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.
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.
We reconstruct entanglement thermodynamics by means of Hessian geometry, since this method exactly generalizes thermodynamics into much wider exponential family cases including quantum entanglement. Starting with the correct first law of entanglement thermodynamics, we derive that a proper choice of the Hessian potential leads to both of the entanglement entropy scaling for quantum critical systems and hyperbolic metric (or AdS space with imaginary time). We also derive geometric representation of the entanglement entropy in which the entropy is described as integration of local conserved current of information flowing across an entangling surface. We find that the entangling surface is equivalent to the domain boundary of the Hessian potential. This feature originates in a special property of critical systems in which we can identify the entanglement entropy with the Hessian potential after the second derivative by the canonical parameters, and this identification guarantees violation of extensive nature of the entropy.
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
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.
The series Time as Phase Flow (Papers I–IV) established that time is not fundamental but emerges from the geometric phase flow of quantum information, providing a unified account of dynamically generated cosmological constants, informational temporal viscosity, and the suppression of large-scale structure. Despite these successes, a fundamental mechanism explaining why a single, stable observable reality emerges from the space of quantum informational states has remained unresolved. In this work, we introduce the framework of Information-Geometric Spacetime (IGS), in which spacetime itself emerges from the geometry of quantum information, entanglement, and modular flow. Within IGS, we define an Actualization Operator that mediates between quantum informational states and observable spacetime geometry. Derived from first principles, the operator combines entropy-based suppression with modular evolution, governed by a universal geometric constant β = 1/(π√3) ≈ 0.1837 fixed by the geometry of causal diamonds and the associated modular Hamiltonian. We show that actualization preserves information conservation and quantum unitarity, acting as a structural filter that selectively stabilizes informational states capable of forming consistent geometric reality, in close analogy with decoherence-driven classical emergence. Under this mechanism, previously identified cosmological effects—most notably the suppression of the matter power spectrum and the S_8 tension—are reinterpreted as observational consequences of information actualization rather than modifications of gravity or exotic matter. This framework also offers a geometric reformulation of the quantum measurement problem, in which the observer is not an external agent but an emergent process associated with modular structure and entanglement thermodynamics. The theory yields testable predictions in controlled quantum systems and forthcoming cosmological surveys, providing a concrete pathway toward empirical tests of spacetime emergence from quantum information.
We provide an explicit connection between the differential generation of entan-glement entropy in a tensor network representation of the ground states of two field theories, and a geometric description of these states based on the Fisher information metric. We show how the geometrical description remains invariant despite there is an irreducible gauge freedom in the definition of the tensor network. The results might help to understand how spacetimes may emerge from distributions of quantum states, or more concretely, from the structure of the quantum entanglement concomitant to those distributions.
We propose a unified geometric framework in which space and time emerge as observer-dependent projections of quantum correlations embedded in the universal wavefunction. In this approach, we formulate six axioms (including a duality between entanglement and time) that establish a correspondence between correlation structures and effective spacetime geometry. We introduce the "correlationhedron", a convex geometric object representing the set of all permissible correlation vectors of a given state, and show how different observer "slicings" (modeled by projection maps) yield distinct emergent spacetimes from this object. An emergent metric is defined on each observer's spacetime via the second derivative of a correlation density (analogous to a Fisher information metric), linking gradients in quantum entanglement to curvature. This paper establishes the conceptual foundations of this approach as the first in a planned series of investigations. We illustrate the framework with conceptual and quantitative examples, including an explicit calculation of the emergent geometry for a two- qubit system in both separable and entangled states. Connections to holography (AdS/CFT) (Maldacena1998, Ryu2006), tensor networks (Swingle2012), and relational quantum approaches (Rovelli1996) are discussed, along with experimental signatures (such as curvature variations with entanglement) that could support the theory. Finally, we outline the limitations of the framework, particularly regarding dynamics and observer-dependent causality, to guide future investigations. This kinematic framework establishes the conceptual skeleton of emergent spacetime; detailed constructions, proofs of smoothness, and dynamical laws will appear in forthcoming work.