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Computational Modelling of the Page Curve and Entanglement Entropy in Simplified Black Hole Systems

Aaryan Wagh

Published Aug 10, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.6084/m9.figshare.33198153

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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.
We investigate the multi-partite entanglement structure of an evaporating black hole and its Hawking radiation by dividing the radiation into finer subsystems. We approximate an evaporating black hole and its radiation with a Haar-random state for this purpose. Using the multi-entropy of these configurations, we define a black hole multi-entropy curve, which describes how the multi-entropy changes during the black hole evaporation. This black hole multi-entropy curve is a natural generalization of the Page curve since the multi-entropy reduces to the entanglement entropy for the bi-partite case. The multi-entropy curve keeps increasing in the early time. It reaches the maximum value at the multi-entropy time, which is later than the Page time, and starts to decrease. However, it does not decrease to zero at the end of the black hole evaporation. This non-zero value of the multi-entropy represents the secret entanglement between Hawking particles.
Quantum computers are emerging technologies expected to become important tools for exploring various aspects of fundamental physics in the future. Therefore, we pose the question of whether quantum computers can help us to study the Page curve and the black hole information dynamics, which has been a key focus in fundamental physics. In this regard, we rigorously examine the qubit transport model, a toy qubit model of black hole evaporation on IBM’s superconducting quantum computers, to shed light on this question. Specifically, we implement the quantum simulation of the scrambling dynamics in black holes using an efficient random unitary circuit. Furthermore, we employ the swap-based many-body interference protocol and the randomized measurement protocol to measure the entanglement entropy of Hawking radiation qubits in this model. Finally, by incorporating quantum error mitigation techniques into our challenging implementation of entanglement entropy measurement protocols on the IBM quantum hardware, we accurately determine the Rényi entropy in the qubit transport model, thus showcasing the utility of quantum computers for future investigations of complex quantum systems.
A bstract We investigate the multi-partite entanglement structure of an evaporating black hole and its Hawking radiation by dividing the radiation into finer subsystems. We approximate an evaporating black hole and its radiation with a Haar-random state for this purpose. Using the multi-entropy of these configurations, we define a black hole multi-entropy curve , which describes how the multi-entropy changes during the black hole evaporation. This black hole multi-entropy curve is a natural generalization of the Page curve since the multi-entropy reduces to the entanglement entropy for the bi-partite case. The multi-entropy curve keeps increasing in the early time. It reaches the maximum value at the multi-entropy time , which is later than the Page time, and starts to decrease. However, it does not decrease to zero at the end of the black hole evaporation. This non-zero value of the multi-entropy represents the secret entanglement between Hawking particles.
A bstract If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, without resorting to typicality. Our argument is based on what we refer to as the “operator gas” approach, which allows one to understand the evolution of the microstate of the black hole from generic features of the Heisenberg evolution of operators. One key feature which leads to the Page curve is the possibility of dynamical processes where operators in the “gas” can “jump” outside the black hole, which we refer to as void formation processes. Such processes are initially exponentially suppressed, but dominate after a certain time scale, which can be used as a dynamical definition of the Page time. In the Hayden-Preskill protocol for young and old black holes, we show that void formation is also responsible for the transfer of information from the black hole to the radiation. We conjecture that void formation may provide a microscopic explanation for the recent semi-classical prescription of including islands in the calculation of the entanglement entropy of the radiation.
In this work, we describe optically inspired models for unitary black hole (BH) evaporation. The goal of these models are (i) to be operationally simple, (ii) approximately preserve the thermal nature of the emitted Hawking Radiation (HR), and (iii) attempt to reproduce the Page Curve that purports that information flows forth from the BH when it has evaporated to approximately half its initial mass. We concentrate on modeling the BH as a single mode squeezed state successively interacting, by means of beam splitters and squeezers, with vacuum modes near the horizon, giving rise to entangled pairs representing the external Hawking radiation and its partner particle inside the horizon. Since all states and operations are Gaussian throughout, we use a symplectic formalism to track the evolution of the composite system through the evolving means and variances of their quadrature operators. This allows us to easily compute correlations and entanglement between the BH and the HR, as well as calculate correlations between the BH at early and late times.
The Page curve plotted by the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole during which the electric charge does not radiate out with a uniform rate macroscopically. In this context the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match with the macroscopic phenomenon of the charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation, and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide more realistic description for the entanglement between the charged black hole and radiation during the process of evaporation.