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Aspects of entanglement entropy, the information paradox and physics-informed deep learning

Filip Landgren

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

This thesis explores entanglement entropy, the black hole information paradox, and applications of physics-informed deep learning, with the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence serving as a central framework. The first part examines entanglement islands as a possible resolution to the information paradox. Islands are surfaces in spacetime that contribute to purifying Hawking radiation in semi-classical gravity. We derive the entanglement entropy for annular entangling regions on d-dimensional AdS black hole backgrounds, and show that islands are highly sensitive to the underlying field theory modeling radiation - not only in details but in their very existence. Furthermore, we show how inhomogeneous transformations of entanglement entropy, via the replica trick, encode details about island formation, reflected in the stress tensor on both replica and base manifolds. We also introduce a framework for computing replica corrections to the thermal stress tensor, with intrinsic interest. To make contact with islands in positively curved spacetimes, we construct a multiverse model using T2-deformed dS wedge holography with dS Jackiw–Teitelboim gravity on infrared end-of-the-world branes, where finite-cutoff observers recover Page curves via islands. The second part investigates physics-informed neural networks (PINNs), which embed physical laws into the loss function. Motivated by the challenge of evaluating holographic entanglement entropy with sparse boundary data, we apply Bayesian PINNs (B-PINNs) to solve the corresponding nonlinear PDEs. The latter exemplifies high-energy theory complexities, extending PINNs beyond traditional engineering uses. We examine overconfidence in these models, attributing it to physical priors in the loss rather than miscalibration, and introduce diagnostic tools to assess this effect. Further, Hessian decomposition reveals how constraints hierarchically influence network behavior and shape the solution space, in a non-trivial way when varying with loss weight adjustments.

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Islands and entanglement entropy in ddd-dimensional curved backgrounds

Filip Landgren, Arvind Shekar

A large part of the discussion on entanglement islands has explored the specific setup of $2d$ JT gravity with a flat heatbath coupled to a $2d$ CFT. In this paper, we consider a more general setup and treatment of islands in a $d-$dimensional AdS black hole background. The quantum fields modeling the Hawking radiation have a scale and are consistently inherited from a conformal parent theory; their symmetries are thus compatible with those of curved backgrounds. We demonstrate explicitly that the existence of islands is sensitive to the choice of CFT used to model the Hawking radiation. We compute the renormalised entanglement entropy of conformal fields on a negatively curved background in $d$ dimensions at zero temperature as well as the thermal regulated entropy of an entangling region near the UV boundary. Using the latter quantity as the entropy of the Hawking radiation, we find that islands never emerge for $d>2$.

arXiv:2401.01653•Jan 3, 2024•0 citations

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We give a pedagogical review of how concepts from quantum information theory build up the gravitational side of the anti-de Sitter/conformal field theory correspondence. The review is self-contained in that it only presupposes knowledge of quantum mechanics and general relativity; other tools-including holographic duality itself-are introduced in the text. We have aimed to give researchers interested in entering this field a working knowledge sufficient for initiating original projects. The review begins with the laws of black hole thermodynamics, which form the basis of this subject, then introduces the Ryu-Takayanagi proposal, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) relation, and subregion duality. We discuss tensor networks as a visualization tool and analyze various network architectures in detail. Next, several modern concepts and techniques are discussed: Rényi entropies and the replica trick, differential entropy and kinematic space, modular Berry phases, modular minimal entropy, entanglement wedge cross-sections, bit threads, and others. We discuss the extent to which bulk geometries are fixed by boundary entanglement entropies, and analyze the relations such as the monogamy of mutual information, which boundary entanglement entropies must obey if a state has a semiclassical bulk dual. We close with a discussion of black holes, including holographic complexity, firewalls and the black hole information paradox, islands, and replica wormholes.

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This thesis is devoted to a possible generalization of the entanglement entropy and its implication to the holography, especially AdS/CFT correspondence.The AdS/CFT correspondence is a way to define the quantum gravity in d + 1-dimensional asymptotically anti-de Sitter (AdS) spacetime by using the conformal field theory (CFT) in d-dimension.In this context, it is still unclear how we can realize the d + 1-dimensional full bulk geometries from the d-dimensional quantum field theories.The main object of this thesis is the minimal surface not necessarily anchored on the boundary, which has not been derived from the field theory analysis.First, we review how we can obtain minimal surfaces anchored on the boundary from the entanglement entropy.Then, we introduce a generalization of the entanglement entropy for mixed states and discuss its basic properties.Finally, we argue that we can extract the aforementioned minimal surfaces ending on the bulk from the generalized version of the entanglement entropy.Our explicit derivations include the geometries in three-dimensional AdS and BTZ black holes.First and foremost, I would like to express my deepest gratitude to Norihiro Iizuka, my supervisor.I could not reach here without his patient guidance and encouragement.

