Entanglement island, miracle operators and the firewall

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.

Page Curves and Bath Deformations

“…Differences in the literature and pending questions aside, a lot can be learned by either slightly deviating from or further probing the typical "brane and nongravitat-ing half-space BCFT bath" construction, as done by [35][36][37][38][39][40][41].…”

The entanglement wedge of unknown couplings

“…A crucial difference is for the complement of sys: the density matrix of ρ ptr is diagonal in the λ basis whereas ρ journal has off-diagonal elements. Furthermore, the entanglement wedge of journal can contain an island (as we will show), whereas a quick argument [12] rules out any possible island for the entanglement wedge of ptr: for the pointer system, the matter conditional entropy is positive S(A|ptr) = S(A ∪ ptr) − S(ptr) ≥ 0 for any bulk region A, so adding any the bulk region A can only increase the entropy. Therefore, any island QES must be non-minimal.…”

Page curve of AdS-Vaidya model for evaporating black holes

“…There were many proposals to try and alleviate the issues posed by the information paradox [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Among the proposals, one route that has taken much attention is through the Ryu and Takayanagi (RT) prescription which studies the fine-grained entropy in a geometrical context through the AdS/CFT correspondence, albeit only for time-independent cases [23,24].…”