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arXiv:2502.01933

Lectures on Quantum Extremal Surfaces and the Page Curve

Raghu Mahajan

Published Feb 4, 2025• 0 citations
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Abstract

This article is a write-up of pedagogical lectures delivered at the Asia Pacific Center for Theoretical Physics online winter school held in January 2021, and also as part of the Quantum Information, Quantum Field Theory and Gravity program held at ICTS, Bengaluru, in August 2024. The topics covered include a brief derivation of Hawking radiation from the perspective of correlation functions, a description of entropy paradoxes in the eternal black hole, and details of the associated entanglement island computations in two-dimensional models.

DOI
10.48550/arxiv.2502.01933

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This article is a write-up of pedagogical lectures delivered at the Asia Pacific Center for Theoretical Physics online winter school held in January 2021, and also as part of the Quantum Information, Quantum Field Theory and Gravity program held at ICTS, Bengaluru, in August 2024. The topics covered include a brief derivation of Hawking radiation from the perspective of correlation functions, a description of entropy paradoxes in the eternal black hole, and details of the associated entanglement island computations in two-dimensional models.
Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.
The quantum extremal island rule allows us to compute the Page curves of Hawking radiation in semi-classical gravity. In this work, we study the connection between these calculations and the thermalisation of chaotic quantum many-body systems, using a coarse-grained description of entanglement dynamics known as the entanglement membrane. Starting from a double-holographic model of eternal two-sided asymptotically AdSd_dd​ (d>2d>2d>2) black hole each coupled to a flat ddd-dimensional bath, we show that the entanglement dynamics in the late-time, large-subregion limit is described by entanglement membrane, thereby establishing a quantitative equivalence between a semi-classical gravity and a chaotic quantum many-body system calculation of the Page curve.
This undergraduate dissertation was completed as part of the BSc Computer Science degree at Newcastle University in May 2026. It investigates whether Page-curve-like behaviour can be reproduced in a simplified computational model of black hole evaporation using concepts from quantum information theory.The black hole information paradox arises from an apparent conflict between Hawking radiation and the unitary evolution required by quantum mechanics. Rather than attempting to simulate the full physical dynamics of an evaporating black hole, this project models the black hole and its emitted radiation as two subsystems of a finite-dimensional multi-qubit quantum system. The aim is to explore the information-theoretic behaviour associated with the Page curve through computational modelling of entanglement entropy.The model was implemented in Python using NumPy and Matplotlib. Random pure quantum states are generated and represented using density matrices, after which partial trace operations are used to obtain the reduced density matrix of the radiation subsystem. The von Neumann entropy of this subsystem is then calculated as the proportion of the overall quantum system assigned to radiation increases. Repeated simulations are averaged to examine the evolution of entanglement entropy over normalised evaporation time.The project evaluates the model using Page-curve shape, estimated Page time, variation of a mass-like evaporation parameter, scaling across different system sizes, statistical stability across repeated simulations, and validation tests using known quantum states. The baseline simulation reproduces the characteristic rise, peak, and fall associated with the Page curve, with maximum radiation entropy occurring close to the midpoint of the normalised evolution. Simulations across different qubit counts also produce broadly consistent Page-curve-like behaviour.The results should be interpreted as a computational demonstration of entanglement-entropy behaviour under a simplified information-theoretic model rather than as a complete physical simulation of black hole evaporation. The model does not include spacetime geometry, quantum field dynamics, back-reaction, or Hamiltonian time evolution. Its purpose is instead to provide a transparent and reproducible computational representation of the relationship between subsystem partitioning, quantum entanglement, entropy, and the Page curve.This record contains the final undergraduate dissertation.Degree: BSc Computer Science Institution: Newcastle University Year: 2026 Supervisor: Dr Jonte HanceThis work is an undergraduate dissertation and has not been peer-reviewed as a journal publication.
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.