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Density Hypercubes, Higher Order Interference and Hyper-decoherence: A Categorical Approach

  • Stefano GogiosoEmail author
  • Carlo Maria Scandolo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11690)

Abstract

In this work, we use the recently introduced double-dilation construction by Zwart and Coecke to construct a new categorical probabilistic theory of density hypercubes. By considering multi-slit experiments, we show that the theory displays higher-order interference of order up to fourth. We also show that the theory possesses hyperdecoherence maps, which can be used to recover quantum theory in the Karoubi envelope.

Keywords

Quantum theory Categorical probabilistic theories Higher-order interference Hyperdecoherence 

Notes

Acknowledgements

SG is supported by a grant on Quantum Causal Structures from the John Templeton Foundation. CMS was supported in the writing of this paper by the Engineering and Physical Sciences Research Council (EPSRC) through the doctoral training grant 1652538 and by the Oxford-Google DeepMind graduate scholarship. CMS is currently supported by the Pacific Institute for the Mathematical Sciences (PIMS) and from a Faculty of Science Grand Challenge award at the University of Calgary. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

References

  1. 1.
    Barnum, H., Lee, C.M., Scandolo, C.M., Selby, J.H.: Ruling out higher-order interference from purity principles. Entropy 19(6), 253 (2017).  https://doi.org/10.3390/e19060253CrossRefGoogle Scholar
  2. 2.
    Barnum, H., Müller, M.P., Ududec, C.: Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys. 16(12), 123029 (2014).  https://doi.org/10.1088/1367-2630/16/12/123029CrossRefGoogle Scholar
  3. 3.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Phys. Rev. A 81, 062348 (2010).  https://doi.org/10.1103/PhysRevA.81.062348CrossRefGoogle Scholar
  4. 4.
    Chiribella, G., Scandolo, C.M.: Entanglement as an axiomatic foundation for statistical mechanics. arXiv:1608.04459 [quant-ph] (2016). http://arxiv.org/abs/1608.04459
  5. 5.
    Chiribella, G., Scandolo, C.M.: Microcanonical thermodynamics in general physical theories. New J. Phys. 19(12), 123043 (2017).  https://doi.org/10.1088/1367-2630/aa91c7CrossRefGoogle Scholar
  6. 6.
    Coecke, B.: Terminality implies no-signalling... and much more than that. New Gener. Comput. 34(1–2), 69–85 (2016).  https://doi.org/10.1007/s00354-016-0201-6CrossRefzbMATHGoogle Scholar
  7. 7.
    Coecke, B., Lal, R.: Causal categories: relativistically interacting processes. Found. Phys. 43(4), 458–501 (2013).  https://doi.org/10.1007/s10701-012-9646-8MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. 23(3), 555–567 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coecke, B., Perdrix, S.: Environment and classical channels in categorical quantum mechanics. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 230–244. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15205-4_20CrossRefGoogle Scholar
  10. 10.
    Dakić, B., Paterek, T., Brukner, Č.: Density cubes and higher-order interference theories. New J. Phys. 16(2), 023028 (2014).  https://doi.org/10.1088/1367-2630/16/2/023028CrossRefGoogle Scholar
  11. 11.
    Gogioso, S., Scandolo, C.M.: Categorical probabilistic theories. In: Coecke, B., Kissinger, A. (eds.) Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3–7 July 2017. Electronic Proceedings in Theoretical Computer Science, vol. 266, pp. 367–385. Open Publishing Association (2018).  https://doi.org/10.4204/EPTCS.266.23MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gogioso, S.: Higher-order CPM constructions. Electron. Proc. Theor. Comput. Sci. 270, 145–162 (2019).  https://doi.org/10.4204/EPTCS.287.8MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jin, F., et al.: Experimental test of born’s rule by inspecting third-order quantum interference on a single spin in solids. Phys. Rev. A 95, 012107 (2017).  https://doi.org/10.1103/PhysRevA.95.012107CrossRefGoogle Scholar
