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Foundations of Physics

, Volume 44, Issue 2, pp 192–212 | Cite as

Local Tomography and the Jordan Structure of Quantum Theory

  • Howard Barnum
  • Alexander WilceEmail author
Article

Abstract

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.

Keywords

General probabilistic theories Local tomography Jordan algebras 

Notes

Acknowledgments

We thank C. M. Edwards for drawing our attention to Hanche-Olsen’s paper. Part of this work was done while the authors were guests of the Oxford University Computing Laboratory, whose hospitality is also gratefully acknowledged. H. B. thanks the Foundational Questions Institute (FQXi) for travel support for the visit. Additional work was done at the Perimeter Institute for Theoretical Physics; work at Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

References

  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS ’04), pp. 415–425 (2004)Google Scholar
  2. 2.
    Araki, H.: On a characterization of the state space of quantum mechanics. Comm. Math. Phys. 75, 1–24 (1980)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baez, J.: Quantum quandaries: a category-theoretic perspective. In: Rickles, D., French, S., Saatsi, J. (eds.) The Structural Foundations of Quantum Gravity. Oxford University Press, Oxford (2006). (Preprint arxiv.org/abs/quant-ph/0404040v2, 2004)Google Scholar
  4. 4.
    Barnum, H., Duncan, R., Wilce, A.: Symmetry, compact closure, and dagger compactness for categories of convex operational models. arxiv:1004.2920 (2010)Google Scholar
  5. 5.
    Barnum, H., Fuchs, C., Renes, J., Wilce, A.: Influence-free states on compound quantum systems. arXiv:quant-ph/0507108 (2005)Google Scholar
  6. 6.
    Barnum, H., Gaebler, P., Wilce, A.: Ensemble steering, weak self-duality, and the structure of probabilistic theories. arXiv:0912.5532 (2009)
  7. 7.
    Barnum, H., Wilce, A.: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum theory, arxiv:0908.2354 (2009)Google Scholar
  8. 8.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: A generalized no-broadcasting theorem. Phys. Rev. Lett. 99, 240501–240504 (2007). (Preprint arxiv:0707.0620, 2007)ADSCrossRefGoogle Scholar
  9. 9.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Phys. Rev. A 81, 062348 (2010). (Preprint arXiv:0908.1583)
  10. 10.
    Dakić, B., Brukner, Č.: Quantum theory and beyond: is entanglement special? arXiv:0911.0695 (2009)
  11. 11.
    Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Comm. Math. Phys. 17, 239–260 (1970)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    de la Torre, G., Masanes, Ll., Short, A., Müller, M.: Deriving quantum theory from its local structure and reversibility, Preprint arXiv:1110:5482 (2011)Google Scholar
  13. 13.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cones. University Press, Oxford (1994)zbMATHGoogle Scholar
  14. 14.
    Foulis, D., Randall, C.: Empirical logic and tensor products. In: Neumann, H. (ed.) Interpretations and Foundations of Quantum Theory. B. I. Wisssencshaft, Mannheim (1981)Google Scholar
  15. 15.
    Goyal, P.: From information geometry to to quantum theory. New J. Phys. 12, 023012 (2010)ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hanche-Olsen, H.: JB-algebras with tensor products are \(C^{\ast }\)-algebras. In: Araki, H. (ed.) Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics 1132. Springer, Berlin (1985)Google Scholar
  17. 17.
    Hardy, L.: Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012 (2000)Google Scholar
  18. 18.
    Holevo, A.: Probabilistic and Statistical Aspects of Quantum Mechanics, 2nd edn. North-Holland, Amsterdam (1982). (Edizioni della Normale, Pisa, 2011)Google Scholar
  19. 19.
    Jordan, P.: Ueber verallgemeinerungsmöglichkeiten des formalismus der quantenmechanik. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. 41, 209–217 (1933)Google Scholar
  20. 20.
    Jordan, P., von Neumann, J., Wigner, E.P.: On an algebraic generalization of the quantum-mechanical formalism. Ann. Math. 35, 29–64 (1934)CrossRefGoogle Scholar
  21. 21.
    Knapp, A.: Lie Groups Beyond an Introduction, 2nd edn. Birkhauser, Basel (2002)zbMATHGoogle Scholar
  22. 22.
    Koecher, M.: Die geodätischen von positivitätsbereichen. Math. Annalen. 135, 192–202 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ludwig, G.: Foundations of Quantum Mechanics. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  24. 24.
    Mackey, G.: Mathematical Foundations of Quantum Mechanics. Addison Wesley, Boston (1963)zbMATHGoogle Scholar
  25. 25.
    Masanes, L.I., Müller, M.: A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011). (arXiv:1004.1483, 2011)
  26. 26.
    Müller, M., Ududec, C.: The computational power of quantum mechanics determines its self-duality. Phys. Rev. Lett. 108, 130401 (2012). (Preprint arXiv:1110:3516, 2011)CrossRefGoogle Scholar
  27. 27.
    Rau, J.: On quantum vs. classical probability. Ann. Phys. 324, 2622–2637 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Selinger, P.: Towards a semantics for higher-order quantum computation. In: Proceedings of the 2nd International Workshop on Quantum Programming Languages, Turku, pp. 127–143. Turku Center for Computer Science, Publication No. 33 (2004)Google Scholar
  29. 29.
    Vinberg, E.B.: Homogeneous cones. Dokl. Acad. Nauk. SSSR 141, 270–273 (1961). (English trans. Soviet Math. Dokl. 2(1961), 1416–1619)MathSciNetGoogle Scholar
  30. 30.
    Wilce, A.: Conjugates, correlation and quantum mechanics, arXiv:1206.2897 (2012)
  31. 31.
    Wilce, A.: Four and a half axioms for finite-dimensional quantum theory. In: Ben-Menahem, Y., Hemmo, M. (eds.) Probability in Physics: Essays in Honor of Itamar Pitowsky. Springer, Berlin (2012). (Preprint arXiv:0912.5530, 2009)
  32. 32.
    Wilce, A.: Symmetry, self-duality, and the Jordan structure of quantum theory. Preprint arXiv:1110.6607 (2011)
  33. 33.
    Wilce, A.: The tensor product in generalized measure theory. Int. J. Theor. Phys. 31, 1915–1928 (1992)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of New MexicoAlbuquerqueUSA
  2. 2.Department of MathematicsSusquehanna UniversitySelinsgroveUSA

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