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arXiv:2311.07655 · Journal of High Energy Physics

Page curves and replica wormholes from random dynamics

Jan De Boer A, Jildou Hollander, Andrew Rolph

Published Jul 3, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep07(2024)023

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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.
A bstract Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description.
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.
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.
Naively, resolving the black hole information paradox requires microscopic details about quantum gravity. Recent work suggests that, instead, a unitary Page curve can be recovered by adding disorder-averaged replica instantons to the path integral, though their origin is unclear. In this Letter, we show how replica instantons and disorder averaging emerge naturally in an effective theory built from typical microscopic states. We relate replica instantons to a moment expansion of simple operators, and find a microcanonical description in terms of wormholes and Euclidean black holes.
Abstract Recent work has shown how to understand the Page curve of an evaporating black hole from replica wormholes. However, more detailed information about the structure of its quantum state is needed to fully understand the dynamics of black hole evaporation. Here we study entanglement negativity, an important measure of quantum entanglement in mixed states, in a couple of toy models of evaporating black holes. We find four phases dominated by different types of geometries: the disconnected, cyclically connected, anti-cyclically connected, and pairwise connected geometries. The last of these geometries are new replica wormholes that break the replica symmetry spontaneously. We also analyze the transitions between these four phases by summing more generic replica geometries using a Schwinger-Dyson equation. In particular, we find enhanced corrections to various negativity measures near the transition between the cyclic and pairwise phase.
In a close look at the replica wormhole derivation of the island conjecture, we note a discrepancy in the derived matrix eigenvalues in the limit where the Hawking radiation entropy goes to zero. The small corrections to the density matrix for the radiation infer that tr(ρR2){\rm tr}(ρ_\textsf{R}^2)tr(ρR2​), equally a sum over the eigenvalues squared, must be small (≪1\ll1≪1) at all times. This is in contradiction with an ensemble dominated by a density matrix with an eigenvalue close to unity. Also, the radiation entropy has been approximated by the ensemble averaged entropy, as in a final state analysis. For ρR=ρR(σ)ρ_\textsf{R}=ρ_\textsf{R}(σ)ρR​=ρR​(σ) with σσσ the set of variables used in the average, the ensemble average is only an accurate estimate of the entropy after σσσ has been measured. At the formation of each Hawking pair, there is a distribution over σσσ that affects the entropy. Without a real time determination of σσσ, the radiation entropy follows Hawking's result during the evaporation. A question remains of how quickly such determinations can be assumed to occur. A complete argument for that the radiation entropy follows the Page curve should address this.