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arXiv:2312.11085

Reduced Half-sided Translations and Islands

Krishna Jalan, Roji Pius

Published Dec 18, 2023• 0 citations
arXivHTMLPDF

Abstract

An AdS$_2$ black hole in equilibrium with a finite temperature non-gravitating bath comes with a Hawking-like information paradox. The resolution of this paradox requires introducing an island for the bath after the Page time. Since the island region contains the black hole interior, the consistency of the island paradigm demands the failure of any interior reconstruction proposal that uses only the operators in the AdS$_2$ region outside the black hole horizon after the Page time. In this paper, we investigate the consistency of the island paradigm using the black hole interior reconstruction proposal due to Leutheusser and Liu. They argued that the operators in the interior of a black hole can be reconstructed by the half-sided translations of the operators outside the black hole horizon. However, in order to illustrate the island paradigm we need to modify the Leutheusser-Liu proposal by introducing the notion of the reduced half-sided translations. The reduced half-sided translations, unlike the half-sided translations, have non-trivial action only on the operators in the AdS$_2$ black hole region. As a result, the reconstructed interior operators are expressed only using the operators in the AdS$_2$ region outside the black hole horizon. We demonstrate the consistency of the island paradigm by showing that the reduced half-sided translation, which successfully reconstruct the operators in the black hole interior before the Page time, fails to reconstruct the interior operators after the Page time.

DOI
10.48550/arxiv.2312.11085

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An AdS2_22​ black hole in equilibrium with a finite temperature non-gravitating bath comes with a Hawking-like information paradox. The resolution of this paradox requires introducing an island for the bath after the Page time. Since the island region contains the black hole interior, the consistency of the island paradigm demands the failure of any interior reconstruction proposal that uses only the operators in the AdS2_22​ region outside the black hole horizon after the Page time. In this paper, we investigate the consistency of the island paradigm using the black hole interior reconstruction proposal due to Leutheusser and Liu. They argued that the operators in the interior of a black hole can be reconstructed by the half-sided translations of the operators outside the black hole horizon. However, in order to illustrate the island paradigm we need to modify the Leutheusser-Liu proposal by introducing the notion of the reduced half-sided translations. The reduced half-sided translations, unlike the half-sided translations, have non-trivial action only on the operators in the AdS2_22​ black hole region. As a result, the reconstructed interior operators are expressed only using the operators in the AdS2_22​ region outside the black hole horizon. We demonstrate the consistency of the island paradigm by showing that the reduced half-sided translation, which successfully reconstruct the operators in the black hole interior before the Page time, fails to reconstruct the interior operators after the Page time.
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.
Abstract In this paper, we obtain some general results on information retrieval from the black hole interior, based on the recent progress on quantum extremal surface formula and entanglement island. We study an AdS black hole coupled to a bath with generic dynamics, and ask whether it is possible to retrieve information about a small perturbation in the interior from the bath system. We show that the one-norm distance between two reduced states in a bath region A is equal to the same quantity in the bulk quantum field theory for region AI where I is the entanglement island of A. This is a straightforward generalization of bulk-boundary correspondence in AdS/CFT. However, we show that a contradiction arises if we apply this result to a special situation when the bath dynamics includes a unitary operation that carries a particular measurement to a region A and send the result to another region W. Physically, the contradiction arises between transferability of classical information during the measurement, and non-transferability of quantum information which determines the entanglement island. We propose that the resolution of the contradiction is to realize that the state reconstruction formula does not apply to the special situation involving interior-information-retrieving measurements. This implies that the assumption of smooth replica AdS geometry with boundary condition set by the flat space bath has to break down when the particular measurement operator is applied to the bath. Using replica trick, we introduce an explicitly construction of such operator, which we name as “miracle operators”. From this construction we see that the smooth replica geometry assumption breaks down because we have to introduce extra replica wormholes connecting with the “simulated blackholes” introduced by the miracle operator. We study the implication of miracle operators in understanding the firewall paradox.
A bstract When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/ 2 π log S BH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlog⁡SBHβ/2π\log S_{BH}β/2πlogSBH​ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
A bstract Recent works have suggested that the entanglement wedge of Hawking radiation coming from an AdS black hole will include an island inside the black hole interior after the Page time. In this paper, we propose a concrete way to extract the information from the island by acting only on the radiation degrees of freedom, building on the equivalence between the boundary and bulk modular flow. We consider examples with black holes in JT gravity coupled to baths. In the case that the bulk conformal fields contain free massless fermion field, we provide explicit bulk picture of the information extraction process, where we find that one can almost pull out an operator from the island to the bath with modular flow.