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Quantum Entanglement Gravity (QEG) - Geometry and Einstein Equations from the Entanglement of a Critical Quantum System

Mohammad Akbar Habib Safi

Published Oct 22, 2026• 0 citations
Publisher

Abstract

Abstract We present RQCG (Relativistic Quantum Cosmology from Geometry via Entanglement), a framework in which spacetime geometry and the Einstein equations emerge from the entanglement structure of a critical quantum many-body system. The microscopic starting point is the open critical transverse-field Ising model, equivalent to free Majorana fermions and, in the continuum, to a c = 1/2 Majorana conformal field theory. From the entanglement entropies of spatial intervals we reconstruct a local speed profile that defines an optical metric. The entanglement first law is verified to high precision (mean ratio ΔS/ΔK = 0.9955 ± 0.016). The Ryu-Takayanagi first law holds exactly in dimensions d = 2…6. Real-time dynamics produce light-cone propagation at the theoretical velocity. In the continuum limit the reconstructed geometry satisfies the Einstein equations in the FLRW and Bianchi-I sectors, including the vacuum Kasner solutions. All claims are confirmed by exact free-fermion numerics and symbolic computation. This work seals the Level-1 theory (1+1 dimensions and cosmological reductions). Keywords: emergent gravity, entanglement, quantum gravity, conformal field theory, Ising model, Einstein equations, cosmology, Ryu-Takayanagi

DOI
10.5281/zenodo.23564874

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Abstract We present RQCG (Relativistic Quantum Cosmology from Geometry via Entanglement), a framework in which spacetime geometry and the Einstein equations emerge from the entanglement structure of a critical quantum many-body system. The microscopic starting point is the open critical transverse-field Ising model, equivalent to free Majorana fermions and, in the continuum, to a c = 1/2 Majorana conformal field theory. From the entanglement entropies of spatial intervals we reconstruct a local speed profile that defines an optical metric. The entanglement first law is verified to high precision (mean ratio ΔS/ΔK = 0.9955 ± 0.016). The Ryu-Takayanagi first law holds exactly in dimensions d = 2…6. Real-time dynamics produce light-cone propagation at the theoretical velocity. In the continuum limit the reconstructed geometry satisfies the Einstein equations in the FLRW and Bianchi-I sectors, including the vacuum Kasner solutions. All claims are confirmed by exact free-fermion numerics and symbolic computation. This work seals the Level-1 theory (1+1 dimensions and cosmological reductions). Keywords: emergent gravity, entanglement, quantum gravity, conformal field theory, Ising model, Einstein equations, cosmology, Ryu-Takayanagi
We examine geometry and dynamics of classical spacetime derived from entanglement spectrum. The spacetime is a kind of canonical parameter space defined by the Fisher information metric. As a concrete example, we focus on the spectrum for free fermions in spatially one dimension. The spectrum has exponential family form like thermal probability distribution owing to mixed-state feature emerging from truncation of environmental degrees of freedom. In this case, the Fisher metric is given by the second derivative of the Hessian potential that can be identified with the entanglement entropy. We emphasize that the canonical parameters are nontrivial functions of partial system size by the truncation, filling fraction of fermions, and time. Then, the precise determination of this nontrivial mapping is necessary to derive the functional form of the Hessian potential that leads to correct entanglement entropy scaling. By this potential, we find that the emergent geometry becomes anti-de Sitter spacetime with imaginary time, and a radial axis as well as spacetime coordinates appears spontaneously. We also find that the information of the UV limit of the original free fermions lives in the boundary of the anti-de Sitter spacetime. These findings strongly suggest that the Hessian potential for free fermions has enough geometrical meaning associated with gauge-gravity correspondence. Furthermore, some deformation of the spectrum near the conformal fixed point is mapped onto spacetime dynamics. The fluctuation of the entanglement entropy embedded into the spacetime behaves like free scaler field, and the dynamics is described by the Einstein equation with a negative cosmological constant. Therefore, the Einstein equation can be regarded as the equation of original quantum state.
This is an idiosyncratic colloquium-style review of the idea that spacetime and gravity can emerge from entanglement. Drawing inspiration from the conjectured duality between quantum gravity in anti de Sitter space and certain conformal field theories, we argue that tensor networks can be used to define a discrete geometry that encodes entanglement geometrically. With the additional assumption that a continuum limit can be taken, the resulting geometry necessarily obeys Einstein's equations. The discussion takes the point of view that the emergence of spacetime and gravity is a mysterious phenomenon of quantum many-body physics that we would like to understand. We also briefly discuss possible experiments to detect emergent gravity in highly entangled quantum systems.
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.
The Ryu-Takayanagi formula directly connects quantum entanglement and geometry. Yet the assumption of static geometry lead to an exponentially small mutual information between far-separated disjoint regions, which does not hold in many systems such as free fermion conformal field theories. In this paper, we proposed a microscopic model by superimposing entanglement features of an ensemble of random tensor networks of different bond dimensions, which can be mapped to a statistical gravity model consisting of a massive scalar field on a fluctuating background geometry. We propose a machine-learning algorithm that recovers the underlying geometry fluctuation from multi-region entanglement entropy data by modeling the bulk geometry distribution via a generative neural network. To demonstrate its effectiveness, we tested the model on a free fermion system and showed mutual information can be mediated effectively by geometric fluctuation. Remarkably, locality emerged from the learned distribution of bulk geometries, pointing to a local statistical gravity theory in the holographic bulk.
We develop a complete geometric framework in which each quantum particle possesses its own private spacetime—a world-block—constructed from Fermi–Walker coordinates. The intrinsic spatial metric on each proper-time slice is a dynamical field governed by an action with a universal stiffness constant A0. A single world-block exhibits kinematic non-locality: the metric perturbation at a point depends on the entire wavefunction through an integral relation, while maintaining dynamic locality via causal wave equations. This duality captures the essential non-locality of quantum mechanics without violating relativistic causality. When two particles interact, their world-blocks are stitched along a common boundary, forming a single compound world-block in ordinary four-dimensional spacetime. The stitching imposes local matching conditions that yield a non-separable strain field, providing a geometric account of quantum entanglement. Measurements correspond to fixing boundary conditions on one part of the compound block; the correlated outcome on the distant part is automatically determined by the shared global geometry, not by any superluminal signal. Thus, the apparent non-locality of EPR correlations is explained as a manifestation of geometric connectivity within a single 4D manifold, consistent with Bell’s theorem because the geometry itself is non-local. In the continuum limit of many overlapping blocks, coarse-graining restores an effective local description, and Newton’s law of universal gravitation emerges exactly, with Newton’s constant given by G = 3c4/(8πA0). The model offers a unified, deterministic, and fully relativistic foundation for quantum mechanics and gravity, without invoking extra dimensions or stochastic elements. Experimental signatures in ultrafast interferometry and possible connections to dark energy are discussed. The framework aligns with recent developments in emergent gravity and provides a concrete geometric realization of spacetime from quantum entanglement.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.