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Operational formulation of time reversal in quantum theory

Nature Physics volume 11, pages 853–858 (2015)Cite this article

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Abstract

The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Born’s rule do not apply backward in time. Here, we resolve this problem within a rigorous operational probabilistic framework. We argue that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations through both pre- and post-selection. We develop the generalized formulation of quantum theory that stems from this approach and give a precise definition of time-reversal symmetry, emphasizing a previously overlooked distinction between states and effects. We prove an analogue of Wigner’s theorem, which characterizes all allowed symmetry transformations in this operationally time-symmetric quantum theory. Remarkably, we find larger classes of symmetry transformations than previously assumed, suggesting a possible direction in the search for extensions of known physics.

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Figure 1: Operation.
Figure 2: Circuit.
Figure 3: Time reversal as an active transformation.
Figure 4: A toy model of the universe between two instants of time.
Figure 5: Preparing non-standard operations without post-selection in a world with non-standard future boundary conditions.

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References

  1. Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959).

    MATH  Google Scholar 

  2. Wigner, E. P. On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939).

    Article  ADS  MathSciNet  Google Scholar 

  3. Schwinger, J. The theory of quantized fields I. Phys. Rev. 82, 914–927 (1951).

    Article  ADS  MathSciNet  Google Scholar 

  4. Lüders, G. On the equivalence of invariance under time reversal and under particle-antiparticle conjugation for relativistic field theories. Mat.-Fys. Medd. K. Dan. Vidensk. Selsk. 28, 1–17 (1954).

    MathSciNet  MATH  Google Scholar 

  5. Pauli, W. in Niels Bohr and the Development of Physics (eds Pauli, W., Rosenfeld, L. & Weisskopf, V.) 30–51 (McGraw-Hill, 1955).

    MATH  Google Scholar 

  6. Bell, J. S. Time reversal in field theory. Proc. R. Soc. Lond. A 231, 479–495 (1955).

    Article  ADS  MathSciNet  Google Scholar 

  7. Watanabe, S. Symmetry of physical laws. Part 3. Prediction and retrodiction. Rev. Mod. Phys. 27, 179–186 (1955).

    Article  ADS  MathSciNet  Google Scholar 

  8. Aharonov, Y., Bergmann, P. G. & Lebowitz, J. L. Time asymmetry in quantum process of measurement. Phys. Rev. B 134, 1410–1416 (1964).

    Article  ADS  Google Scholar 

  9. Holster, A. T. The criterion for time symmetry of probabilistic theories and the reversibility of quantum mechanics. New J. Phys. 5, 130 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  10. Busch, P., Gabrowski, M. & Lahti, P. J. Operational Quantum Physics (Springer, 1995).

    Book  Google Scholar 

  11. Ludwig, G. Foundations of Quantum Mechanics Vol. 1 (Springer, 1983).

    Book  Google Scholar 

  12. Hardy, L. Quantum theory from five reasonable axioms. Preprint at http://arXiv.org/abs/quant-ph/0101012 (2001).

  13. Barrett, J. Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007).

    Article  ADS  Google Scholar 

  14. Hardy, L. Operational Structures as a Foundation for Probabilistic Theories (2009); http://pirsa.org/09060015.

  15. Chiribella, G., D’Ariano, G. M. & Perinotti, P. Probabilistic theories with purification. Phys. Rev. A 81, 062348 (2010).

    Article  ADS  Google Scholar 

  16. Dakic, B. & Brukner, Č. in Deep Beauty: Understanding the Quantum World Through Mathematical Innovation (ed. Halvorson, H.) (Cambridge Univ. Press, 2011).

    MATH  Google Scholar 

  17. Masanes, L. & Müller, M. P. A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011).

    Article  ADS  Google Scholar 

  18. Chiribella, G., D’Ariano, G. M. & Perinotti, P. Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011).

    Article  ADS  Google Scholar 

  19. Hardy, L. Reformulating and reconstructing quantum theory. Preprint at http://arXiv.org/abs/1104.2066 (2011).

  20. Masanes, L., Mueller, M. P., Augusiak, R. & Perez-Garcia, D. Existence of an information unit as a postulate of quantum theory. Proc. Natl Acad. Sci. USA 110, 16373–16377 (2013).

    Article  ADS  Google Scholar 

  21. Coecke, B. Quantum picturialism. Contemp. Phys. 51, 59–83 (2010).

    Article  ADS  Google Scholar 

  22. Gammelmark, S., Julsgaard, B. & Mølmer, K. Past quantum states of a monitored system. Phys. Rev. Lett. 111, 160401 (2013).

    Article  ADS  Google Scholar 

  23. Hawking, S. W. Arrow of time in cosmology. Phys. Rev. D 32, 2489–2495 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  24. Mlodinow, L. & Brun, T. A. Relation between the psychological and thermodynamic arrows of time. Phys. Rev. E 89, 052102 (2014).

    Article  ADS  Google Scholar 

  25. Aharonov, Y. & Rohrlich, D. Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005).

    Book  Google Scholar 

  26. Griffiths, R. B. Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 219–272 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  27. Gell-Mann, M. & Hartle, J. B. in Physical Origins of Time Asymmetry (eds Halliwell, J. J., Perez-Mercader, J. & Zurek, W. H.) 311–345 (Cambridge Univ. Press, 1994).

    Google Scholar 

  28. Hartle, J. B. Unitarity and causality in generalized quantum mechanics for non-chronal spacetimes. Phys. Rev. D 49, 6543–6555 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  29. Horowitz, G. T. & Maldacena, J. The black hole final state. J. High Energy Phys. 02, 008 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  30. Gottesmann, D. & Preskill, J. Comment on “The black hole final state”. J. High Energy Phys. 2004, 026 (2004).

