, for finite-dimensional classical systems, the probabilistic structure itself strongly constrains the available geometry. Related uniqueness results have been extended to broader infinite-dimensional settings [8, 9, 10], further reinforcing the distinguished role of the Fisher–Rao metric tensor in the classical case.

The quantum case, by contrast, exhibits a richer and more varied geometric landscape. On manifolds of positive-definite quantum states over a finite-dimensional Hilbert space \( \mathcal{H}_{n}\) , the uniqueness of the metric is lost. Instead of a single Riemannian geometry as in the classical case, Morozova and Čencov understood that quantum Markov morphisms are not enough to grant uniqueness [11], and Petz classified the family of admissible Riemannian geometries through a family of symmetric and normalized operator monotone functions [4]. All these Riemannian metric tensors are invariant under the action of the unitary group on quantum states, and monotone with respect to quantum Markov morphisms, i.e., completely-positive and trace-preserving (CPTP) maps. Among them, Symmetric Logarithmic Derivative (SLD) metric tensor is closely related to the Fisher information and to the quantum Cram
specialChar{39}er–Rao bound for quantum parameter estimation [3, 12, 13, 14, 15, 16].

A peculiar place is occupied by the Fubini–Study metric tensor on pure quantum states, which is the unique (up to a positive constant) Riemannian metric tensor invariant under the unitary group, both in finite and infinite dimensions [2, 17]. This uniqueness is reminiscent of the classical case [18], but appears in a completely different mathematical landscape, namely, that of Hilbert spaces.

Indeed, classical and quantum information geometry are traditionally developed with rather different mathematical languages. Classical information geometry is rooted in probability and measure theory, where systems are described using sample spaces, random variables, and probability distributions. Quantum information geometry, on the other hand, is rooted in operator theory on Hilbert spaces, where systems are described using Hilbert spaces, quantum observables (self-adjoint operators), and quantum states (density operators). Despite these differences, both theories are concerned with families of states and with their geometric structures encoding operational distinguishability. This suggests that the contrast between the classical and quantum settings need not prevent a common treatment. It is already well known how positive-operator-valued measures (POVMs) allow one to determine a classical statistical model from a quantum one [12, 3, 19], but POVMs provide a bridge between the two formalisms rather than a unifying mathematical framework.

On the other hand, the language of \( C^*\) -algebras provides a natural setting for classical and quantum information geometry to be formulated “simultaneously” [20, 21, 22, 23, 24, 25, 26]. In this setting, the focus is shifted from the classical outcome space or the quantum Hilbert space to a \( C^*\) -algebra \( \mathscr{A}\) , that is, a complex Banach algebra with a continuous anti-linear involution \( \dagger\) that satisfies the so-called \( C^*\) identity \( \Vert \mathbf{x}^{\dagger}\,\mathbf{x}\Vert = \Vert \mathbf{x}\Vert^{2}\) . Probability measures and quantum states are recovered as states on the algebra \( \mathscr{A}\) , that is, as bounded linear functionals in \( \mathscr{A}^*\) such that \( \rho(\mathbf{x}^{\dagger}\mathbf{x})\geq 0\) and \( \Vert\rho\Vert=1\) . The space of all states of \( \mathscr{A}\) is denoted by \( \mathcal{S}(\mathscr{A})\) and it is a convex set that is compact in the weak*-topology of \( \mathscr{A}^*\) whenever \( \mathscr{A}\) has an identity element. The classical case may be recovered using commutative algebras like \( \mathcal{L}^{\infty}(\Omega,\mu)\) where \( (\Omega,\mu)\) is a \( \sigma\) -finite measure space, or \( C_{0}(X)\) where \( X\) is a locally-compact Hausdorff space, in which cases \( \mu\) -absolutely continuous probability measures on \( (\Omega,\mu)\) and Radon measures on \( X\) are states in the \( C^*\) -algebraic sense. Standard quantum mechanical systems are recovered when \( \mathscr{A}=\mathcal{B}(\mathcal{H})\) , in which case density operators are states in the \( C^*\) -algebraic sense.

In particular, both classical and quantum parametric (statistical) models may be described by a triple \( (M,\mathrm{i},\mathscr{A})\) where \( M\) is a real smooth manifold, \( \mathscr{A}\) is a \( C^{*}\) -algebra, and \( \mathrm{i}\colon M\rightarrow\mathcal{S}(\mathscr{A})\) is the map determining the model [20]. From this perspective, one is led to ask whether apparently different geometric structures such as the Fisher–Rao, SLD, and Fubini–Study tensors may be understood as different realizations of a common background construction. A partial positive answer, valid in finite dimensions, is given in [21, 27], where these tensors are obtained from a coadjoint-orbit-like construction for the Jordan product of self-adjoint elements. That construction, however, presupposes a smooth ambient manifold of states and does not extend beyond finite dimensions.

The present paper differs from that approach in three structural respects. First, it replaces the ambient-manifold picture by the GNS fibration, so the construction never requires \( \mathcal S(\mathscr A)\) to be a manifold and remains meaningful when the state space is stratified or has corners. Second, it operates in genuine infinite-dimensional examples: for faithful states the relevant representative is the Hilbert–Schmidt amplitude rather than the possibly unbounded SLD operator, and the compatibility condition becomes an explicit Hilbert–Schmidt integrability requirement, verified below for displaced thermal states. Third, alongside the metric tensor it naturally produces a skew-symmetric two-form \( \Omega\) , whose closedness is analyzed structurally in Section 7.

