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Introducing the Qplex: a novel arena for quantum theory

  • Marcus Appleby
  • Christopher A. FuchsEmail author
  • Blake C. Stacey
  • Huangjun Zhu
Open Access
Topical Review

Abstract

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC.

Graphical abstract

References

  1. 1.
    C.A. Fuchs, R. Schack, Rev. Mod. Phys. 85, 1693 (2013)ADSCrossRefGoogle Scholar
  2. 2.
    C.A. Fuchs, R. Schack, Found. Phys. 41, 345 (2011)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    C.A. Fuchs, arXiv:1003.5209 [quant-ph] (2010)
  4. 4.
    C.A. Fuchs, in Quantum Theory: Reconsideration of Foundations, edited by A. Khrennikov (Växjö University Press, 2002), pp. 463–543Google Scholar
  5. 5.
    C.A. Fuchs, Coming of Age with Quantum Information: Notes on a Paulian Idea (Cambridge University Press, Cambridge, UK, 2010).Google Scholar
  6. 6.
    A. Peres, Am. J. Phys. 46, 745 (1978)ADSCrossRefGoogle Scholar
  7. 7.
    G. Zauner, Int. J. Quantum Inf. 9, 445 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.M. Renes, R. Blume-Kohout, A.J. Scott, C.M. Caves, J. Math. Phys. 45, 2171 (2004)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    C.A. Fuchs, My Struggles with the Block Universe (2014); foreword by M. Schlosshauer, edited by B.C. Stacey, arXiv:1405.2390 [quant-ph]
  10. 10.
    M. Appleby, S. Flammia, G. McConnell, J. Yard, Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines, arXiv:1604.06098 [quant-ph] (2016)
  11. 11.
    A.J. Scott, M. Grassl, J. Math. Phys. 51, 042203 (2010)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    D.M. Appleby, H. Yadsan-Appleby, G. Zauner, Quantum Inf. Comput. 13, 672 (2013)MathSciNetGoogle Scholar
  13. 13.
    S.T. Flammia, Unpublished (2004)Google Scholar
  14. 14.
    N.S. Jones, N. Linden, Phys. Rev. A 71, 012324 (2005)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    D.M. Appleby, Å. Ericsson, C.A. Fuchs, Found. Phys. 41, 564 (2011)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, 1985)Google Scholar
  17. 17.
    D.M. Appleby, S.T. Flammia, C.A. Fuchs, J. Math. Phys. 52, 022202 (2011)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    D.M. Appleby, C.A. Fuchs, H. Zhu, Quantum Inf. Comput. 15, 61 (2015)MathSciNetGoogle Scholar
  19. 19.
    B. Grünbaum, Convex Polytopes (Springer, 2003)Google Scholar
  20. 20.
    G.M. Ziegler, Lectures on Polytopes (Springer, 1998)Google Scholar
  21. 21.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms (Springer, 1993)Google Scholar
  22. 22.
    A. Wiles, Ann. Math. 141, 443 (1995)MathSciNetCrossRefGoogle Scholar
  23. 23.
    C.A. Fuchs, B.C. Stacey, in Quantum Theory: informational Foundations and Foils (Springer, 2016)Google Scholar
  24. 24.
    B. Coecke, R. Duncan, A. Kissinger, Q. Wang, in Quantum Theory: Informational Foundations and Foils (Springer, 2016)Google Scholar
  25. 25.
    P.A. Hoehn, C. Wever, Phys. Rev. A 95, 012102 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    L. Masanes, M.P. Müller, New J. Phys. 13, 063001 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012v4 (2001)
  28. 28.
    R. Schack, Found. Phys. 33, 1461 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    H. Barnum, M.P. Müller, C. Ududec, New J. Phys. 16, 123029 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    C.A. Fuchs, R. Schack, in Probability and Physics, edited by Y. Ben-Menahem, B. Hemmo (Springer, 2012), pp. 233–47Google Scholar
  31. 31.
    B.C. Stacey, Multiscale Structure in Eco-Evolutionary Dynamics, Ph.D. thesis, Brandeis University (2015)Google Scholar
  32. 32.
    E.T. Jaynes, Phys. Rev. 108, 171 (1957)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    W. Ochs, Erkenntnis 16, 339 (1981)CrossRefGoogle Scholar
  34. 34.
    C.M. Caves, C.A. Fuchs, R. Schack, J. Math. Phys. 43, 4537 (2002)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    B.C. Stacey, Philos. Trans. R. Soc. A 374, 2068 (2016)CrossRefGoogle Scholar
  36. 36.
    M. Plávala, Phys. Rev. A 94, 042108 (2016)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1993)Google Scholar
  38. 38.
    R. Haag, Lect. Notes Phys. 153, 168 (1982)ADSCrossRefGoogle Scholar
  39. 39.
    H. Araki, Commun. Math. Phys. 75, 1 (1980)ADSCrossRefGoogle Scholar
  40. 40.
