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 $C<sup>*$-algebras. Since the state space of a $C<sup>*$-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.

• The paper constructs a unified Hermitian tensor from the dual GNS inner product, recovering the Fisher–Rao metric, Fubini–Study geometry, SLD quantum Fisher metric, and mean Uhlmann curvature.
• The framework applies to regular finite- and infinite-dimensional models, with explicit GNS-boundedness conditions and formulas for classical densities, pure states, faithful quantum states, qubits, and displaced thermal states.
• The induced two-form measures noncommutativity but need not be closed, and its closedness is governed by the covariant variation of the canonical GNS lift rather than by the existence of a closed form on the total space.

Motivation and scope

The paper develops a unified operator-algebraic construction of the principal geometric tensors of classical and quantum information geometry — the Fisher–Rao metric, the Fubini–Study metric and symplectic form, and the SLD (symmetric logarithmic derivative) metric together with the mean Uhlmann curvature — starting from a single object: the Hermitian product of the Gelfand–Naimark–Segal (GNS) construction, transported to the dual GNS Hilbert space and pulled back along a parametric statistical model. The setting is that of parametric models over a $C^*$-algebra $\mathscr A$, described by a triple $(M,\mathrm i,\mathscr A)$ with $M$ a real smooth manifold and $\mathrm i\colon M\to\mathcal S(\mathscr A)$ the model map. The central technical obstacle is that the state space $\mathcal S(\mathscr A)$ is not a smooth manifold — even in finite dimensions it is a manifold with boundary, with corners, or a stratified space — so the standard pullback of tensor fields from an ambient state manifold is unavailable. The construction replaces the ambient-manifold picture with the GNS fibration and its dual, which are shown to be non-locally-trivial Hilbert fibrations over $\mathcal S(\mathscr A)$, built via the construction theorem for Banach fibrations of Fell–Doran using the tautological sections $\Psi_a(\rho)=[a]_\rho$.

GNS fibrations and the induced tensors

For each state $\rho$, the GNS Hilbert space $\mathcal H_\rho$ is the completion of $\mathscr A$0 with inner product $\mathscr A$1. Realifying $\mathscr A$2 yields canonical real-bilinear forms: the symmetric $\mathscr A$3 and the skew $\mathscr A$4, related by the complex structure $\mathscr A$5. The disjoint unions of the $\mathscr A$6 and $\mathscr A$7 over $\mathscr A$8, topologized via the tautological sections, are Hilbert fibrations (the GNS fibration and dual GNS fibration); the paper proves they are in general not locally trivial.

A parametric model is called GNS-smooth if all expectation-value maps $\mathscr A$9 are smooth, if derivatives of expectations satisfy a boundedness condition $(M,\mathrm i,\mathscr A)$0, and if the expectation functionals separate tangent vectors. Under the boundedness condition, each tangent vector admits a unique canonical representative $(M,\mathrm i,\mathscr A)$1 in the real closed span $(M,\mathrm i,\mathscr A)$2 of the GNS vectors $(M,\mathrm i,\mathscr A)$3 with $(M,\mathrm i,\mathscr A)$4 self-adjoint, characterized by $(M,\mathrm i,\mathscr A)$5. Pulling back the dual GNS Hermitian product along the induced complex-linear lift $(M,\mathrm i,\mathscr A)$6 produces a Hermitian tensor $(M,\mathrm i,\mathscr A)$7 on $(M,\mathrm i,\mathscr A)$8 whose real and imaginary parts give, under the regularity conditions of the paper, a smooth weak Riemannian metric $(M,\mathrm i,\mathscr A)$9 (strong when $M$0 is finite-dimensional) and a smooth two-form $M$1. Regularity is preserved under restriction to immersed submanifolds, with $M$2, $M$3, $M$4.

The three canonical cases

Commutative dominated models. For faithful dominated $M$5-integrable statistical models with regular density function (in the sense of Ay–Jost–Lê–Schwachhöfer), with $M$6, the GNS fiber is $M$7 and the canonical representative of a tangent vector is the score $M$8. The induced metric is exactly the Fisher–Rao tensor, and the two-form vanishes identically, $M$9.

Pure states. For $\mathrm i\colon M\to\mathcal S(\mathscr A)$0 with $\mathrm i\colon M\to\mathcal S(\mathscr A)$1, the GNS fiber over $\mathrm i\colon M\to\mathcal S(\mathscr A)$2 is identified with $\mathrm i\colon M\to\mathcal S(\mathscr A)$3, and the canonical representative of a tangent vector represented by horizontal lift $\mathrm i\colon M\to\mathcal S(\mathscr A)$4 is $\mathrm i\colon M\to\mathcal S(\mathscr A)$5. Consequently $\mathrm i\colon M\to\mathcal S(\mathscr A)$6, so $\mathrm i\colon M\to\mathcal S(\mathscr A)$7 and $\mathrm i\colon M\to\mathcal S(\mathscr A)$8: the construction recovers the Fubini–Study metric and symplectic form up to normalization and sign/order convention imposed by the dual GNS pairing, and coincides (up to conventions) with the quantum geometric tensor. Here $\mathrm i\colon M\to\mathcal S(\mathscr A)$9 is closed and symplectic.