It is worth noting that the problem of classifying admissible Riemannian geometries on statistical models can be recast in a categorical language reminiscent of Čencov’s original works [5, 11]. In recent work [28], the notion of a field of covariances has been introduced as a contravariant functor from a category of non-commutative probability spaces to Hilbert spaces. Such functors generalize the classical statistical covariance, and their classification in finite dimensions recovers both Čencov’s uniqueness theorem and the Morozova–Čencov–Petz classification of monotone quantum metrics as particular instances, providing also an extension to non-faithful states that are not pure generalizing the radial procedure in [29]. Among all fields of covariances, the one arising from the Gelfand–Naimark–Segal (GNS) construction is distinguished: it associates to each state \( \rho\) the GNS Hilbert space \( \mathcal{H}_\rho\) with its canonical inner product \( \langle\cdot,\cdot\rangle_\rho\) , and it corresponds, in the Morozova–Čencov–Petz classification, to the SLD metric. The present paper may be understood as a detailed geometric study of this particular field of covariances from a dual perspective: we develop the pullback construction through which the dual GNS inner product induces geometric tensors on parametric models, and we work out its consequences in the classical, pure-state, and faithful quantum settings. The extension of this programme to other fields of covariances is a natural direction for future work.

The paper provides a GNS-based unifying mechanism for several central geometries of classical and quantum information geometry that is valid also for parametric models over infinite-dimensional \( C^{*}\) -algebras, when the regularity assumptions are verified. Our main idea is that familiar geometric structures in classical and quantum information geometry can be traced back to a single mathematical object, namely, the (dual of the) GNS inner product. Seen from this angle, the metrics mentioned above do not appear as isolated constructions, but rather as different geometric manifestations of the same operator-algebraic background structure. The idea is very close to the standard construction of pull-back metrics in differential geometry, where a Riemannian metric tensor is pulled back along a smooth immersion.

However, a basic obstacle appears immediately. The state space \( \mathcal S(\mathscr{A})\) is not a smooth manifold. Even in finite dimensions, it is either a manifold with boundary when \( \mathscr{A}\cong \mathbb{C}^{2}\) or \( \mathscr{A}\cong \mathcal{M}_{2}(\mathbb{C})\cong\mathcal{B}(\mathbb{C}^{2})\) , or a manifold with corners when \( \mathscr{A}\cong \mathbb{C}^{n}\) with \( n>2\) , or a stratified space whose strata are smooth manifolds [30, 31, 32]. Because of this, the usual differential-geometric procedure of pulling back covariant tensor fields is not directly available. This leads us to look for another geometric object attached to each state and varying coherently enough across the state space to support a pullback construction along statistical models.

The GNS construction suggests such an object. To each state \( \rho\in\mathcal{S}(\mathscr{A})\) it associates a Hilbert space \( \mathcal{H}_\rho\) and its dual \( \mathcal{H}_{\rho}^{*}\) . These Hilbert spaces glue together to give rise to a non-locally-trivial Hilbert fibration in the sense of [33, § II.13.14] (see Proposition 1).

This point of view gives us a geometric environment that remains workable even though the state space is not a manifold. If a statistical model \( (M,\mathrm{i},\mathscr{A})\) satisfies additional regularity conditions (see Definitions 2 and 3), we can pull back the real and imaginary parts of the dual GNS Hermitian structure to define a smooth weak Riemannian metric tensor \( G\) and a smooth \( 2\) -form \( \Omega\) on \( M\) . When \( M\) is finite-dimensional, the weak Riemannian metric is automatically strong.

A relevant feature of this construction is that it recovers, within the same general mechanism, several of the fundamental geometries of information theory, as explained in sections 4, 5, and 6. In the classical commutative case, the regularity of the model is related to the existence of an \( \mathcal{L}^{2}\) -integrable score function, and the induced metric tensor is the Fisher–Rao metric tensor, while the \( 2\) -form identically vanishes. For pure quantum states, the construction yields the Fubini–Study metric tensor together with its symplectic form up to normalization and sign/order convention. For faithful states on \( \mathcal B(\mathcal{H})\) , the regularity of the model is related to the existence of a Hilbert–Schmidt amplitude associated with the weak SLD equation, which in finite dimensions reduces to the usual SLD and the induced metric is the SLD metric (equivalently, four times the Bures-Helstrom metric) [13, 15]. Moreover, for finite-dimensional faithful models, the two-form is proportional to the expected commutator of the SLD representatives, also known as mean Uhlmann curvature [34, 35, 36, 37, 38, 39] (see 13). In particular, models with pairwise commuting SLDs are isotropic for \( \Omega\) , although the converse need not hold (see remark 12).

The behaviour of the two-form \( \Omega\) is sensitive to the model. It vanishes in the commutative case (see Section 4); it is closed (and symplectic) in the case of pure quantum states (see 5); but there are models of faithful quantum states in which it is not closed (see Section 6). In section 7, we give a structural explanation of this phenomenon under additional regularity assumptions on the model (see Definition 5). The idea is to lift the analysis to the total space of the realified dual GNS bundle, where the fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, in analogy with the finite-dimensional theory of closed forms on symplectic fibrations [40]. The closedness of the two-form on the model is controlled by the covariant exterior derivative of the lift used to define the two-form.

