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.
Similar content being viewed by others
Notes
We note that (3.15) demonstrates that the vector \(\sqrt{p_{0}}\left| 0\right\rangle _{A} \otimes \vert \psi _{0}\rangle _{RSCC^{\prime }}\) and the operator \(\text {Tr}_{C_{1}\ldots C_{n}}\{ V_{AC_{1}\ldots C_{n}}\}\) saturate the bound given in [11, Eq. (10)]. That is, to saturate [11, Eq. (10)], we can therein set \(P = \text {Tr}_{C_{1}\ldots C_{n}}\{ V_{AC_{1}\ldots C_{n}}\}\), \(\vert \psi \rangle = \sqrt{p_{0}}\left| 0\right\rangle _{A} \otimes \vert \psi _{0}\rangle _{RSCC^{\prime }}\), the chronology-respecting systems to be \(ARSCC^{\prime }\), and the chronology-violating systems to be \(C_{1}\ldots C_{n}\). We thank John-Mark Allen for pointing this out to us.
References
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)
Bonnor, W.B.: The rigidly rotating relativistic dust cylinder. J. Phys. A 13(6), 2121 (1980)
Gott, J.R.: Closed timelike curves produced by pairs of moving cosmic strings: exact solutions. Phys. Rev. Lett. 66(9), 1126–1129 (1991)
Deutsch, D.: Quantum mechanics near closed timelike lines. Phys. Rev. D 44(10), 3197–3217 (1991)
Bennett, CH., Schumacher, B. talk at QUPON, Wien. http://www.research.ibm.com/people/b/bennetc/ (2005)
Svetlichny, G. Effective quantum time travel. arXiv:0902.4898 (2009)
Svetlichny, George: Time travel: Deutsch vs. teleportation. Int. J. Theor. Phys. 50(12), 3903–3914 (2011)
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
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
Jozsa, R. Illustrating the concept of quantum information. IBM J. Res. Dev., 48(1):79–85, (2004). arXiv:quant-ph/0305114
Allen, John-Mark A.: Treating time travel quantum mechanically. Phys. Rev. A 90(4), 042107 (2014). arXiv:1401.4933
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)
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
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
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
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
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
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
Cavalcanti, E.G., Menicucci, N.C.: Verifiable nonlinear quantum evolution implies failure of density matrices to represent proper mixtures. (2010). arXiv:1004.1219
Cavalcanti, E.G., Menicucci, N.C., Pienaar, J.L.: The preparation problem in nonlinear extensions of quantum theory. (2012). arXiv:1206.2725
Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A, 461(2063):3473–3482 (2005). arXiv:quant-ph/0412187
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
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
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
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
Ralph, Timothy C., Downes, Tony G.: Relativistic quantum information and time machines. Contemp. Phys. 53(1), 1–16 (2012). arXiv:1111.2648
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
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
Wolf, M.M.: Guided tour, July, Quantum channels & operations (2012)
Dicke, Robert H.: Coherence in spontaneous radiation processes. Phys. Rev. 93(1), 99–110 (1954)
Berinde, V.: Iterative Approximation of Fixed Points. Springer, Berlin (2007)
Blume-Kohout, R., Croke, S., Zwolak, M.: Quantum data gathering. Sci. Rep. 3, 1800 (2013). arXiv:1201.6625
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.”
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brun, T.A., Wilde, M.M. Simulations of Closed Timelike Curves. Found Phys 47, 375–391 (2017). https://doi.org/10.1007/s10701-017-0066-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-017-0066-7