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.
Highlights
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
As a consequence of this general state reconstruction formula, we show that a region in the bath that is only classically correlated with the rest of the system can never reconstruct information that is encoded to the system by applying a unitary in the interior, because the interior is space-like separated from the bath
After the measurement, we show that the state reconstruction formula suggests the state of W still has no correlation with the interior information, which is contradictory with the fact that W knows the measurement result from A
Summary
We begin by an overview of the island formula, from the point of view of replica calculation. [8, 9] For concreteness we consider a single-sided AdS black hole formed from collapse, which is coupled with a flat space bath. The discussion here is entirely parallel with the proof of the HRT formula in ordinary AdS/CFT [7, 15,16,17], except that the gravitational theory has a different boundary condition set by the radiation. Whether the radiation is a flat space CFT or a quantum computer, we expect the quantum extremal surface derivation above to hold, as long as the gravity in the AdS region remains semi-classical. We can compute quantity such as tr ρnA−mσAm and expect that it is dominated by the same saddle point as tr (ρnA) In this case one obtains tr ρnA−mσAm e−An tr ρnA−I mσmAI ;. Consider ρ as the state of an evaporating black hole coupled with the bath, in which a region of the radiation A has an entanglement island I. We will discuss a more explicit setup for extracting such information, which leads to an apparent contradiction
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