The paper is organized as follows. Section 2 recalls the GNS construction and introduces the GNS fibration together with its dual. Section 3 defines GNS-smooth and regular parametric models, constructs the real GNS lift, the real dual lift, and the induced Hermitian tensor \( K\) , with real and imaginary parts \( G\) and \( \Omega\) . Sections 4, 5, and 6 analyze the main examples: commutative dominated models, pure states, and faithful quantum states, including faithful qubits and displaced thermal states. Finally, Section 7 studies the closedness problem for \( \Omega\) by introducing the stronger bundle-regularity condition and by separating closed extensions on the real dual GNS bundle from closedness of the induced two-form on the parameter manifold.

We now build two non-locally trivial Hilbert fibrations over the space of states \( \mathcal{S}(\mathscr{A})\) in the sense of [33, II.13 and II.14] Differently from [33], we use the term fibration rather than bundle to avoid suggesting local triviality that is often associated to the word bundle in differential geometric contexts.. The starting point is the GNS construction associated with a state we now briefly recall.

Let \( \mathscr{A}\) be a \( C^{*}\) -algebra and \( \rho: \mathscr{A}\to \mathbb{C}\) a state on \( \mathscr{A}\) . The Gelfand-Naimark-Segal (GNS) construction associates to the pair \( (\mathscr{A},\rho)\) a triple \( (\mathcal{H}_\rho,\pi_{\rho}, \Omega^{\rho})\) , where \( \mathcal{H}_\rho\) is a complex Hilbert space called the GNS Hilbert space of \( \rho\) , \( \pi_{\rho}\) is a \( *\) -homomorphism of \( \mathscr{A}\) in \( \mathcal{B}(\mathcal{H}_\rho)\) called the GNS representation of \( \mathscr{A}\) associated with \( \rho\) , and \( \Omega^{\rho}\in\mathcal{H}_\rho\) is a vector that is cyclic with respect to \( \pi_{\rho}\) (which means that \( \mathrm{span}\left\{\pi_{\rho}(a)\Omega^{\rho}\mid a\in\mathscr{A}\right\}\) is dense in \( \mathcal{H}_\rho\) ). Specifically, the GNS Hilbert space of \( \rho\) is defined as

where

is the so-called GNS ideal of \( \rho\) , and the closure is taken with respect to the inner product

which is well defined because it does not depend on the representative of the classes. We denote by \( \psi_a^\rho\) the equivalence class \( [a]_\rho\) when immersed in the closure \( \mathcal{H}_\rho\) . Generic elements of \( \mathcal{H}_\rho\) will be denoted by \( \psi^{\rho} , \eta^{\rho} ,\xi^{\rho} \) , etc. The GNS representation is defined by

on \( \mathscr{A}/\mathcal{N}_{\rho}\) , and then extended by continuity. In particular, if \( \mathscr A\) has an identity element \( \mathbb I\) , then the cyclic vector \( \Omega^\rho\) is just \( \psi^\rho_{\mathbb I}\) . If \( \mathscr A\) is nonunital, one may either pass to the unitization, or, equivalently, obtain the cyclic vector as the Hilbert-space limit of \( \psi^\rho_{e_\lambda}\) for any approximate identity \( (e_\lambda)_\lambda\) of \( \mathscr A\) ; the resulting vector is independent of the chosen approximate identity. In the present paper, however, the construction of the GNS fibration only uses the vectors \( \psi^\rho_a\) with \( a\in\mathscr A\) .

Sometimes, one practically starts from a set-theoretic surjection \( p\colon A\rightarrow X\) for which each fiber is a Banach space, and would like to build a Banach fibration over \( X\) in the sense of Definition 1. It turns out such a procedure is possible once a suitably regular set of sections is introduced.

We now exploit Theorem 1 to define the GNS fibration over \( \mathcal{S}(\mathscr{A})\) and its dual. We start with the set-theoretical disjoint union

equipped with the natural projection

and its dual

with the projection

For each \( a\in\mathscr{A}\) , we define the set-theoretical section of \( (\mathcal{H}(\mathscr{A}),p,\mathcal{S}(\mathscr{A}))\) given by

We shall call these sections tautological sections of \( \mathcal{H}(\mathscr{A})\) . Analogously, for each \( a\in\mathscr{A}\) , we can also define the dual section

where \( R_\rho\) is the conjugate-linear Riesz identification

Equivalently, for every \( \xi\in\mathcal{H}_\rho\) one has

We shall call the sections \( \Psi_a^\ast\) the tautological sections of \( \mathcal{H}^\ast(\mathscr{A})\) .

It is worth noting that the complexified tangent bundle of a real smooth manifold provides a natural example of a locally trivial Banach fibration in the sense of Definition 1. Let \( M\) be a real smooth manifold. For every \( m\in M\) , we denote by

the complexification of the real tangent space \( T_mM\) . Every element of \( T_m^{\mathbb C}M\) can be written uniquely in the form

which we shall denote simply by \( v_m+iw_m\) . This notation identifies the underlying real vector space of \( T_m^{\mathbb C}M\) with \( T_mM\oplus T_mM\) . This decomposition should not be understood as a decomposition into complex subspaces. The complexified tangent space also carries the canonical complex-antilinear involution determined by

Equivalently, if \( z_m=v_m+iw_m\) , then \( \overline{z_m}=v_m-iw_m\) . Its fixed-point set is the canonical copy of \( T_mM\) inside \( T_m^{\mathbb C}M\) , i.e., \( \{z_m \in T_m^\mathbb{C} : \overline{z_m} = z_m\} = \{v_m \otimes 1 : v_m \in T_mM\}\) .

