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The GFI–HLM Bridge: Entropy Operator as the Quadratic Seed of Local Modular Information Geometry

David Gogishvili

Published Jun 24, 2026• 0 citations
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Abstract

This note identifies a direct conceptual bridge between the entropy operator formalism of Hollands–Longo–Morsella (2026) and the information-geometric framework of the Geometric Foundations of Information (GFI) programme.Starting from the HLM entropy operator �, local entropy is expressed as a positive quadratic form  S(\Phi|H)=\mathrm{Re}\langle\Phi,E_H\Phi\rangle.  We observe that its natural polarization defines a symmetric positive bilinear form �, which in the GFI interpretation serves as a local information metric in the wave-packet sector.This yields the chain  E_H ;\longrightarrow; \delta^2 S ;\longrightarrow; g_{\rm info},  linking entropy, information geometry, and modular dynamics.We further relate the HLM ant formula to a modular clock identity,  \dot S_{\rm mod}=2\pi E_{\rm mod}^{\rm min},  suggesting a unified entropy–metric–clock structure underlying the GFI programme.The note does not claim a full identification with the BKM/Kubo–Mori metric on the state space of a von Neumann algebra. Rather, it provides a concrete one-particle realization of the corresponding information-geometric structure and highlights the entropy operator as the quadratic seed of local modular information geometry.

DOI
10.5281/zenodo.20835414

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This note identifies a direct conceptual bridge between the entropy operator formalism of Hollands–Longo–Morsella (2026) and the information-geometric framework of the Geometric Foundations of Information (GFI) programme.Starting from the HLM entropy operator �, local entropy is expressed as a positive quadratic form  S(\Phi|H)=\mathrm{Re}\langle\Phi,E_H\Phi\rangle.  We observe that its natural polarization defines a symmetric positive bilinear form �, which in the GFI interpretation serves as a local information metric in the wave-packet sector.This yields the chain  E_H ;\longrightarrow; \delta^2 S ;\longrightarrow; g_{\rm info},  linking entropy, information geometry, and modular dynamics.We further relate the HLM ant formula to a modular clock identity,  \dot S_{\rm mod}=2\pi E_{\rm mod}^{\rm min},  suggesting a unified entropy–metric–clock structure underlying the GFI programme.The note does not claim a full identification with the BKM/Kubo–Mori metric on the state space of a von Neumann algebra. Rather, it provides a concrete one-particle realization of the corresponding information-geometric structure and highlights the entropy operator as the quadratic seed of local modular information geometry.
This preprint investigates structural relations between relative entropy, modular theory, and information geometry in algebraic quantum field theory (AQFT). In finite‑dimensional quantum systems, the Hessian of Umegaki relative entropy coincides with the Bogoliubov–Kubo–Mori (BKM) metric, and modular automorphisms act by isometries of this geometry; these standard results are reviewed as a mathematically controlled reference model. Motivated by this setting, the work formulates minimal structural conditions under which a perturbative modular information‑geometric programme for Type III1_11 von Neumann algebras could become mathematically meaningful. The analysis focuses on admissible perturbation sectors, Araki relative entropy, modular covariance, and the functional‑analytic obstructions caused by unbounded modular operators. A general consequence of positive semidefiniteness of the perturbative entropy Hessian is recorded: whenever the Hessian defines a well‑defined positive semidefinite bilinear form, its null directions annihilate the full form. In the Rindler wedge model, the corresponding regularity problem for candidate perturbations is formulated in terms of spectral properties and domain issues of the Lorentz boost generator. Throughout, explicit separation is maintained between rigorous results, conditional statements, heuristic motivations, and unresolved analytic problems. A conjecture concerning degenerate modular directions is stated together with a necessary falsifiable condition. This document is exploratory in nature and has not been peer reviewed.
This document presents a synthesis of research within the "Geometry from Information" (GFI) programme, specifically focusing on Paper 18 and Paper 27. We establish that the Bogoliubov-Kubo-Mori (BKM) metric is the Hessian of relative entropy within the quantum Wasserstein-2 geometry (W_{2}^{Q}) at the de Sitter point. The work demonstrates that the locality of the BKM metric is a consequence of the W_{2}^{Q} geodesic structure. Additionally, it explores the equivalence between the Island formula extremization and W_{2}^{Q}-optimal partitioning, providing a quantum Type II_{\infty} analogue to the Mondino-Suhr theory.
We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massive Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the massive Legendre differential operator and the Green integral for the Helmholtz operator. The quadratic form of the massive modular Hamiltonian is expressed in terms of an integral of the energy density with parabolic distribution and of a Yukawa potential, that here appears intrinsically. We then get the formula for the local entropy of a Klein-Gordon wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we have a criterion for a Bogoliubov automorphism to be locally weakly inner and we provide the passivity characterisation of the modular Hamiltonian within the standard subspace set up.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1)U(1)U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.
This work proposes a theoretical framework in which entropy is treated as a geometric degree of freedom in an extended quantum state space. By introducing a resolution parameter defined as the inverse of von Neumann entropy, an informational metric is constructed on the space of density matrices. Within this framework, a causal structure emerges from a null condition on the metric, defining a light-cone-like boundary for the propagation of quantum correlations. Entanglement propagation is derived and shown to depend on entropy evolution, leading to a dynamical causal structure. In the near-equilibrium limit, the model reproduces a Lieb–Robinson-type bound with finite correlation speed. Connections to holography are established by interpreting entropy as a radial coordinate controlling scale, and an analogue of the Ryu–Takayanagi relation is derived, where entropy corresponds to the length of minimal curves in the informational geometry. Black hole horizons are interpreted as maximal entropy surfaces corresponding to a collapse of distinguishability in quantum state space. The framework suggests that spacetime geometry, causality, and gravitational behavior may emerge from entropy-weighted quantum information structure. This work is an independent theoretical exploration aimed at providing a unified perspective linking quantum information, geometry, and gravity.
Building on Paper A (canonical emergence of the density operator) and Paper B (relative modular dynamics and CPTP completion), we introduce phase structure and the emergence of locality in a conservative, information-theoretic manner. A phase is specified by an access structure encoded by an optimal partition P , the corresponding accessible algebra AF , and a pointer/center structure ZF . The reference state σF within a macro-constraint class is defined canonically via a maximum-entropy (MaxEnt) construction under macroscopic constraints extracted from AF . We emphasize the equivalence between MaxEnt references and relative-entropy projections, which provides a canonical and minimal notion of phase stability. We introduce a canonical information-geometric metric (BKM/Fishertype) on the state manifold and connect these constructions to modular spectral diagnostics used in numerical protocols, such as spectral quantiles kq = −log λq and commutator-based probes.