    Article  Google Scholar 

  31. Bennett, C. H. Simulated Time Travel, Teleportation Without Communication, and How to Conduct a Romance with Someone who has Fallen into a Black Hole (QUPON, 2005); http://web.archive.org/web/20070206131550/http://www.research.ibm.com/people/b/bennetc/QUPONBshort.pdf.

    Google Scholar 

  32. Svetlichny, G. Effective quantum time travel. Preprint at http://arXiv.org/abs/0902.4898 (2009).

  33. Lloyd, S. et al. Closed timelike curves via post-selection: Theory and experimental demonstration. Phys. Rev. Lett. 106, 040403 (2011).

    Article  ADS  Google Scholar 

  34. Hardy, L. Probability theories with dynamic causal structure: A new framework for quantum gravity. Preprint at http://arXiv.org/abs/gr-qc/0509120 (2005).

  35. Chiribella, G., D’Ariano, G. M., Perinotti, P. & Valiron, B. Quantum computations without definite causal structures. Phys. Rev. A 88, 022318 (2013).

    Article  ADS  Google Scholar 

  36. Oreshkov, O., Costa, F. & Brukner, Č. Quantum correlations with no causal order. Nature Commun. 3, 1092 (2012).

    Article  ADS  Google Scholar 

  37. Oreshkov, O. & Cerf, N. J. Operational quantum theory without predefined time. Preprint at http://arXiv.org/abs/1406.3829 (2014).

  38. Leifer, M. S. & Spekkens, R. W. Towards a formulation of quantum theory as a causally neutral theory of bayesian inference. Phys. Rev. A 88, 052130 (2013).

    Article  ADS  Google Scholar 

  39. Arrighi, P., Nesme, V. & Werner, R. Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77, 372–378 (2011).

    Article  MathSciNet  Google Scholar 

  40. Colbeck, R. & Renner, R. No extension of quantum theory can have improved predictive power. Nature Commun. 2, 411 (2011).

    Article  ADS  Google Scholar 

  41. Coecke, B. & Lal, R. Time-asymmetry of probabilities versus relativistic causal structure: An arrow of time. Phys. Rev. Lett. 108, 200403 (2012).

    Article  ADS  Google Scholar 

  42. Wood, C. J. & Spekkens, R. W. The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning. New J. Phys. 17, 033002 (2015).

    Article  ADS  Google Scholar 

  43. Fritz, T. Beyond Bell’s theorem II: Scenarios with arbitrary causal structure. Preprint at http://arXiv.org/abs/1404.4812 (2014).

  44. Bancal, J.-D. et al. Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling. Nature Phys. 8, 867–870 (2012).

    Article  ADS  Google Scholar 

  45. Rudolph, T. Quantum causality: Information insights. Nature Phys. 8, 860–861 (2012).

    Article  ADS  Google Scholar 

  46. Baumeler, Ä., Feix, A. & Wolf, S. Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios. Phys. Rev. A 90, 042106 (2014).

    Article  ADS  Google Scholar 

  47. Brukner, Č. Quantum causality. Nature Phys. 10, 259–263 (2014).

    Article  ADS  Google Scholar 

  48. Coecke, B. Terminality implies non-signalling. Electron. Proc. Theor. Comput. Sci. 172, 27–35 (2014).

    Article  MathSciNet  Google Scholar 

  49. Pienaar, J. & Brukner, Č. A graph-separation theorem for quantum causal models. Preprint at http://arXiv.org/abs/1406.0430 (2014).

  50. Ried, K. et al. A quantum advantage for inferring causal structure. Nature Phys. 11, 411–420 (2015).

    Article  ADS  Google Scholar 

  51. Davies, E. & Lewis, J. An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970).

    Article  ADS  MathSciNet  Google Scholar 

  52. Kraus, K. States, Effects and Operations (Springer, 1983).

    MATH  Google Scholar 

  53. Aharonov, Y. & Vaidman, L. The two-state vector formalism: An updated review. Lect. Notes Phys. 734, 399–447 (2008).

    Article  ADS  Google Scholar 

  54. Barnett, S. M., Pegg, D. T. & Jeffers, J. Bayes’ theorem and quantum retrodiction. J. Mod. Opt. 47, 1779–1789 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  55. Wolf, M. M. Quantum channels & operations: Guided tour. Preprint at http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf (2012).

  56. Wick, G. C., Wightman, A. S. & Wigner, E. P. The intrinsic parity of elementary particles. Phys. Rev. 88, 101–105 (1952).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the European Commission under the Marie Curie Intra-European Fellowship Programme (PIEF-GA-2010-273119) and by the F.R.S.–FNRS under the Chargé de recherches (CR) Fellowship Programme.

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Authors and Affiliations

  1. QuIC, Ecole Polytechnique de Bruxelles, CP 165, Université Libre de Bruxelles, 1050 Brussels, Belgium

    Ognyan Oreshkov & Nicolas J. Cerf

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  1. Ognyan Oreshkov
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O.O. conceived the project, developed the concepts and wrote the manuscript. N.J.C. supervised the project, discussed the results and edited the manuscript.

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Correspondence to Ognyan Oreshkov.

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The authors declare no competing financial interests.

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Oreshkov, O., Cerf, N. Operational formulation of time reversal in quantum theory. Nature Phys 11, 853–858 (2015). https://doi.org/10.1038/nphys3414

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