An additional Hermitian structure can be defined on \( T_m^{\mathbb{C}}M\) for instance by choosing a Riemannian metric on \( M\) and extending it Hermitianly to \( T^{\mathbb{C}}M\) . More generally, suppose that for every \( m\in M\) we are given a positive definite Hermitian form

depending smoothly on \( m\) . Then each fiber \( T_m^{\mathbb C}M\) becomes a complex Hilbert space, and the triple

is a locally trivial complex Hilbert fibration where

is the canonical projection. The Hermitian tensor \( K\) determines canonically two real tensors on the underlying real manifold \( M\) , namely

for all \( v_m,w_m\in T_mM\subset T_m^{\mathbb C}M\) . In this way, the real part of \( K\) defines a symmetric bilinear form on \( TM\) , while its imaginary part defines a skew-symmetric \( 2\) -form.

This construction provides the natural geometric setting in which the Hermitian structure carried by the GNS fibers can be related to the geometry of a smooth parametric model. More precisely, under suitable compatibility and regularity assumptions, the Hermitian product of the GNS fibers will induce canonically a Hermitian tensor on the complexified tangent bundle of the model. In the examples considered later, this point of view will allow us to recover familiar geometric structures of information geometry from a single Hermitian tensor: in the classical case, the real part gives the Fisher–Rao metric tensor, while in the quantum cases it yields, according to the model, the Fubini–Study and SLD metric tensors together with their associated imaginary parts.

Recall that in the \( C^{*}\) -algebraic approach to classical and quantum information geometry, a parametric (statistical) model may be described by a triple \( (M,\mathrm{i},\mathscr{A})\) where \( M\) is a real smooth manifold, \( \mathscr{A}\) is a \( C^{*}\) -algebra, and \( \mathrm{i}\colon M\rightarrow\mathcal{S}(\mathscr{A})\) taking values in the set of states is the map determining the model [20]. If \( \mathrm{i}\) is continuous with respect to the topology on \( M\) and the weak\( ^*\) -topology on \( \mathcal{S}(\mathscr{A})\) , we can build two non-locally-trivial Hilbert fibrations over \( M\) from the GNS and the dual GNS fibrations introduced in Section 2 following [33, § II.13.17]. Specifically, we consider

and the obvious projections over \( M\) coming from the left projections on \( M\times \mathcal{H}(\mathscr{A})\) and \( M\times \mathcal{H}^{*}(\mathscr{A})\) .

The purpose of this section is to introduce a class of parametric models for which the complexified tangent space \( T_m^{\mathbb C}M\) admits a canonical lift to the fibers of the pullback dual GNS fibration \( \mathrm{i}^*\mathcal{H}^*(\mathscr{A})\) . This makes it possible to induce on \( T_m^{\mathbb C}M\) a Hermitian form coming from the dual GNS inner product. Its real and imaginary parts then determine, on the underlying real tangent space \( T_mM\) , a symmetric bilinear form and a skew-symmetric \( 2\) -form, respectively.

In many examples of practical interest, the map \( \mathrm{i}\colon M\rightarrow\mathcal{S}(\mathscr{A})\) is such that, for every self-adjoint element \( a\in \mathscr{A}_{\mathrm{sa}}\) , the expectation-value map \( \ell_a \colon M \to \mathbb{R}\) given by \( \ell_a(m) := \mathrm{i}(m)(a)\) is smooth. When this holds, every tangent vector \( v_m \in T_m M\) determines a real-linear functional \( \eta^{v_m}\) on the real subspace of \( \mathcal{H}_{\mathrm{i}(m)}^\mathbb{R}\) generated by vectors of the form \( \psi_a^{\mathrm{i}(m)}\) with \( a \in \mathscr{A}_{\mathrm{sa}}\) , according to:

In general, this functional need not extend to a continuous real-linear functional on the whole GNS Hilbert space. This motivates the following definition.

Let \( (M,\mathrm{i},\mathscr{A})\) be a GNS-smooth parametric model as in Definition 2. From Proposition 2, we first obtain the real-linear map

The vector \( \Xi_m(v_m)\) is referred to as the real GNS vector lift of \( v_{m}\) . Then, applying the real Riesz isomorphism, we obtain the linear map

The linear functional in equation (28) is referred to as the real GNS lift of \( v_{m}\) .

Consider the canonical complex structure \( J_{\mathrm{i}(m)}\) defined in Since \( \mathcal{H}_{\mathrm{i}(m)}^{\mathbb{R}}\) as in equation (5). We shall identify real-dual representatives with complex-dual representatives through the canonical real-linear isomorphism

where, with an evident abuse of notation, whenever \( \zeta\in\mathcal H_{\mathrm i(m)}\) appears as an argument of \( \alpha\) or \( J_{\mathrm i(m)}\) , it is understood as the same vector regarded as an element of the realification \( \mathcal H_{\mathrm i(m)}^{\mathbb R}\) . Equivalently, \( \mathfrak C_{\mathrm{i}(m)}(\alpha)\) is the unique complex-linear functional on \( \mathcal{H}_{\mathrm{i}(m)}\) whose real part is \( \alpha\) . We then set

