Foundations of Physics

, Volume 42, Issue 3, pp 341–361 | Cite as

Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves



Bennett and Schumacher’s postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch’s model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between “Deutschian” CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.


Postselected closed time-like curves State distinguishability Paradoxical computation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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) ADSMATHCrossRefGoogle Scholar
  2. 2.
    Bonnor, W.B.: The rigidly rotating relativistic dust cylinder. J. Phys. A, Math. Gen. 13(6), 2121 (1980) MathSciNetADSCrossRefGoogle 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) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Hartle, J.B.: Unitarity and causality in generalized quantum mechanics for nonchronal spacetimes. Phys. Rev. D, Part. Fields 49(12), 6543–6555 (1994) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Morris, M.S., Thorne, K.S., Yurtsever, U.: Wormholes, time machines, and the weak energy condition. Phys. Rev. Lett. 61(13), 1446–1449 (1988) ADSCrossRefGoogle Scholar
  6. 6.
    Deutsch, D.: Quantum mechanics near closed timelike lines. Phys. Rev. D, Part. Fields 44(10), 3197–3217 (1991) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Bacon, D.: Quantum computational complexity in the presence of closed timelike curves. Phys. Rev. A 70(3), 032309 (2004) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Aaronson, S., Watrous, J.: Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A 465(2102), 631–647 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Brun, T.A., Harrington, J., Wilde, M.M.: Localized closed timelike curves can perfectly distinguish quantum states. Phys. Rev. Lett. 102(21), 210402 (2009) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    DeJonghe, R., Frey, K., Imbo, T.: Discontinuous quantum evolutions in the presence of closed timelike curves. Phys. Rev. D, Part. Fields 81, 087501 (2010). arXiv:0908.2655 ADSCrossRefGoogle Scholar
  11. 11.
    Pati, A.K., Chakrabarty, I., Agrawal, P.: Purification of mixed state with closed timelike curve is not possible. May 2010. arXiv:1003.4221
  12. 12.
    Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum channel. Probl. Inf. Transm. 9, 177–183 (1973) MathSciNetGoogle Scholar
  13. 13.
    Bennett, C.H., Leung, D., Smith, G., Smolin, J.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) MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Bennett, C.H., Leung, D., Smith, G., Smolin, J.: The impotence of nonlinearity: Why closed timelike curves and nonlinear quantum mechanics don’t improve quantum state discrimination, and haven’t been shown to dramatically speed up computation, if computation is defined in a natural, adversarial way. In: Rump Session Presentation at the 13th Workshop on Quantum Information Processing, Zurich, Switzerland, January 2010 (2010) Google Scholar
  15. 15.
    Ralph, T.C., Myers, C.R.: Information flow of quantum states interacting with closed timelike curves. March 2010. arXiv:1003.1987
  16. 16.
    Cavalcanti, E.G., Menicucci, N.C.: Verifiable nonlinear quantum evolution implies failure of density matrices to represent proper mixtures. April 2010. arXiv:1004.1219
  17. 17.
    Wallman, J.J., Bartlett, S.D.: Revisiting consistency conditions for quantum states of systems on closed timelike curves: an epistemic perspective. May 2010. arXiv:1005.2438
  18. 18.
    Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V., Shikano, Y., Pirandola, S., Rozema, L.A., Darabi, A., Soudagar, Y., Shalm, L.K., Steinberg, A.M.: Closed timelike curves via post-selection: theory and experimental demonstration. May 2010. arXiv:1005.2219
  19. 19.
    Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V., Shikano, Y.: The quantum mechanics of time travel through post-selected teleportation. July 2010. arXiv:1007.2615
  20. 20.
    Svetlichny, G.: Effective quantum time travel. February 2009. arXiv:0902.4898
  21. 21.
    Bennett, C.H.: Talk at QUPON, Wien., May 2005
  22. 22.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993) MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A 461(2063), 3473–3482 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Brun, T.A.: Computers with closed timelike curves can solve hard problems. Found. Phys. Lett. 16, 245–253 (2003) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chefles, A.: Unambiguous discrimination between linearly independent quantum states. Phys. Lett. A 239, 339–347 (1998) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Bennett, C.H.: Private communication. In: 12th Workshop on Quantum Information Processing, Albuquerque, New Mexico, January 2009 (2009) Google Scholar
  27. 27.
    Smith, G.: Private communication. In: International Symposium on Information Theory, Austin, Texas, June 2010 (2010) Google Scholar
  28. 28.
    Horowitz, G.T., Maldacena, J.: The black hole final state. J. High Energy Phys. 2004(02), 008 (2004) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lloyd, S.: Almost certain escape from black holes in final state projection models. Phys. Rev. Lett. 96(6), 061302 (2006) MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  31. 31.
    Bennett, C.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992) MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Bennett, C., Brassard, G.: Quantum cryptography: Public-key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, December 1984, pp. 175–179 (1984) Google Scholar
  33. 33.
    Scarani, V., Acin, A., Ribordy, G., Gisin, N.: Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulse implementations. Phys. Rev. Lett. 92, 057901 (2004) ADSCrossRefGoogle Scholar
  34. 34.
    d’Espagnat, B.: On Physics and Philosophy. Princeton University Press, Princeton (2006). ISBN:978-0-691-11964-9 Google Scholar
  35. 35.
    Cook, S.A.: The complexity of theorem proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on the Theory of Computing, pp. 151–158 (1971) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Communication Sciences InstituteUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations