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From Hawking to Marchenko–Pastur: A Guide to the Page Curve and Beyond

Bernd Johannes Wuebben

Published Oct 11, 2026• 0 citations
Publisher

Abstract

This tutorial develops the Page curve program from quantum entanglement and Hawking radiation to holographic entropy, replica wormholes, and random matrix theory. We derive the normalized random-state benchmark and its Marchenko–Pastur limit, explaining the roles of non-crossing partitions and spectral resolvents. We then distinguish averaging conditional entropies from calculating the entropy of a state with fluctuating energy or area. Gaussian broadening, finite reservoirs, and a quartic critical model illustrate how the preparation and the fluctuation law enter this distinction. Exact finite-dimensional examples separate averaged spectra from ensemble fluctuations and radiation data for one input from information about a quantum channel. Normalized annealed and quenched Rényi entropies have a common von Neumann limit, while their finite-order difference and connected replica moments measure additional ensemble information. Throughout, explicit calculations are distinguished from gravitational applications that require further assumptions. The resulting account provides a self-contained introduction to the established methods and to the remaining questions about microscopic information recovery.

DOI
10.5281/zenodo.23643249

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•Aug 9, 2026•0 citations
This tutorial develops the Page curve program from quantum entanglement and Hawking radiation to holographic entropy, replica wormholes, and random matrix theory. We derive the normalized random-state benchmark and its Marchenko–Pastur limit, explaining the roles of non-crossing partitions and spectral resolvents. We then distinguish averaging conditional entropies from calculating the entropy of a state with fluctuating energy or area. Gaussian broadening, finite reservoirs, and a quartic critical model illustrate how the preparation and the fluctuation law enter this distinction. Exact finite-dimensional examples separate averaged spectra from ensemble fluctuations and radiation data for one input from information about a quantum channel. Normalized annealed and quenched Rényi entropies have a common von Neumann limit, while their finite-order difference and connected replica moments measure additional ensemble information. Throughout, explicit calculations are distinguished from gravitational applications that require further assumptions. The resulting account provides a self-contained introduction to the established methods and to the remaining questions about microscopic information recovery.
Abstract In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Rényi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half of the circle size), the dominant saddle connects the replicas in a non-trivial way using the singular boundary geometry. The result extends the ‘island conjecture’ to a non-evaporating setting.
Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulas for two important statistical ensembles with a U(1)U(1)U(1)-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the subleading finite-size corrections.
A bstract We show how to capture both the non-unitary Page curve and replica wormhole-like contributions that restore unitarity in a toy quantum system with random dynamics. The motivation is to find the simplest dynamical model that captures this aspect of gravitational physics. In our model, we evolve with an ensemble of Hamiltonians with GUE statistics within microcanonical windows. The entropy of the averaged state gives the non-unitary curve, the averaged entropy gives the unitary curve, and the difference comes from matrix index contractions in the Haar averaging that connect the density matrices in a replica wormhole-like manner.
We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group SNS_NSN​ . The paper is self-contained, providing all the necessary mathematical background.
The interior of a pure-state black hole is known to be reconstructed from the Petz map by collecting a sufficiently large amount of the emitted Hawking radiation. This was established based on the Euclidean replica wormhole, which comes from an ensemble averaging over gravitational theories. On the other hand, this means that the Page curve and the interior reconstruction are both ensemble averages; thus, there is a possibility of large errors. In the previous study [R. Bousso and M. Miyaji, Fluctuations in the entropy of Hawking radiation], it was shown that the entropy of the Hawking radiation has fluctuation of order e−SBH , thus is typical in the ensemble. In the present article, we show that the fluctuations in the information recovery are similarly suppressed by the factor e−SBH by studying the fluctuations of the relative entropy difference in the encoding map and the entanglement fidelity of the Petz map. This result establishes the typicality of the information recovery in the ensemble. In addition, we also compute the entanglement loss of the encoding map. Published by the American Physical Society 2024
The Page curve, the rise and fall of radiation entropy during unitary evaporation, is now measurable on quantum simulators: trapped-ion and superconducting platforms routinely prepare scrambled states of 10–50 qubits and estimate subsystem Rényi entropies by randomized measurements. Semiclassical gravity, including the quantum-extremal-surface and replica-wormhole machinery that derives the Page curve, is constructed in the G → 0 limit and makes no prediction at finite qubit number. We state a set of exact finite-N laws for a concrete evaporation protocol — the location and determinism of the fine-grained transition, its first-order (whole-island) character, a quantized entanglement barrier with a per-qubit mechanism identity, the barrier's monotone response to scrambling depth, and a one-particle "mode books" formula that predicts the transition time under partial (graybody) emission — together with the complete experimental design: register sizes, gate budgets (~84–348 two-qubit gates at 12–24 qubits), a one-line noise mitigation certified to 3× typical hardware error rates, a measurement plan in which no register larger than max(t, N_I − t + 1) qubits is ever measured, and a total shot budget of order 10⁵. Every prediction is stated with the condition that falsifies it. All numbers were fixed before any experiment; the reproduction package included with this deposit reproduces every claim, including the exact hardware-numbers table, in under a minute.