By construction, for every \( a\in\mathscr{A}_{\mathrm{sa}}\) , it holds

By complexification, we obtain a complex-linear map

for each \( m\in M\) . For real tangent vectors one has

where \( \Xi_m(v_m)\) is seen as an element of \( \mathcal{H}_{\mathrm{i}(m)}\) rather than of its realification \( \mathcal H_{\mathrm i(m)}^{\mathbb R}\) . Since the Riesz map \( R_{\mathrm i(m)}:\mathcal{H}_{\mathrm i(m)}\to\mathcal{H}_{\mathrm i(m)}^*\) is conjugate-linear with our convention for the GNS inner product, the Riesz vector corresponding to \( L_m^{\mathbb C}(v_m+\mathrm i w_m)\) is

and it holds

For \( m\in M\) , the maps in equation (32) assemble into a fiberwise complex-linear lift

where \( \mathrm{i}^*(\mathcal H^*(\mathscr A))\) denotes the pullback of the dual GNS fibration along \( \mathrm{i}\) . We now pull back the fiberwise Hermitian products of the dual GNS fibration along \( L^{\mathbb C}\) . The Hilbert product on the dual GNS Hilbert space is the one transported by the conjugate-linear Riesz identification \( R_\rho(\xi)(\zeta)=\langle\xi,\zeta\rangle_\rho\) , namely

For every \( m\in M\) , with \( \rho=\mathrm{i}(m)\) , define

Equivalently, using equation (34), it holds

Restricting the real and imaginary parts of \( K_m\) to the real tangent space \( T_mM\subseteq T_m^{\mathbb C}M\) , we obtain

The bilinear forms are well defined by the construction of \( L_m\) in equation (30), or equivalently by the canonical construction of the Riesz-vector map \( \Xi_m\) in equation (27). Condition (III) in Definition 2 implies that \( L_m\) is injective, and hence \( G_m\) is positive definite. Indeed, if \( G_m(v_m,v_m)=0\) , then \( L_m(v_m)=0\) , and therefore

By condition (III) in Definition 2, this gives \( v_m=0\) . Moreover, by construction, \( G_m\) is symmetric and \( \Omega_m\) is skew-symmetric.

In principle, there is no guarantee that the forms in equation (42) glue together to smooth tensor fields on \( M\) , and we are thus led to the following definition.

An immediate consequence of the construction is that regularity is stable under restriction to immersed submanifolds.

The purpose of this section is to show that an important class of statistical models of probability measures used in classical information geometry are Hermitian regular GNS-smooth parametric models in the sense of Definitions 2 and 3.

In classical information geometry, one usually works with statistical models satisfying regularity assumptions strong enough to make logarithmic derivatives and the Fisher–Rao metric tensor well defined. A natural and sufficiently broad framework is provided by the \( k\) -integrable statistical models of Ay, Jost, L{^e}, and Schwachhöfer [43, 9]. In particular, for \( k=2\) , logarithmic derivatives are square-integrable and their \( \mathcal{L}^2\) -pairing gives the Fisher–Rao tensor (see [43, Def. 4.4 and Def. 4.5] and [9, Ch. 4]).

In this section we consider the dominated faithful case where all probability measures in the model are absolutely continuous with respect to a fixed \( \sigma\) -finite reference measure \( \nu\) , and their densities are strictly positive \( \nu\) -almost everywhere. This is precisely the setting naturally associated with normal faithful states on \( \mathscr{A}=\mathcal{L}^\infty(X,\nu)\) .

Every faithful dominated \( 2\) -integrable statistical model determines a parametric model \( (M,\mathrm{i},\mathscr{A})\) of faithful normal states in the sense of the operator-algebraic approach to information geometry [20]. Indeed, set \( \mathscr{A}=\mathcal{L}^\infty(X,\nu)\) , and, for each \( m\in M\) , define a state \( \rho_m\equiv \mathrm{i}(m)\) on \( \mathscr{A}\) by

Since \( p(\cdot;m)>0\) \( \nu\) -almost everywhere, \( \rho_m\) is a normal faithful state on \( \mathscr{A}\) .

The next proposition shows that, when the faithful dominated \( 2\) -integrable statistical model is nondegenerate, the associated operator-algebraic model is GNS-smooth and Hermitian regular in the sense of Definitions 2 and 3. Moreover, the canonical GNS representative is the classical score, the induced Riemannian metric is the Fisher–Rao metric, and the induced skew-symmetric tensor vanishes identically.

We now consider the model of pure normal states on the algebra of bounded operators on a separable complex Hilbert space \( \mathcal{H}\) , where the Hilbert product is conjugate-linear in the first argument and linear in the second. The purpose of the section is to show that these states form a Hermitian regular GNS-smooth parametric model as in Definitions 2 and 3, and that the resulting geometric tensors coincide (up to normalization and sign/order convention) with the standard Fubini–Study metric tensor and symplectic form.

Let \( \mathscr{A}=\mathcal{B}(\mathcal{H})\) and \( M=\mathbb{CP}(\mathcal{H})\) , where \( \mathbb{CP}(\mathcal{H}):=\mathbb{S}(\mathcal{H})/U(1)\) denotes the projective Hilbert space of \( \mathcal{H}\) , with \( \mathbb{S}(\mathcal{H})\) the unit sphere in \( \mathcal{H}\) . To make contact with parametric models in information geometry which use real manifolds, we regard \( M=\mathbb{CP}(\mathcal{H})\) as a real smooth Hilbert manifold, although it also carries a natural complex Hilbert manifold structure. Define the map \( \mathrm{i}\colon M \longrightarrow \mathcal{S}(\mathscr{A})\) setting

where \( \psi\in\mathcal{H}\) is any normalized representative of the ray \( [\psi]\) . This map is well defined because \( \rho_{e^{\mathrm i\theta}\psi}=\rho_\psi\) .