  14. 14.
    Kauten, T., Keil, R., Kaufmann, T., Pressl, B., Brukner, Č., Weihs, G.: Obtaining tight bounds on higher-order interferences with a 5-path interferometer. New J. Phys. 19(3), 033017 (2017).  https://doi.org/10.1088/1367-2630/aa5d98CrossRefGoogle Scholar
  15. 15.
    Krumm, M., Barnum, H., Barrett, J., Müller, M.P.: Thermodynamics and the structure of quantum theory. New J. Phys. 19(4), 043025 (2017).  https://doi.org/10.1088/1367-2630/aa68efCrossRefGoogle Scholar
  16. 16.
    Lee, C.M., Selby, J.H.: Deriving Grover’s lower bound from simple physical principles. New J. Phys. 18(9), 093047 (2016).  https://doi.org/10.1088/1367-2630/18/9/093047CrossRefGoogle Scholar
  17. 17.
    Lee, C.M., Selby, J.H.: Generalised phase kick-back: the structure of computational algorithms from physical principles. New J. Phys. 18(3), 033023 (2016).  https://doi.org/10.1088/1367-2630/18/3/033023CrossRefGoogle Scholar
  18. 18.
    Lee, C.M., Selby, J.H.: Higher-order interference in extensions of quantum theory. Found. Phys. 47(1), 89–112 (2017).  https://doi.org/10.1007/s10701-016-0045-4MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lee, C.M., Selby, J.H., Barnum, H.: Oracles and query lower bounds in generalised probabilistic theories. arXiv:1704.05043 [quant-ph] (2017). https://arxiv.org/abs/1704.05043
  20. 20.
    Lee, C.M., Selby, J.H.: A no-go theorem for theories that decohere to quantum mechanics. Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 474(2214), 20170732 (2018).  https://doi.org/10.1098/rspa.2017.0732MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Niestegge, G.: Three-slit experiments and quantum nonlocality. Found. Phys. 43(6), 805–812 (2013).  https://doi.org/10.1007/s10701-013-9719-3MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Park, D.K., Moussa, O., Laflamme, R.: Three path interference using nuclear magnetic resonance: a test of the consistency of Born’s rule. New J. Phys. 14(11), 113025 (2012).  https://doi.org/10.1088/1367-2630/14/11/113025CrossRefGoogle Scholar
  23. 23.
    Sinha, A., Vijay, A.H., Sinha, U.: On the superposition principle in interference experiments. Sci. Rep. 5, 10304 (2015).  https://doi.org/10.1038/srep10304CrossRefGoogle Scholar
  24. 24.
    Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., Weihs, G.: Ruling out multi-order interference in quantum mechanics. Science 329(5990), 418–421 (2010).  https://doi.org/10.1126/science.1190545MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sorkin, R.D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(33), 3119–3127 (1994).  https://doi.org/10.1142/S021773239400294XMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sorkin, R.D.: Quantum Measure Theory and its Interpretation. In: Quantum Classical Correspondence: The 4th Drexel Symposium on Quantum Nonintegrability, pp. 229–251. International Press, Boston (1997)Google Scholar
  27. 27.
    Ududec, C.: Perspectives on the formalism of quantum theory. Ph.D. thesis, University of Waterloo (2012)Google Scholar
  28. 28.
    Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. Found. Phys. 41(3), 396–405 (2011).  https://doi.org/10.1007/s10701-010-9429-zMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zwart, M., Coecke, B.: Double dilation \(\ne \) double mixing (extended abstract). In: Coecke, B., Kissinger, A. (eds.) Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3–7 July 2017. Electronic Proceedings in Theoretical Computer Science, vol. 266, pp. 133–146. Open Publishing Association (2018).  https://doi.org/10.4204/EPTCS.266.9MathSciNetCrossRefGoogle Scholar
  30. 30.
    Życzkowski, K.: Quartic quantum theory: an extension of the standard quantum mechanics. J. Phys. A 41(35), 355302 (2008).  https://doi.org/10.1088/1751-8113/41/35/355302MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of OxfordOxfordUK
  2. 2.University of CalgaryCalgaryCanada

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