Synthese

, Volume 192, Issue 7, pp 1989–2008

A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer

Article

Abstract

There exists a growing literature on the so-called physical Church–Turing thesis in a relativistic spacetime setting. The physical Church–Turing thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, Németi and Dávid (Appl Math Comput 178:118–142, 2006) have shown how an appreciation of the physical Church–Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour István’s seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in Németi and Dávid (Appl Math Comput 178:118–142, 2006) might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo–Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church–Turing thesis.

Keywords

Hypercomputation Relativistic computers Malament–Hogarth spacetimes Timelike entanglement 

References

  1. Aharonov, Yakir, Anandan, Jeeva, Maclay, G. Jordan, & Suzuki, Jun. (2004). Model for entangled states with spin-spin interaction. Physical Review A, 70, 052114.CrossRefGoogle Scholar
  2. Andréka, Hajnal, Németi, István, & Németi, Péter. (2009). General relativistic hypercomputing and foundation of mathematics. Natural Computing, 8, 499–516.CrossRefGoogle Scholar
  3. Bennett, Charles H., Brassard, Gilles, Crépeau, Claude, Jozsa, Richard, Peres, Asher, & Wooters, William K. (1993). Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Physical Review Letters, 70, 1895–1899.CrossRefGoogle Scholar
  4. Berkovitz, J. (2008). Action at a distance in quantum mechanics. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Available at http://plato.stanford.edu/entries/qm-action-distance/, Winter 2008 edition.
  5. Butterfield, Jeremy N., Fleming, Gordon N., Ghirardi, GianCarlo C., & Grassi, Renata. (1993). Parameter dependence in dynamical models for state reductions. International Journal of Theoretical Physics, 32, 2287–2303.Google Scholar
  6. Earman, John. (1995). Bangs, crunches, whimpers, and shrieks. New York: Oxford University Press.Google Scholar
  7. Earman, John, & Norton, John. (1993). Forever is a day: Supertasks in Pitowsky and Malament–Hogarth spacetimes. Philosophy of Science, 60, 22–42.CrossRefGoogle Scholar
  8. Earman, John, & Norton, John. (1996). Infinite pains: The trouble with supertasks. In A. Morton & S. Stich (Eds.), Benaceraff and his Critics (pp. 231–261). Cambridge, MA: Blackwell.Google Scholar
  9. Etesi, Gábor, & Németi, István. (2002). Turing computability and Malament–Hogarth spacetimes. International Journal of Theoretical Physics, 41, 342–370.CrossRefGoogle Scholar
  10. Fuentes-Schuller, Ivette, & Mann, Robert B. (2005). Alice falls into a black hole: Entanglement in noninertial frames. Physical Review Letters, 95, 120404.CrossRefGoogle Scholar
  11. Hogarth, Mark. (1992). Does general relativity allow an observer to view an eternity in a finite time? Foundations of Physics Letters, 5, 173–181.CrossRefGoogle Scholar
  12. Kennedy, John Bernard. (1995). On the empirical foundations of the quantum no-signalling proofs. Philosophy of Science, 62, 543–560.CrossRefGoogle Scholar
  13. Manchak, John Byron. (2009). Can we know the global structure of spacetime? Studies in the History and Philosophy of Modern Physics, 40, 53–56.CrossRefGoogle Scholar
  14. Manchak, John Byron. (2010). On the possibility of supertasks in general relativity. Foundations of Physics, 40, 276–288.CrossRefGoogle Scholar
  15. Maudlin, Tim. (2002). Quantum non-locality and relativity (2nd ed.). Malden: Blackwell.CrossRefGoogle Scholar
  16. Németi, István, & Andréka, Hajnal. (2006). Can general relativistic computers break the Turing barrier? In A. Beckmann, U. Berger, B. Löwe, & J. V. Tucker (Eds.), Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006, volume 3988 of Lecture Notes in Computer Science, pp. 398–412, Berlin: Springer.Google Scholar
  17. Németi, István, & Dávid, Gyula. (2006). Relativistic computers and the Turing barrier. Applied Mathematics and Computation, 178, 118–142.CrossRefGoogle Scholar
  18. Nielsen, Michael A., & Chuang, Isaac L. (2000). Quantum computation and quantum information. Cambridge: Cambridge University Press.Google Scholar
  19. Olson, S. Jay., & Ralph, Timothy. C. (2011). Entanglement between the future and the past in the quantum vacuum. Physical Review Letters, 106, 110404.Google Scholar
  20. Olson, S. Jay., & Ralph, Timothy. C. (2012). Extraction of timelike entanglement from the quantum vacuum. Physical Review A, 85, 012306.Google Scholar
  21. O’Neill, Barrett. (1995). The geometry of Kerr black holes. Wellesley: A K Peters.Google Scholar
  22. Peacock, Kent. A. (2009). The no-signalling theorems: A nitpicking distinction. Manuscript.Google Scholar
  23. Peacock, Kent. A., & Hepburn, Brian. (2000). Begging the signalling question: Quantum signalling and the dynamics of multiparticle systems. In B. Brown & J. Woods (Eds.), Logical consequences: Rival approaches and new studies in exact philosophy: Logics, mathematics, and science (Vol. II, pp. 279–292). Oxford: Hermes Science Publishers.Google Scholar
  24. Peres, Asher, & Terno, Daniel R. (2004). Quantum information and relativity theory. Reviews of Modern Physics, 76, 93–123.CrossRefGoogle Scholar
  25. Piccinini, Gualtiero. (2011). Physical Church–Turing thesis: modest or bold? British Journal for the Philosophy of Science, 62, 733–769.CrossRefGoogle Scholar
  26. Pitowsky, Itamar. (1990). The physical Church–Turing thesis and physical computational complexity. Iyyun, 39, 81–99.Google Scholar
  27. Shagrir, Oron, & Pitowsky, Itamar. (2003). Physical hypercomputation and the Church–Turing thesis. Minds and Machines, 13, 87–101.CrossRefGoogle Scholar
  28. Wald, Robert M. (1984). General relativity. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  29. Welch, Philip D. (2008). The extent of computation in Malament–Hogarth spacetimes. British Journal for the Philosophy of Science, 59, 659–674.CrossRefGoogle Scholar
  30. Wiltshire, David L, Visser, Matt, & Scott, Susan M (Eds.). (2009). The Kerr spacetime: Rotating black holes in general relativity. New York: Cambridge University Press.Google Scholar
  31. Wüthrich, Christian. (1999). On time machines in Kerr–Newman spacetimes. Master’s thesis, Bern: University of Bern.Google Scholar
  32. Xian-Hui, Ge, & You-Gen, Shen. (2005). Quantum teleportation and Kerr–Newman spacetime. Chinese Physics, 14, 1512–1516.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of CaliforniaSan Diego, La JollaUSA

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