For each normalized representative \( \psi\) of the ray \( [\psi]\) , the GNS Hilbert space \( \mathcal{H}_{\rho_\psi}\) may be identified with \( \mathcal{H}\) by the map

Indeed, it holds

so \( U_\psi\) is an isometry on the dense subspace of GNS vectors. Its range is all of \( \mathcal{H}\) , because for every \( \varphi\in\mathcal{H}\) the rank-one operator

satisfies \( \theta_{\varphi,\psi}\psi=\varphi\) . Hence \( U_\psi\) extends to a unitary isomorphism. This identification depends on the choice of normalized representative because, if \( \widetilde\psi=e^{\mathrm i\theta}\psi\) , then

The tensors obtained below are independent of this choice. Under the identification induced by \( U_\psi\) , the distinguished real subspace \( V_{\rho_\psi}\) in equation (21) is

Indeed, if \( \varphi=U_{\psi}(\psi_{a}^{\rho_{\psi}})=a\psi\) with \( a=a^*\) , then

Conversely, if \( \varphi\in\mathcal{H}\) satisfies \( \langle\psi,\varphi\rangle\in\mathbb R\) , we can write

and the self-adjoint operator

satisfies \( U_{\psi}(\psi_{a}^{\rho_{\psi}})=a\psi=\varphi\) . Moreover, it also holds

which is precisely the vertical phase direction.

Let \( v_{[\psi]}\in T_{[\psi]}\mathbb{CP}(\mathcal{H})\) . Choose a smooth curve of normalized vectors \( \psi(t)\) such that

and set

Since \( \|\psi(t)\|=1\) , it follows that

We define the horizontal component of \( \dot\psi\) by

so that

If another normalized lift is chosen, say

then

Together with \( U_{e^{\mathrm i\theta}\psi}=e^{\mathrm i\theta}U_\psi\) , this shows that the GNS vector determined below does not depend on the chosen normalized representative.

Recall that the projective Hilbert space \( \mathbb{CP}(\mathcal{H})=M\) carries its standard Fubini–Study K
specialChar{34}ahler structure [17, 44, 2, 45]. With the convention that the Hilbert product is conjugate-linear in the first argument and linear in the second, we normalize the Fubini–Study Hermitian tensor by

where \( \dot\psi_{\mathrm h}\) and \( \dot\phi_{\mathrm h}\) are the horizontal components of normalized lifts representing \( v_{[\psi]}\) and \( w_{[\psi]}\) . Its real and imaginary parts are denoted by

and are, respectively, the Fubini–Study metric tensor and symplectic form. Other normalizations of the Fubini–Study metric differ from this one by an overall positive constant.

We now turn to the case of parametric models of faithful normal states on \( \mathscr{A}\cong\mathcal B(\mathcal{H})\) , where \( \mathcal H\) is a separable complex Hilbert space. A normal state on \( \mathcal B(\mathcal H)\) is uniquely represented by a positive trace-class operator of unit trace. Thus, a normal parametric model in this setting consists of a smooth manifold \( M\) together with a map \( \mathrm{i}:M\to \mathscr S(\mathscr{A})\) , \( m\mapsto \rho_m\) such that, for each \( m\in M\) , there exists a density operator

where \( \mathcal{B}_1(\mathcal H)\) denotes the space of trace-class operators on \( \mathcal{H}\) , satisfying

The state \( \rho_m\) is faithful precisely when \( \varrho_m\) is injective. Notice that, in infinite dimension, injectivity does not imply that \( \varrho_m^{-1}\) is bounded; equivalently, a faithful density operator need not be bounded below.

Fix \( m\in M\) and write

Because \( \rho\) is faithful, its GNS ideal in equation (1) is trivial. Indeed, if \( a\in\mathscr{A}\) satisfies

then \( a\varrho^{1/2}=0\) . Since \( \varrho\) is injective, \( \varrho^{1/2}\) is injective and has dense range, and hence \( a=0\) . Therefore the GNS Hilbert space \( \mathcal H_\rho\) is the completion of \( \mathscr{A}=\mathcal B(\mathcal H)\) with respect to the inner product

There is a canonical Hilbert–Schmidt realization of this GNS space. On the dense subspace of GNS vectors define

where \( \mathcal{B}_2(\mathcal H)\) is the space of Hilbert–Schmidt operators on \( \mathcal{H}\) . For all \( a,b\in\mathscr{A}\) one has

so \( U_\varrho\) is an isometry on a dense subspace. It remains to observe that its range is dense in \( \mathcal{B}_2(\mathcal H)\) . Since \( \varrho^{1/2}\) is injective, one has

Therefore, if \( |u\rangle\langle v|\) is a rank-one operator, we may choose a sequence \( (w_n)_n\subset\mathcal H\) such that \( \varrho^{1/2}w_n\to v\) , and then

in Hilbert–Schmidt norm. Hence all rank-one operators, and therefore all finite-rank operators, belong to the Hilbert–Schmidt closure of

Since finite-rank operators are dense in \( \mathcal{B}_2(\mathcal H)\) , the isometry \( U_\varrho\) extends uniquely to a unitary isomorphism

We now transport the distinguished real subspace \( V_\rho\) of equation (21) through the unitary map \( U_\varrho\) . Since \( U_\varrho\) is a real-linear homeomorphism on the realified Hilbert spaces, one obtains

This closed real Hilbert subspace is the faithful quantum analogue of the real \( \mathcal{L}^2\) space in which the classical score function lives.

