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.
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.
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.
In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.
A bstract Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For Möbius driving by the SL(2 , ℝ) subgroup, the BKM metric becomes the hyperbolic metric on the disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via Möbius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.
We unify two emergent aspects of quantum theory within a single information-geometricframework based on the Araki relative entropy and the associated Bogoliubov–Kubo–Mori(BKM) metric on a von Neumann algebra state manifold.First, we show that in the commutative reduction to a spatial density subalgebra, theBKM metric becomes the classical Fisher information metric. Its trace gives, after a singlenormalization choice, the expectation value of the Bohmian quantum potential, and thecorresponding Bohmian trajectories arise as Jacobi geodesics of a conformally related metric.The guidance equation is then recovered from the polar decomposition of states satisfyingthe emergent Schr¨odinger equation.Second, we show that the modular flow of the reference state yields the Heisenbergand von Neumann equations, and that the physical Schr¨odinger equation follows under astandard unitary state implementation. In the finite regulated K¨ahler sector, geometricprequantization of the BKM state manifold leads to a discrete horizon area spectrum and touncertainty relations, with an induced gravitational coupling derived from the Seeley–DeWittcoefficient [1].No continuum limit is taken, and no identification with the observed Newton constantis assumed. The results exhibit a common information-geometric framework for Bohmianmechanics, quantum dynamics, and emergent gravitational structures.
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) 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 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.