Advertisement

Foundations of Physics

, Volume 47, Issue 3, pp 375–391 | Cite as

Simulations of Closed Timelike Curves

  • Todd A. Brun
  • Mark M. WildeEmail author
Article

Abstract

Proposed models of closed timelike curves (CTCs) have been shown to enable powerful information-processing protocols. We examine the simulation of models of CTCs both by other models of CTCs and by physical systems without access to CTCs. We prove that the recently proposed transition probability CTCs (T-CTCs) are physically equivalent to postselection CTCs (P-CTCs), in the sense that one model can simulate the other with reasonable overhead. As a consequence, their information-processing capabilities are equivalent. We also describe a method for quantum computers to simulate Deutschian CTCs (but with a reasonable overhead only in some cases). In cases for which the overhead is reasonable, it might be possible to perform the simulation in a table-top experiment. This approach has the benefit of resolving some ambiguities associated with the equivalent circuit model of Ralph et al. Furthermore, we provide an explicit form for the state of the CTC system such that it is a maximum-entropy state, as prescribed by Deutsch.

Keywords

Closed timelike curves Teleportation Simulation PSPACE Deutschian CTCs Postselected CTCs Transition probability CTCs 

Notes

Acknowledgements

We are especially grateful to Tom Cooney for many enlightening discussions on fixed points of CPTP linear maps. We thank John-Mark Allen for helpful feedback that improved the manuscript. We also acknowledge Jonathan Dowling for his help in obtaining FQXI funds to support this research. Finally, we acknowledge support from the Department of Physics and Astronomy at Louisiana State University and the Foundational Questions Institute (FQXI) for supporting the grant “Closed timelike curves and quantum information processing.”