Let now \( (M,\mathrm i,\mathscr{A})\) be a GNS-smooth parametric model of normal faithful states as in Definition 2. For every \( m\in M\) and every \( v_m\in T_m M\) , Proposition 2 gives the real GNS vector lift \( \xi_{v_{m}}^{\rho_{m}}=\Xi_m(v_m)\in V_{\rho_m}\subset\mathcal{H}_{\rho_{m}}^{\mathbb{R}}\) , and we define its Hilbert–Schmidt representative by

By construction, it holds

Moreover, in Hilbert–Schmidt form, equation (22) becomes

for all \( a\in\mathscr{A}_{\mathrm{sa}}=\mathcal{B}(\mathcal{H})_{\mathrm{sa}}\) , where \( \ell_a(m):=\rho_m(a)=\operatorname{Tr}(\varrho_m a)\) .

This formula should be read as the faithful quantum counterpart of the classical score identity in equation (69). The role of the classical score \( s_{v_m}\) is now played first by the Hilbert–Schmidt operator \( X_{v_m}\) .

When the map \( \varrho:M\longrightarrow \mathcal B_1(\mathcal H)_{\mathrm{sa}}\) determined by the model through \( m\mapsto \varrho_m\) is smooth, where \( \mathcal B_1(\mathcal H)_{\mathrm{sa}}\) is regarded as a real Banach space with the trace norm, then for every \( v_m\in T_mM\) one has

Moreover, since \( \operatorname{Tr}(\varrho_m)=1\) for all \( m\in M\) , differentiating the normalization condition gives

For every \( a=a^*\in\mathcal B(\mathcal H)\) , the map

is a continuous real-linear functional. Hence, for the expectation-value map

one obtains

for all \( a\in\mathscr{A}_{\mathrm{sa}}=\mathcal B(\mathcal H)_{\mathrm{sa}}\) . Combining this identity with equation (136) leads to the weak symmetric logarithmic derivative relation for the Hilbert–Schmidt representative \( X_{v_m}\) . This is made precise in the proposition following the Lemma.

6.1 Faithful quantum states in finite dimensions

6.2 Displaced thermal states and fixed-temperature submodels

The examples of the preceding sections show that the two-form \( \Omega\) induced by the imaginary part of the GNS Hermitian tensor has no uniform closedness property. It vanishes in the commutative case, it is closed for pure states because it is, up to normalization and sign/order convention, the Fubini–Study symplectic form, but it is not closed for faithful qubits and for the full displaced thermal model. The purpose of this section is to explain this behaviour from a structural point of view.

The main point is that there are two related but distinct geometric questions. The first one is a total-space question: after imposing enough regularity so that the real dual of the realified pullback GNS fibration becomes a smooth real Hilbert bundle over \( M\) , the fiberwise symplectic forms induced by the dual GNS Hermitian structure define a vertical two-form, and one may ask whether this vertical form admits a closed extension to the total space. The second one is a base-space question: whether the two-form \( \Omega\) induced on the parameter manifold is closed. These two questions are not equivalent. Indeed, \( \Omega\) is not obtained as the pullback of a closed two-form on the total space along a section of the bundle. Rather, it is obtained by pairing the canonical real dual GNS lift \( L^{\mathbb R}\) with itself through the fiberwise symplectic form. Consequently, the closedness of \( \Omega\) is controlled by the covariant exterior derivative of \( L^{\mathbb R}\) , not merely by the existence of a closed extension on the total space.

Throughout this section, differential forms on smooth Hilbert manifolds are understood as smooth sections of the bundles of continuous alternating multilinear forms. Thus, for a smooth Hilbert manifold \( N\) , we write

All exterior derivatives, pullbacks, and contractions below are understood in this category. When we use de Rham cohomological language for Hilbert manifolds, it is only in settings where the standard smooth homotopy invariance of the chosen de Rham complex is available.

7.1 Bundle-regular models and the vertical symplectic form

7.2 Closed extensions on the total space

7.3 Closedness of the induced two-form on the parameter manifold

7.4 Examples and further remarks

We have developed a GNS-based construction of geometric tensors on smooth parametric models over \( C^*\) -algebras. The central point is that the real and imaginary parts of the dual GNS Hermitian product, organized fiberwise over the state space and pulled back along suitable parametric models, produce a Hermitian tensor on the complexified tangent bundle of the model. Under the regularity assumptions of Definition 3, its real and imaginary parts assemble into a smooth weak Riemannian metric tensor \( G\) and a smooth two-form \( \Omega\) .

This construction gives a common operator-algebraic origin for several standard geometries of classical and quantum information theory. In the commutative dominated case it recovers the Fisher–Rao metric tensor and the two-form vanishes. For pure quantum states it recovers the Fubini–Study metric tensor together with the Fubini–Study symplectic form, up to the normalization and sign/order conventions imposed by the dual GNS pairing. For faithful quantum states it gives the SLD metric tensor (equivalently, four times the Bures–Helstrom metric in the convention used here), while the corresponding two-form is proportional, in finite dimensions, to the expected commutator of the SLD representatives, or equivalently to the mean Uhlmann curvature.

A distinctive feature of the construction is that it does not require the state space \( \mathcal S(\mathscr A)\) itself to be a smooth manifold. The usual ambient-manifold pullback picture is replaced by the GNS fibration and its dual. Tangent vectors to the parameter manifold are represented by continuous functionals on the realified GNS fibers, and the ambiguity of such representatives is removed by passing to the distinguished real subspace generated by self-adjoint observables. This produces the canonical real dual GNS lift from which the tensors are defined. In the faithful quantum case, this viewpoint gives the SLD a natural geometric meaning: it is not introduced as an external ansatz, but appears as the canonical representative of a tangent direction.