•Mar 25, 2019•0 citations

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Entanglement entropy is crucial for understanding the link between quantum mechanics and information theory. This thesis investigates how energy fluctuations and acceleration affect entanglement entropy through three key scenarios. First, we analyze entanglement between disjoint Rindler observers in de Sitter spacetime. We introduce 1-loop corrections to the partition function of the conformal field theory, disrupting the previously flat entanglement spectrum and resulting in an area law for entanglement entropy, while preserving the finiteness of the Hilbert space. Next, we extend the holographic entanglement entropy framework to field theories with global $U(1)$ symmetry, considering non-linearly charged theories and their thermal and quantum fluctuations. We find that the dual theory experiences spontaneous symmetry breaking and a reduction in the leading-order coefficient of the $\langle TT\rangle$ correlator due to quantum corrections. Using conformal electrodynamics coupled with three-dimensional gravity as an example, we explore a black hole solution and its connection to two-dimensional free bosons. Finally, we investigate three-dimensional accelerating black holes, revealing that increased acceleration reduces the accessible boundary region and, consequently, the entanglement entropy. This suggests a loss of information in the dual theory due to acceleration.

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Where in the Island is the Information?

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The Island Formula for Quantum Extremal Surfaces of large AdS black holes, which are made to evaporate by coupling the CFT of the AdS/CFT correspondence to an external system, demonstrates that the space time picture can reproduce the quantum mechanically required Page curve of the CFT/bath system if a region in the interior of the black hole has become entangled with the external radiation by the Page time. In an effective quantum field theory (EFT) description in ``nice slice" coordinates, this appears to imply paradoxical entanglement between degrees of freedom that have been causally disconnected since long before the Page time. We will give two arguments that all of the quantum information in the island is in fact concentrated in a tiny space-like portion of the island very close to the black hole horizon at the Page time. The first applies the covariant entropy bound\cite{fsb} (CEB) to the causal diamonds of geodesics originating on the space-like region that becomes the island, but on a time slice just before the Page time. All points on the island except those in a stretched horizon region very close to the horizon have entropy bounds that are tiny fractions of the black hole entropy. The second argument uses standard quantum statistical mechanics combined with the tensor network renormalization group regularization of CFT to argue that the wave function of a typical pure state in the microcanonical evaporating black hole always has most of its probability concentrated in a small range of wavelengths corresponding to a small range of radii in the dual tensor network/AdS slice.

arXiv:2610.02406•Oct 1, 2026•0 citations

Quantum Information Approaches to Holographic Dualities

A. Jahn

The AdS/CFT correspondence, a remarkable duality between certain gravitational theories in anti-de Sitter (AdS) spacetime and quantum field theories with conformal symmetry (CFT), has had a profound effect on the development of theoretical physics in the past two decades. Recently, many connections of AdS/CFT to quantum information theory have been found, in particular by providing gravitationally dual descriptions of various entanglement measures. Understanding these manifestions of AdS/CFT --- or more generally, the conjectured holographic principle encompassing it --- requires the combination of tools from both high-energy theory and quantum information physics. In this cumulative thesis, the convergence between these two fields is approached from two fronts: First, by calculations within the dual gravitational theory, and second, using a tensor network ansatz to describe the quantum states suspected to possess such a gravitational description. In the first approach, using the gravitational side of AdS/CFT, entanglement entropies of complicated 2+1-dimensional excited CFTs are computed, thus showing how the holographic approach provides access to systems previously out of reach of practical methods, while introducing new numerical methods that this approach necessitates. The second approach is given by tensor networks, a highly successful ansatz for computing properties of one- and two-dimensional quantum systems. Efficiently computable classes of tensor networks are tested in their ability to represent simple holographic systems, successfully reproducing both hyperbolic geometrical features as well as critical boundary states. In addition, the general properties of tensor networks on regular hyperbolic tesselations are considered, leading to new connections to models not previously considered in the context of holography. This interplay of different approaches to quantum information holography showcases the richness of this new field and suggests that a wide range of physical phenomena is accessible via the new tools now at our disposal.

