The AMPS paradox challenges black hole complementarity by apparently constructing a way for an observer to bring information from the outside of the black hole into its interior if there is no drama at its horizon, making manifest a violation of monogamy of entanglement. We propose a new resolution to the paradox: this violation cannot be explicitly checked by an infalling observer in the finite proper time they have to live after crossing the horizon. Our resolution depends on a weak relaxation of the no-drama condition (we call it “little-drama”) which is the “complementarity dual” of scrambling of information on the stretched horizon. When translated to the description of the black hole interior, this implies that the fine-grained quantum information of infalling matter is rapidly diffused across the entire interior while classical observables and coarse-grained geometry remain unaffected. Under the assumption that information has diffused throughout the interior, we consider the difficulty of the information-theoretic task that an observer must perform after crossing the event horizon of a Schwarzschild black hole in order to verify a violation of monogamy of entanglement. We find that the time required to complete a necessary subroutine of this task, namely the decoding of Bell pairs from the interior and the late radiation, takes longer than the maximum amount of time that an observer can spend inside the black hole before hitting the singularity. Therefore, an infalling observer cannot observe monogamy violation before encountering the singularity.
S. Aaronson, Computational complexity underpinnings of the Harlow-Hayden argument, unpublished.
K.S. Thorne, R.H. Price and D.A. Macdonald, Black Holes: The Membrane Paradigm, Yale University Press, New Haven, U.S.A. (1986), pg. 367.
D.N. Page, Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole, Phys. Rev. D 13 (1976) 198 [INSPIRE].
D.N. Page, Particle Emission Rates from a Black Hole. 2. Massless Particles from a Rotating Hole, Phys. Rev. D 14 (1976) 3260 [INSPIRE].
F.G.S.L. Brandao, A. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. Lett. A 80 (2009) 012304.
J. Shore and R. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inf. Theory 26 (1980) 26.
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ArXiv ePrint: 1607.05141
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Bao, N., Bouland, A., Chatwin-Davies, A. et al. Rescuing complementarity with little drama. J. High Energ. Phys. 2016, 26 (2016). https://doi.org/10.1007/JHEP12(2016)026
- Black Holes
- Models of Quantum Gravity