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International Journal of Physics and Applications

Entanglement-induced geometry deformations: A holographic mapping from boundary entropy to bulk curvature

Vinod A Rode

Published Jul 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.33545/26647575.2025.v7.i2e.218

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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.
We show that the gradient of entanglement entropy across a holographic boundary, as described by the Ryu–Takayanagi formula, generates an effective force thatobeys Einstein’s field equations. By representing the entanglement structure of aquantum many-body system as a MERA tensor network, we demonstrate that thevariation of the entanglement entropy with respect to the boundary metric yieldsthe Einstein tensor. The result provides a constructive link between quantum information theory and classical gravity, in which spacetime curvature emerges asa thermodynamic consequence of entanglement renormalization. A testable prediction is derived: the entanglement-driven acceleration of a test particle near aholographic screen deviates from the Newtonian prediction by a factor proportionalto the mutual information density.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of a minimal surface in its holographic dual. We show that this relation can be inverted for any state in the conformal field theory to compute the bulk stress-energy tensor near the boundary of the bulk spacetime, reconstructing the local data in the bulk from the entanglement on the boundary. We also show that positivity, monotonicity, and convexity of the relative entropy for small spherical domains between the reduced density matrices of any state and of the ground state of the conformal field theory are guaranteed by positivity conditions on the bulk matter energy density. As positivity and monotonicity of the relative entropy are general properties of quantum systems, this can be interpreted as a derivation of bulk energy conditions in any holographic system for which the Ryu-Takayanagi prescription applies. We discuss an information theoretical interpretation of the convexity in terms of the Fisher metric.
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT[over ¯]. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT[over ¯] deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT[over ¯]-deformed CFT is the holographic dual of a finite region of spacetime.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.