## Abstract

The black hole information problem provides important clues for trying to piece together a quantum theory of gravity. Discussions on this topic have generally assumed that in a consistent theory of gravity and quantum mechanics, quantum theory is unmodified. In this review, we discuss the black hole information problem in the context of generalisations of quantum theory. In this preliminary exploration, we examine black holes in the setting of generalised probabilistic theories, in which quantum theory and classical probability theory are special cases. We are able to calculate the time it takes information to escape a black hole, assuming that information is preserved. In quantum mechanics, information should escape pure state black holes after half the Hawking photons have been emitted, but we find that this get’s modified in generalisations of quantum mechanics. Likewise the black-hole mirror result of Hayden and Preskill, that information from entangled black holes can escape quickly, also get’s modified. We find that although information exits the black hole as predicted by quantum theory, it is fairly generic that it fails to appear outside the black hole at this point—something impossible in quantum theory due to the no-hiding theorem. The information is neither inside the black hole, nor outside it, but is delocalised.

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## Notes

For background reading, we refer the reader to the review of John Preskill [3].

Modulo certain assumptions, the thermodynamical entropy of any theory is given in terms of the states N which can be distinguished by a physical device (based on an adaptation of the argument of von Neumann, used to justify the fact that the von Neumann entropy has a thermodynamical interpretation [34]). Since the Bekenstein bound is derived from thermodynamical considerations, logN is the entropy which should appear in the entropy bound for potential theories of quantum gravity which go beyond ordinary quantum theory.

## References

Hawking, S.W.: Breakdown of predictability in gravitational collapse. Phys. Rev. D

**14**, 2460 (1976). doi:10.1103/PhysRevD.14.2460Hawking, S.W.: The unpredictability of quantum gravity. Commun. Math. Phys.

**87**, 395 (1982)Preskill, J: Do black holes destroy information? hep-th/9209058

Banks, T., Peskin, M.E., Susskind, L.: Difficulties for the evolution of pure states into mixed states. Nucl. Phys. B

**244**, 125 (1984)Unruh, W.G., Wald, R.M.: On evolution laws taking pure states to mixed states in quantum field theory. Phys. Rev. D

**52**, 2176 (1995)Oppenheim, J., Reznik, B.: Fundamental destruction of information and conservation laws (2009, preprint). arXiv:0902.2361

Hardy, L.: Quantum theory from five reasonable axioms (2001). arXiv:quant-ph/0101012v4

Khalfin, L.A., Tsirelson, B.S.: Quantum and Quasi-Classical Analogs of Bell Inequalities, p. 441. World Scientific, Singapore (1985)

Tsirelson, B.S.: Some results and problems on quantum bell-type inequalities. Hadron. J. Suppl.

**8**(4), 329 (1993)Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys.

**24**, 379 (1994)Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A

**75**, 032304 (2007)Mielnik, B.: Generalized quantum mechanics. Commun. Math. Phys.

**37**, 221 (1974)t’Hooft, G.: On the quantum structure of a black hole. Nucl. Phys. B

**256**, 727 (1985)Aharonov, Y., Casher, A., Nussinov, S.: The unitarity puzzle and Planck mass stable particles. Phys. Lett. B

**191**, 51 (1987)Carlitz, R., Willey, R.: The lifetime of a black hole. Phys. Rev. D

**36**, 2336 (1987)Almheiri, A., Marolf, D., Polchinski, J., Sully, J.: Black holes: complementarity or firewalls? J. High Energy Phys.

**2013**, 62 (2013)Braunstein, S.L., Zyczkowski, K.: Better Late than Never: Information Retrieval from Black Holes (2009, preprint). arXiv:0907.1190v2

Page, D.: Is black hole evaporation predictable? Phys. Rev. Lett.

**44**, 301 (1980)t’Hooft, G.: The black hole interpretation of string theory. Nucl. Phys. B

**335**, 138 (1990)Susskind, L., Thorlacius, L., Uglum, J.: The stretched horizon and black hole complementarity. Phys. Rev. D

**48**, 3743 (1993). doi:10.1103/PhysRevD.48.3743Smolin, J.A., Oppenheim, J.: Locking information in black holes. Phys. Rev. Lett.

**96**, 081302 (2006). hep-th/0507287Hayden, P., Preskill, J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys.

**0709**, 120 (2007)Braunstein, S.L., Zyczkowski, K.: Entangled black holes as ciphers of hidden information (2009). arXiv:0907.1190

Sekino, Y., Susskind, L.: Fast scramblers. J. High Energy Phys.

**2008**, 065 (2008)Horodecki, M., Oppenheim, J., Winter, A.: Quantum state merging and negative information. Comm. Math. Phys.

**269**, 107 (2006). quant-ph/0512247Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A. The mother of all protocols: restructuring quantum information?s family tree (2006). quant-ph/0606225

Braunstein, S.L., Pati, A.K.: Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. Phys. Rev. Lett.

**98**, 080502 (2007). doi:10.1103/PhysRevLett.98.080502Uhlmann, A.: The? transition probability? in the state space of a*-algebra. Rep. Math. Phys.

**9**, 273 (1976)Müller, M.P., Oppenheim, J., Dahlsten, O.C.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories. J. High Energy Phys.

**2012**, 1 (2012)Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys.

**16**, 391 (1986)Braunstein, S.L., Patra, M.K.: Black hole evaporation rates without spacetime. Phys. Rev. Lett.

**107**, 071302 (2011). doi:10.1103/PhysRevLett.107.071302Page, D.N.: Information in black hole radiation. Phys. Rev. Lett.

**71**, 3743 (1993)Müller, M.P.: J. Oppenheim. (in preparation)

Müller, M.P., Ududec, C.: Structure of reversible computation determines the self-duality of quantum theory. Phys. Rev. Lett.

**108**, 130401 (2012). doi:10.1103/PhysRevLett.108.130401Bennett, C., Wiesner, S.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett

**69**, 20 (1992)

## Acknowledgments

We would like to thank the organisers of the Horizons of Quantum Theory conference for the opportunity to present our work, and their wonderful hospitality. We gratefully acknowledge feedback on a draft from Samuel Braunstein, David Jennings, Don Page and Arun Pati. J.O. and M.M. are grateful to the Aspen Center for Physics and NSF Grant 1066293, where some of this research was completed. J.O. would like to thank the Royal Society for their support. O.D. acknowledges support from the National Research Foundation (Singapore). O.D. was frequently visiting Imperial College whilst undertaking this research. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

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Müller, M.P., Oppenheim, J. & Dahlsten, O.C.O. Hiding Information in Theories Beyond Quantum Mechanics, and It’s Application to the Black Hole Information Problem.
*Found Phys* **44**, 829–842 (2014). https://doi.org/10.1007/s10701-014-9809-x

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DOI: https://doi.org/10.1007/s10701-014-9809-x