•Jan 1, 2021•0 citations
This thesis explores entanglement entropy, the black hole information paradox, and applications of physics-informed deep learning, with the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence serving as a central framework. The first part examines entanglement islands as a possible resolution to the information paradox. Islands are surfaces in spacetime that contribute to purifying Hawking radiation in semi-classical gravity. We derive the entanglement entropy for annular entangling regions on d-dimensional AdS black hole backgrounds, and show that islands are highly sensitive to the underlying field theory modeling radiation - not only in details but in their very existence. Furthermore, we show how inhomogeneous transformations of entanglement entropy, via the replica trick, encode details about island formation, reflected in the stress tensor on both replica and base manifolds. We also introduce a framework for computing replica corrections to the thermal stress tensor, with intrinsic interest. To make contact with islands in positively curved spacetimes, we construct a multiverse model using T2-deformed dS wedge holography with dS Jackiw–Teitelboim gravity on infrared end-of-the-world branes, where finite-cutoff observers recover Page curves via islands. The second part investigates physics-informed neural networks (PINNs), which embed physical laws into the loss function. Motivated by the challenge of evaluating holographic entanglement entropy with sparse boundary data, we apply Bayesian PINNs (B-PINNs) to solve the corresponding nonlinear PDEs. The latter exemplifies high-energy theory complexities, extending PINNs beyond traditional engineering uses. We examine overconfidence in these models, attributing it to physical priors in the loss rather than miscalibration, and introduce diagnostic tools to assess this effect. Further, Hessian decomposition reveals how constraints hierarchically influence network behavior and shape the solution space, in a non-trivial way when varying with loss weight adjustments.
A large part of the discussion on entanglement islands has explored the specific setup of 2d2d2d JT gravity with a flat heatbath coupled to a 2d2d2d CFT. In this paper, we consider a more general setup and treatment of islands in a d−d-d−dimensional AdS black hole background. The quantum fields modeling the Hawking radiation have a scale and are consistently inherited from a conformal parent theory; their symmetries are thus compatible with those of curved backgrounds. We demonstrate explicitly that the existence of islands is sensitive to the choice of CFT used to model the Hawking radiation. We compute the renormalised entanglement entropy of conformal fields on a negatively curved background in ddd dimensions at zero temperature as well as the thermal regulated entropy of an entangling region near the UV boundary. Using the latter quantity as the entropy of the Hawking radiation, we find that islands never emerge for d>2d>2d>2.
We give a pedagogical review of how concepts from quantum information theory build up the gravitational side of the anti-de Sitter/conformal field theory correspondence. The review is self-contained in that it only presupposes knowledge of quantum mechanics and general relativity; other tools-including holographic duality itself-are introduced in the text. We have aimed to give researchers interested in entering this field a working knowledge sufficient for initiating original projects. The review begins with the laws of black hole thermodynamics, which form the basis of this subject, then introduces the Ryu-Takayanagi proposal, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) relation, and subregion duality. We discuss tensor networks as a visualization tool and analyze various network architectures in detail. Next, several modern concepts and techniques are discussed: Rényi entropies and the replica trick, differential entropy and kinematic space, modular Berry phases, modular minimal entropy, entanglement wedge cross-sections, bit threads, and others. We discuss the extent to which bulk geometries are fixed by boundary entanglement entropies, and analyze the relations such as the monogamy of mutual information, which boundary entanglement entropies must obey if a state has a semiclassical bulk dual. We close with a discussion of black holes, including holographic complexity, firewalls and the black hole information paradox, islands, and replica wormholes.
This thesis is devoted to a possible generalization of the entanglement entropy and its implication to the holography, especially AdS/CFT correspondence.The AdS/CFT correspondence is a way to define the quantum gravity in d + 1-dimensional asymptotically anti-de Sitter (AdS) spacetime by using the conformal field theory (CFT) in d-dimension.In this context, it is still unclear how we can realize the d + 1-dimensional full bulk geometries from the d-dimensional quantum field theories.The main object of this thesis is the minimal surface not necessarily anchored on the boundary, which has not been derived from the field theory analysis.First, we review how we can obtain minimal surfaces anchored on the boundary from the entanglement entropy.Then, we introduce a generalization of the entanglement entropy for mixed states and discuss its basic properties.Finally, we argue that we can extract the aforementioned minimal surfaces ending on the bulk from the generalized version of the entanglement entropy.Our explicit derivations include the geometries in three-dimensional AdS and BTZ black holes.First and foremost, I would like to express my deepest gratitude to Norihiro Iizuka, my supervisor.I could not reach here without his patient guidance and encouragement.