The skew-symmetric tensor \( \Omega\) is not merely a secondary byproduct of the metric construction. It detects genuinely noncommutative features of the model, although its vanishing is weaker than pairwise commutativity of the SLDs and should not be identified outright with the usual quasiclassical condition. The examples of faithful qubits and displaced thermal states show that \( \Omega\) need not be closed. Section 7 explains this phenomenon structurally: for bundle-regular models the fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, but the closedness of the induced two-form on the parameter manifold is governed by the covariant exterior derivative of the canonical real dual GNS lift.

The framework developed in this paper suggests several natural directions for further investigation.

Other fields of covariances and the Morozova–Čencov–Petz classification. As noted in the introduction, the GNS inner product is a particular instance of a field of covariances in the sense of [28]. The finite-dimensional classification of fields of covariances in [28] provides a unification of the classical Čencov’s result on the uniqueness of the Fisher–Rao metric tensor with the quantum Morozova–Čencov–Petz characterization of monotone metric tensors that extends also to non-faithful states, thus applying to all states. It is natural to ask whether the pullback-like construction developed here can be carried out for the other fields of covariances, at least in the finite-dimensional case. The idea would be that each operator monotone function \( f\) appearing in the classification in [28] determines a different fiberwise inner product on a Hilbert fibration over the state space that is a sort of deformation of the original GNS one. The same Riesz-representation procedure should produce a symmetric tensor \( G^f\) and a skew-symmetric tensor \( \Omega^f\) on any regular parametric model. In the faithful finite-dimensional case, the symmetric part \( G^f\) would recover the corresponding monotone metric in the Morozova–Čencov–Petz family, while the skew-symmetric part \( \Omega^f\) would be a new invariant depending on \( f\) . It is worth noting that the operator monotone functions in Petz’s work [4] satisfy a symmetricity condition that makes the imaginary part of the associated Hermitian product vanish, while the operator monotone functions appearing in [28] do not necessarily satisfy this symmetricity condition, thus allowing a non-zero \( \Omega^{f}\) . Moreover, again at least in finite dimensions, the closed-extension machinery of Section 7 applies to any such Hilbert bundle.

Behavior under quantum channels. A central property of the monotone quantum metrics in the Morozova–Čencov–Petz classification is their monotonicity under completely positive trace-preserving maps, or equivalently under unital completely positive maps in the Heisenberg picture. The present paper does not address this behavior. For the metric part \( G\) , one may expect the pullback construction is compatible with the usual monotonicity of the SLD metric tensor under quantum channels, because quantum channels leads to contraction maps between GNS Hilbert spaces. For the skew-symmetric tensor \( \Omega\) , the appropriate statement is less clear, since a two-form is not ordered in the way a positive symmetric tensor is. A more natural problem is to formulate a functoriality, covariance, or contraction property for \( \Omega\) under suitable state-preserving completely positive maps, and to understand how such a property interacts with quantum Cram
specialChar{39}er–Rao attainability and optimal POVMs.

Nonparametric extensions and the work of Jenčov
specialChar{39}a. The set of faithful normal states \( \mathcal{S}_{nf}(\mathcal{M})\) on a von Neumann algebra \( \mathcal{M}\) can be endowed with the structure of Banach manifold using quantum Orlicz spaces [23, 56], providing an infinite-dimensional analogue of the Pistone–Sempi exponential manifold [57]. This construction is based on relative entropy and state perturbation, and admits exponential and mixture connections as a dual pair. If \( \mathrm{i}\) is the canonical inclusion of \( \mathcal{S}_{nf}(\mathcal{M})\) in \( \mathcal{S}(\mathcal{M})\) , it would be of considerable interest to understand if \( (\mathcal{S}_{nf}(\mathcal{M}),\mathrm{i},\mathcal{M})\) is regular in the sense of Definition 3. Furthermore, the manifold \( \mathcal{S}_{nf}(\mathcal{M})\) is endowed with a canonical divergence satisfying a Pythagorean relation and is invariant under sufficient channels [56], which suggests a possible bridge to the monotonicity questions mentioned above.

The \( 2\) -form \( \Omega\) and quantum estimation theory. The definition of the skew-symmetric tensor \( \Omega\) from the dual GNS Hermitian product is a novel construction of the paper. Its vanishing on quasiclassical models and its relation to the expected commutator of SLD operators (mean Uhlmann curvature) for finite-dimensional faithful quantum states suggest that it is related to an intrinsic obstruction to simultaneous quantum parameter estimation. A precise formulation of this connection would require relating \( \Omega\) to the Holevo bound or to the incompatibility of optimal measurements in multiparameter quantum estimation [14, 16]. Making this link explicit, and understanding how the closedness properties of \( \Omega\) studied in Section 7 bear on the estimation-theoretic picture, is an interesting open problem at the interface of differential geometry and quantum statistics.

This work was partially supported by the Spanish Ministry of Economy and Competitiveness through the research projects PID2024-160539NB-I00 and PID2024-156578NB-I00, and by the Madrid Government through the project TEC-2024/COM-84 QUITEMAD-CM. A.I. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa Program for Centers of Excellence in RD (SEV-2015/0554). This article is based upon work from COST Action CaLISTA CA21109, supported by COST (European Cooperation in Science and Technology). F.M.C., A.I., and L.G.-B. acknowledge their participation in COST Action CaLISTA CA21109.