Faithful quantum states. For faithful normal states on $\mathcal S(\mathscr A)$0, the GNS Hilbert space is realized unitarily on the Hilbert–Schmidt class $\mathcal S(\mathscr A)$1 via $\mathcal S(\mathscr A)$2. The canonical representative is a Hilbert–Schmidt operator $\mathcal S(\mathscr A)$3 satisfying the weak SLD relation

$\mathcal S(\mathscr A)$4

and the induced metric is $\mathcal S(\mathscr A)$5. In finite dimensions $\mathcal S(\mathscr A)$6 with $\mathcal S(\mathscr A)$7 the unique SLD, and $\mathcal S(\mathscr A)$8 is the SLD quantum Fisher metric, i.e., four times the Bures metric in the infinitesimal-distance normalization. The two-form is $\mathcal S(\mathscr A)$9, proportional to the expected commutator of the SLDs, i.e., to minus twice the mean Uhlmann curvature. Pairwise commuting SLDs imply $\mathcal S(\mathscr A)$0, but the converse fails: vanishing of $\mathcal S(\mathscr A)$1 is strictly weaker than quasiclassicality.

Non-closedness of the two-form

Two explicit examples show that $\mathcal S(\mathscr A)$2 need not be closed. Faithful qubits: for the full three-dimensional Bloch-ball model, the induced tensors are $\mathcal S(\mathscr A)$3 and

$\mathcal S(\mathscr A)$4

with $\mathcal S(\mathscr A)$5 — a sharp contrast with the pure-state case, despite the model being finite-dimensional and fully regular. Displaced thermal states: for the infinite-dimensional model $\mathcal S(\mathscr A)$6 on $\mathcal S(\mathscr A)$7, the paper proves GNS-smoothness and Hermitian regularity via the Glauber–Sudarshan representation and explicit Hilbert–Schmidt representatives, obtaining

$\mathcal S(\mathscr A)$8

with $\mathcal S(\mathscr A)$9 for all $\Psi_a(\rho)=[a]_\rho$0. On fixed-temperature submodels, however, $\Psi_a(\rho)=[a]_\rho$1 is a constant multiple of the standard symplectic form on $\Psi_a(\rho)=[a]_\rho$2, hence closed — an application of the restriction stability of regularity.

Closed extensions and the structural explanation

The final section introduces bundle-regular models, for which the real dual of the realified pullback GNS fibration is a smooth Hilbert bundle $\Psi_a(\rho)=[a]_\rho$3 with fiberwise strong symplectic forms $\Psi_a(\rho)=[a]_\rho$4, the canonical real dual lift $\Psi_a(\rho)=[a]_\rho$5 is smooth, and a connection $\Psi_a(\rho)=[a]_\rho$6 preserving the transported metric and complex structure exists. The main results are:

• Total-space closedness: for every such connection, $\Psi_a(\rho)=[a]_\rho$7, where $\Psi_a(\rho)=[a]_\rho$8 is a connection-dependent Liouville-type one-form, is a closed extension of the vertical symplectic form to $\Psi_a(\rho)=[a]_\rho$9, restricting to $\rho$0 on each fiber. Since Hilbert-space fibers are contractible, the cohomological obstruction familiar from the finite-dimensional symplectic-fibration theory of Gotay–Lashof–Śniatycki–Weinstein is absent. These extensions are non-unique, and their horizontal–horizontal block involves the curvature $\rho$1; nondegenerate ("fat") extensions require a nondegenerate base form.
• Base-space closedness: the induced two-form satisfies

$\rho$2

so $\rho$3 is closed precisely when the covariant exterior derivative of the canonical lift vanishes. This separates the total-space question from the base question and explains the non-closedness of the qubit and displaced-thermal examples without contradiction: $\rho$4 is obtained by pairing $\rho$5 with itself through the fiberwise symplectic form, not by pulling back a closed form on $\rho$6. The pure-state, finite-dimensional faithful, displaced thermal, and fixed-temperature models are all shown to be bundle-regular.

Limitations and open questions

The paper is explicit about the scope of its hypotheses. The GNS-boundedness condition is automatic only in finite dimensions; in infinite dimensions it becomes the Hilbert–Schmidt integrability condition $\rho$7, a genuine restriction on admissible tangent directions, and regularity must be verified case by case (as done for displaced thermal states). The commutative dominated models satisfy the tensor-producing regularity but are not shown to be bundle-regular. The construction covers only the GNS field of covariances; extension to other Morozova–\v{C}encov–Petz fields, to monotonicity under CPTP maps (particularly for $\rho$8, where the appropriate covariance or contraction property is unclear), and to the Jen\v{c}ov\'a Banach manifold of faithful normal states on a von Neumann algebra are all left open. Finally, the precise link between $\rho$9, the Holevo bound, and the incompatibility of optimal measurements in multiparameter quantum estimation is identified as an open problem rather than established.

Conclusion

The paper establishes that the Fisher–Rao, Fubini–Study, and SLD geometries, together with the mean Uhlmann curvature, arise from a single mechanism: the pullback of the dual GNS Hermitian product along a regular parametric model, organized through the GNS fibration rather than an ambient state manifold. The construction is valid for infinite-dimensional $\mathcal H_\rho$0-algebras under explicit analytic hypotheses, and it produces a skew-symmetric tensor $\mathcal H_\rho$1 that detects noncommutativity of the model, with closedness governed by the covariant variation of the canonical GNS lift rather than by the symplectic geometry of the total space.