Entanglement island, miracle operators and the firewall

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what is the main message of this paper?

AI: The main message of this paper is that the smooth replica AdS geometry assumption with boundary conditions set by the bath breaks down when a particular measurement operator is applied to the bath, leading to the proposal of "miracle operators." These operators introduce extra replica wormholes connecting with the "simulated blackholes" and have implications for understanding the firewall paradox. The paper also discusses information retrieval from the black hole interior and the contradiction that arises when applying a state reconstruction formula to the special situation involving interior-information-retrieving measurements.

what is the connection to quantum error correction?

AI: There is no explicit mention or connection to quantum error correction in the given text.


Xiao-Liang Qi


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 derive a state reconstruction formula based on one norm. 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.

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1 comment


Xiao-liang, thank you for sharing your insightful work. Below is an extended AI summary FYI and for reader's information. Quick layman question (general):
It's interesting to consider how cosmological theories involving Big Bang address the information issue. It appears that before the Big Bang the information content was zero, now obviously its not entirely zero (us typing these messages is the evidence), Isn't it the proof in and by itself that Unitary cosmology (and by induction gravity) is impossible, This Cosmological information paradox seems opposite to the Black Hole Information paradox (BHIP_). Questions: do you agree that the "inverse BHIP" is a valid question and if so can your methods rooted in quantum information provide any insight on it? Can you turn your arguments around?
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Longer AI summary of the work:

  1. This research explores the paradoxical aspects of the quantum entanglement structure, particularly in the context of black holes. Understanding how quantum entanglement operates in the gravity field of black holes can help unravel fundamental issues in quantum gravity and potentially address the long-standing black hole information paradox. The Holy Grail problems in this field include discerning whether and how information about a black hole's interior can be accessed from its exterior, reconciling the principles of quantum mechanics with black hole physics, and understanding the nature and behavior of quantum entanglement across the event horizon.

  2. Previous attempts to solve these problems have utilized various theoretical constructs such as the Quantum Extremal Surface (QES) formula, AdS/CFT correspondence, and the concept of "miracle operators." However, despite numerous insightful results, a comprehensive understanding has remained elusive. In particular, existing frameworks have struggled with explaining the apparent contradiction between the information preservation requirement of quantum mechanics and the apparent destruction of information in the black hole evaporation process as implied by general relativity.

  3. The author's approach builds upon the QES formula, the idea of "miracle operators," and introduces the notion of a "branchcut" (a method to treat multiple copies of the universe in a so-called replica geometry) to probe deeper into the conundrum. The strategy is to investigate how a measurement in the exterior of the black hole can affect the quantum entanglement structure, potentially leading to qualitative changes in the bulk geometry, and the implications of these measurements for an observer falling into a black hole.

  4. The author employs a technical framework involving sophisticated mathematical structures and concepts from quantum field theory. Two main setups are considered: one in which the quantum bit of interest is outside the black hole, and another in which it has fallen into the black hole. The research introduces an operator that distinguishes two states optimally, and demonstrates that its application can lead to substantial geometric fluctuations in a replica calculation. This represents a theoretical confirmation of the possible existence of "miracle operators" that can trigger significant changes in the bulk geometry. The work concludes with a variety of intriguing future research directions, emphasizing the need for more careful analysis and the possibility of defining "miracle operators" in even more complex situations.

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