Entanglement entropy is crucial for understanding the link between quantum mechanics and information theory. This thesis investigates how energy fluctuations and acceleration affect entanglement entropy through three key scenarios. First, we analyze entanglement between disjoint Rindler observers in de Sitter spacetime. We introduce 1-loop corrections to the partition function of the conformal field theory, disrupting the previously flat entanglement spectrum and resulting in an area law for entanglement entropy, while preserving the finiteness of the Hilbert space. Next, we extend the holographic entanglement entropy framework to field theories with global U(1)U(1)U(1) symmetry, considering non-linearly charged theories and their thermal and quantum fluctuations. We find that the dual theory experiences spontaneous symmetry breaking and a reduction in the leading-order coefficient of the ⟨TT⟩\langle TT\rangle⟨TT⟩ correlator due to quantum corrections. Using conformal electrodynamics coupled with three-dimensional gravity as an example, we explore a black hole solution and its connection to two-dimensional free bosons. Finally, we investigate three-dimensional accelerating black holes, revealing that increased acceleration reduces the accessible boundary region and, consequently, the entanglement entropy. This suggests a loss of information in the dual theory due to acceleration.
The Island Formula for Quantum Extremal Surfaces of large AdS black holes, which are made to evaporate by coupling the CFT of the AdS/CFT correspondence to an external system, demonstrates that the space time picture can reproduce the quantum mechanically required Page curve of the CFT/bath system if a region in the interior of the black hole has become entangled with the external radiation by the Page time. In an effective quantum field theory (EFT) description in ``nice slice" coordinates, this appears to imply paradoxical entanglement between degrees of freedom that have been causally disconnected since long before the Page time. We will give two arguments that all of the quantum information in the island is in fact concentrated in a tiny space-like portion of the island very close to the black hole horizon at the Page time. The first applies the covariant entropy bound\cite{fsb} (CEB) to the causal diamonds of geodesics originating on the space-like region that becomes the island, but on a time slice just before the Page time. All points on the island except those in a stretched horizon region very close to the horizon have entropy bounds that are tiny fractions of the black hole entropy. The second argument uses standard quantum statistical mechanics combined with the tensor network renormalization group regularization of CFT to argue that the wave function of a typical pure state in the microcanonical evaporating black hole always has most of its probability concentrated in a small range of wavelengths corresponding to a small range of radii in the dual tensor network/AdS slice.
The AdS/CFT correspondence, a remarkable duality between certain gravitational theories in anti-de Sitter (AdS) spacetime and quantum field theories with conformal symmetry (CFT), has had a profound effect on the development of theoretical physics in the past two decades. Recently, many connections of AdS/CFT to quantum information theory have been found, in particular by providing gravitationally dual descriptions of various entanglement measures. Understanding these manifestions of AdS/CFT --- or more generally, the conjectured holographic principle encompassing it --- requires the combination of tools from both high-energy theory and quantum information physics. In this cumulative thesis, the convergence between these two fields is approached from two fronts: First, by calculations within the dual gravitational theory, and second, using a tensor network ansatz to describe the quantum states suspected to possess such a gravitational description. In the first approach, using the gravitational side of AdS/CFT, entanglement entropies of complicated 2+1-dimensional excited CFTs are computed, thus showing how the holographic approach provides access to systems previously out of reach of practical methods, while introducing new numerical methods that this approach necessitates. The second approach is given by tensor networks, a highly successful ansatz for computing properties of one- and two-dimensional quantum systems. Efficiently computable classes of tensor networks are tested in their ability to represent simple holographic systems, successfully reproducing both hyperbolic geometrical features as well as critical boundary states. In addition, the general properties of tensor networks on regular hyperbolic tesselations are considered, leading to new connections to models not previously considered in the context of holography. This interplay of different approaches to quantum information holography showcases the richness of this new field and suggests that a wide range of physical phenomena is accessible via the new tools now at our disposal.