References

  1. 1.
    Gödel, K.: An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21(3), 447–450 (1949)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonnor, W.B.: The rigidly rotating relativistic dust cylinder. J. Phys. A 13(6), 2121 (1980)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Gott, J.R.: Closed timelike curves produced by pairs of moving cosmic strings: exact solutions. Phys. Rev. Lett. 66(9), 1126–1129 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Deutsch, D.: Quantum mechanics near closed timelike lines. Phys. Rev. D 44(10), 3197–3217 (1991)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bennett, CH., Schumacher, B. talk at QUPON, Wien. http://www.research.ibm.com/people/b/bennetc/ (2005)
  6. 6.
    Svetlichny, G. Effective quantum time travel. arXiv:0902.4898 (2009)
  7. 7.
    Svetlichny, George: Time travel: Deutsch vs. teleportation. Int. J. Theor. Phys. 50(12), 3903–3914 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lloyd, Seth, Maccone, Lorenzo, Garcia-Patron, Raul, Giovannetti, Vittorio, Shikano, Yutaka: The quantum mechanics of time travel through post-selected teleportation. Phys. Rev. D 84(2), 025007 (2011). arXiv:1007.2615 ADSCrossRefGoogle Scholar
  9. 9.
    Lloyd, Seth, Maccone, Lorenzo, Garcia-Patron, Raul, Giovannetti, Vittorio, Shikano, Yutaka, Pirandola, Stefano, Rozema, Lee A., Darabi, Ardavan, Soudagar, Yasaman, Shalm, Lynden K., Steinberg, Aephraim M.: Closed timelike curves via post-selection: theory and experimental demonstration. Phys. Rev. Lett. 106(4), 040403 (2011). arXiv:1005.2219 ADSCrossRefGoogle Scholar
  10. 10.
    Jozsa, R. Illustrating the concept of quantum information. IBM J. Res. Dev., 48(1):79–85, (2004). arXiv:quant-ph/0305114
  11. 11.
    Allen, John-Mark A.: Treating time travel quantum mechanically. Phys. Rev. A 90(4), 042107 (2014). arXiv:1401.4933 ADSCrossRefGoogle Scholar
  12. 12.
    Bennett, Charles H., Brassard, Gilles, Crépeau, Claude, Jozsa, Richard, Peres, Asher, Wootters, William K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Aaronson, S., Watrous, J.: Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A 465(2102), 631–647 (2009). arXiv:0808.2669
  14. 14.
    Brun, Todd A., Harrington, Jim, Wilde, Mark M.: Localized closed timelike curves can perfectly distinguish quantum states. Phys. Rev. Lett. 102(21), 210402 (2009). arXiv:0811.1209 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Brun, Todd A., Wilde, Mark M., Winter, Andreas: Quantum state cloning using Deutschian closed timelike curves. Phys. Rev. Lett. 111(19), 190401 (2013). arXiv:1306.1795 ADSCrossRefGoogle Scholar
  16. 16.
    Yuan, X., Assad, S.M., Thompson, J., Haw, J.Y., Vedral, V., Ralph, T.C., Lam, P.K., Weedbrook, C., Gu, M.: Replicating the benefits of closed timelike curves without breaking causality. npj Quantum. Inf. 1, 15007 (2015). arXiv:1412.5596
  17. 17.
    Pienaar, Jacques L., Ralph, Timothy C., Myers, Casey R.: Open timelike curves violate Heisenberg’s uncertainty principle. Phys. Rev. Lett. 110(6), 060501 (2013). arXiv:1206.5485 ADSCrossRefGoogle Scholar
  18. 18.
    Bennett, Charles H., Leung, Debbie, Smith, Graeme, Smolin, John A.: Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems? Phys. Rev. Lett. 103(17), 170502 (2009). arXiv:0908.3023 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Cavalcanti, E.G., Menicucci, N.C.: Verifiable nonlinear quantum evolution implies failure of density matrices to represent proper mixtures. (2010). arXiv:1004.1219
  20. 20.
    Cavalcanti, E.G., Menicucci, N.C., Pienaar, J.L.: The preparation problem in nonlinear extensions of quantum theory. (2012). arXiv:1206.2725
  21. 21.
    Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A, 461(2063):3473–3482 (2005). arXiv:quant-ph/0412187
  22. 22.
    Brun, Todd A., Wilde, Mark M.: Perfect state distinguishability and computational speedups with postselected closed timelike curves. Found. Phys. 42(3), 341–361 (2012). arXiv:1008.0433 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Genkina, Dina, Chiribella, Giulio, Hardy, Lucien: Optimal probabilistic simulation of quantum channels from the future to the past. Phys. Rev. A 85(2), 022330 (2012). arXiv:1112.1469 ADSCrossRefGoogle Scholar
  24. 24.
    Ringbauer, M., Broome, M.A., Myers, C.R., White, A.G., Ralph, T.C.: Experimental simulation of closed timelike curves. Nat. Commun., 5:4145 (2014). arXiv:1501.05014
  25. 25.
    Ralph, Timothy C., Myers, Casey R.: Information flow of quantum states interacting with closed timelike curves. Phys. Rev. A 82(6), 062330 (2010). arXiv:1003.1987 ADSCrossRefGoogle Scholar
  26. 26.
    Ralph, Timothy C., Downes, Tony G.: Relativistic quantum information and time machines. Contemp. Phys. 53(1), 1–16 (2012). arXiv:1111.2648 ADSCrossRefGoogle Scholar
  27. 27.
    Hayden, P., Jozsa, R., Petz, D., Winter, A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys., 246(2):359–374, (2004). arXiv:quant-ph/0304007
  28. 28.
    Milán Mosonyi. Entropy, information and structure of composite quantum states. PhD Thesis, Katholieke Universiteit Leuven (2005). Available at https://lirias.kuleuven.be/bitstream/1979/41/2/thesisbook9
  29. 29.
    Wolf, M.M.: Guided tour, July, Quantum channels & operations (2012)Google Scholar
  30. 30.
    Dicke, Robert H.: Coherence in spontaneous radiation processes. Phys. Rev. 93(1), 99–110 (1954)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Berinde, V.: Iterative Approximation of Fixed Points. Springer, Berlin (2007)zbMATHGoogle Scholar
  32. 32.
    Blume-Kohout, R., Croke, S., Zwolak, M.: Quantum data gathering. Sci. Rep. 3, 1800 (2013). arXiv:1201.6625

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Ming Hsieh Department of Electrical Engineering, Center for Quantum Information Science and TechnologyUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Physics and AstronomyHearne Institute for Theoretical Physics, Center for Computation and Technology, Louisiana State UniversityBaton RougeUSA

Personalised recommendations