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arXiv:2403.09165

The Algebraic Page Curve

César Gómez

Published Mar 14, 2024• 0 citations
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Abstract

The Page curve describing the process of black hole evaporation is derived in terms of a family, parametrized in terms of the evaporation time, of finite type II_1 factors, associated, respectively, to the entanglement wedges of the black hole and the radiation. The so defined Page curve measures the relative continuous dimension of the black hole and the radiation along the evaporation process. The transfer of information is quantitatively defined in terms of the Murray von Neumann parameter describing the change of the spatial properties of the factors during the evaporation. In the simplest case the generator of the evaporation process is defined in terms of the action of the fundamental group of the hyperfinite type II_1 factor. In this setup the Page curve describes a phase transition with the transfer of information as order parameter. We discuss the limits of either a type I or a type III description of the black hole evaporation.

DOI
10.48550/arxiv.2403.09165

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The Page curve describing the process of black hole evaporation is derived in terms of a family, parametrized in terms of the evaporation time, of finite type II_1 factors, associated, respectively, to the entanglement wedges of the black hole and the radiation. The so defined Page curve measures the relative continuous dimension of the black hole and the radiation along the evaporation process. The transfer of information is quantitatively defined in terms of the Murray von Neumann parameter describing the change of the spatial properties of the factors during the evaporation. In the simplest case the generator of the evaporation process is defined in terms of the action of the fundamental group of the hyperfinite type II_1 factor. In this setup the Page curve describes a phase transition with the transfer of information as order parameter. We discuss the limits of either a type I or a type III description of the black hole evaporation.
A bstract In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
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.
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.
Abstract The Page curve plotted using 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 macroscopically radiate out with a uniform rate. 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 the macroscopic phenomenon of 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 that 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 a more realistic description for the entanglement between the charged black hole and radiation during the evaporation process.
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.