The ghost in the radiation: robust encodings of the black hole interior
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Springer
Received: March 20, 2020 Accepted: May 13, 2020 Published: June 3, 2020
The ghost in the radiation: robust encodings of the black hole interior
a
Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305, U.S.A. b Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena CA 91125, U.S.A. c School of Physics, The University of Sydney, Sydney, Australia
E-mail: [email protected], [email protected], [email protected] Abstract: We reconsider the black hole firewall puzzle, emphasizing that quantum errorcorrection, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation. Keywords: Black Holes, Black Holes in String Theory, AdS-CFT Correspondence ArXiv ePrint: 2003.05451
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)031
JHEP06(2020)031
Isaac Kim,a,c Eugene Tangb and John Preskillb
Contents 1 Introduction
1
2 Probing the radiation
6 10
4 Quantum pseudorandomness
15
5 Is Hawking radiation pseudorandom?
18
6 Pseudorandomness and decoupling
21
7 Black hole as a quantum error-correcting code 7.1 Correcting low-complexity errors 7.2 Including the probe
26 28 31
8 Theory of ghost logical operators 8.1 Exact ghost operators 8.2 Approximate ghost operators 8.3 Firewall revisited 8.4 State dependence
33 36 40 44 48
9 Inside the black hole
49
10 Conclusion
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A Approximate embedding
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B Complete set of ghost operators implies correctability
56
C Complexity of controlled unitary
60
D What if the radiation is not pseudorandom?
60
1
Introduction
The discovery that black holes emit Hawking radiation raised deep puzzles about the quantum physics of black holes [1]. What happens to quantum information tha
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