    M. Zorn, Bull. Am. Math. Soc. 41, 667 (1935)CrossRefGoogle Scholar
  41. 41.
    G. Kimura, A. Kossakowski, Open Syst. Inf. Dyn. 12, 207 (2005)MathSciNetCrossRefGoogle Scholar
  42. 42.
    D.M. Appleby, Opt. Spectrosc. 103, 416 (2007)ADSCrossRefGoogle Scholar
  43. 43.
    M.S. Klamkin, G.A. Tsintsifas, Math. Mag. 52, 20 (1979)MathSciNetCrossRefGoogle Scholar
  44. 44.
    E.P. Wigner, Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren (Friedrich Vieweg und Sohn, 1931), pp. 251–254. Translation by J.J. Griffin in Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959), pp. 233–236Google Scholar
  45. 45.
    K. Mccrimmon, Bull. Am. Math. Soc. 84, 612 (1978)CrossRefGoogle Scholar
  46. 46.
    N.D. Mermin, Rev. Mod. Phys. 65, 803 (1993)ADSCrossRefGoogle Scholar
  47. 47.
    N.D. Mermin, Rev. Mod. Phys. 88, 039902 (2016)ADSCrossRefGoogle Scholar
  48. 48.
    C. Rovelli, Int. J. Theor. Phys. 35, 1637 (1996)CrossRefGoogle Scholar
  49. 49.
    M.A. Nielsen, I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2011)Google Scholar
  50. 50.
    A. Harrow, Quantum Inf. Comput. 8, 715 (2008)MathSciNetGoogle Scholar
  51. 51.
    B.C. Stacey, Sporadic SICs and the Normed Division Algebras, arXiv:1605.01426 [quant-ph] (2016)
  52. 52.
    B.C. Stacey, Geometric and Information-Theoretic Properties of the Hoggar Lines, arXiv:1609.03075 [quant-ph] (2016)
  53. 53.
    N.D. Mermin, Phys. Today 42, 9 (1989)CrossRefGoogle Scholar
  54. 54.
    C.M. Caves, C.A. Fuchs, Ann. Israel Phys. Soc. 12, 226 (1996)Google Scholar
  55. 55.
    R.W. Spekkens, Phys. Rev. A 75, 032110 (2007)ADSCrossRefGoogle Scholar
  56. 56.
    S.D. Bartlett, T. Rudolph, R.W. Spekkens, Phys. Rev. A 86, 012103 (2012)ADSCrossRefGoogle Scholar
  57. 57.
    R.W. Spekkens, Quasi-quantization: classical statistical theories with an epistemic restriction, arXiv:1409.5041 [quant-ph] (2014)
  58. 58.
    R. Clifton, J. Bub, H. Halvorson, Found. Phys. 33, 1561 (2003)MathSciNetCrossRefGoogle Scholar
  59. 59.
    H. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa, Phys. Rev. Lett. 76, 2818 (1996)ADSCrossRefGoogle Scholar
  60. 60.
    G. Chiribella, G.M. D’Ariano, P. Perinotti, Phys. Rev. A 84, 012311 (2011)ADSCrossRefGoogle Scholar
  61. 61.
    L. Disilvestro, D. Markham, Quantum protocols within Spekkens’ toy model, arXiv:1608.09012 [quant-ph] (2016)
  62. 62.
    A. Cabello, Interpretations of quantum theory: a map of madness, arXiv:1509.04711 [quant-ph] (2015)
  63. 63.
    C.A. Fuchs, in Information & Interaction: Eddington, Wheeler, and the Limits of Knowledge, edited by I.T. Durham, D. Rickles (2017)Google Scholar
  64. 64.
    C.A. Fuchs, N.D. Mermin, R. Schack, Am. J. Phys. 82, 749 (2014)ADSCrossRefGoogle Scholar
  65. 65.
    A. Zeilinger, Nature 438, 743 (2005)ADSCrossRefGoogle Scholar
  66. 66.
    J. Kofler, A. Zeilinger, Eur. Rev. 18, 469 (2010)CrossRefGoogle Scholar
  67. 67.
    D.M. Appleby, Mind and Matter, arXiv:1305.7381 [physics.hist-ph] (2013)
  68. 68.
    C.M. Caves, C.A. Fuchs, R. Schack, Phys. Rev. A 66, 062111 (2002)ADSCrossRefGoogle Scholar
  69. 69.
    B.C. Stacey, Mathematics 4, 36 (2016)CrossRefGoogle Scholar
  70. 70.
    C.A. Fuchs, R. Schack, Phys. Scripta 90, 015104 (2014)ADSCrossRefGoogle Scholar
  71. 71.
    T. Leinster, C.A. Cobbold, Ecology 93, 477 (2012)CrossRefGoogle Scholar
  72. 72.
    W.K. Wootters, Found. Phys. 16, 391 (1986)ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1950), translated from the German by H.P. RobertsonGoogle Scholar
  74. 74.
    H. Zhu, J. Phys. A 43, 305305 (2010)MathSciNetCrossRefGoogle Scholar
  75. 75.
    D.M. Appleby, H.B. Dang, C.A. Fuchs, Entropy 16, 1484 (2014)ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Marcus Appleby
    • 1
  • Christopher A. Fuchs
    • 2
    • 3
    Email author
  • Blake C. Stacey
    • 4
  • Huangjun Zhu
    • 5
  1. 1.Centre for Engineered Quantum Systems, School of Physics, University of SydneySydneyAustralia
  2. 2.Physics Department, University of Massachusetts BostonBostonUSA
  3. 3.Max Planck Institute for Quantum OpticsGarchingGermany
  4. 4.Department of PhysicsUniversity of Massachusetts BostonBostonUSA
  5. 5.Institute for Theoretical Physics, University of CologneCologneGermany

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