Four and a Half Axioms for Finite-Dimensional Quantum Probability

  • Alexander WilceEmail author
Part of the The Frontiers Collection book series (FRONTCOLL)


It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus per se. However, from this angle, the rather special C *-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis of which this apparatus can be reconstructed. In this paper, I explore one route to such a derivation of finite-dimensional quantum mechanics, by means of a set of strong, but probabilistically intelligible, axioms. Stated very informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of (basic, discrete) measurements as orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space – in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.


Real Hilbert Space Jordan Algebra Hermitian Operator Test Space Classical Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I wish to thank Howard Barnum for reading and commenting on an earlier draft of this paper, and, more especially, for introducing me to the papers of Koecher and Vinberg, on which the present exercise depends. Thanks also to C. M. Edwards for pointing out the paper [21] of Hanche-Olsen.


  1. 1.
    von Neumann, J.: Mathematical Foundations of Quantum Mechnanics. Springer, Berlin (1932); English translation Princeton University Press (1952)Google Scholar
  2. 2.
    Mackey, G.: Mathematical Foundations of Quantum Mechanics. Addison Wesley, Reading (1963)zbMATHGoogle Scholar
  3. 3.
    Birkhoff and von Neumann: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Zierler, N.: Axioms for non-relativistic quantum mechanics. Pacific J. Math. 11, 1151–1169 (1961)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Piron, C.: Axiomatique quantique. Helvetica Phys. Acta 37, 439–468 (1964)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Ludwig, G.: An Axiomatic Basis for Quantum Mechanics, vol. I, II. Springer, Berlin (1985). 1987CrossRefzbMATHGoogle Scholar
  7. 7.
    Gunson, J.: On the algebraic structure of quantum mechanics. Commun. Math. Phys. 6, 262–285 (1967)CrossRefzbMATHADSMathSciNetGoogle Scholar
  8. 8.
    Mielnik, B.: Theory of filters. Commun. Math. Phys. 15, 1–46 (1969)CrossRefzbMATHADSMathSciNetGoogle Scholar
  9. 9.
    Araki, H.: On a characterization of the state space of quantum mechanics. Commun. Math. Phys. 75, 1–24 (1980)CrossRefzbMATHADSMathSciNetGoogle Scholar
  10. 10.
    Bell, J.: Against “measurement”. Physics World, pp. 33–40 (1990)Google Scholar
  11. 11.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Cloning and broadcasting in generic probabilistic theories, arXiv: 0611295; also a general no-broadcasting theorem. Phys. Rev. Lett. 99, 240501–240505 (2007)Google Scholar
  12. 12.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Teleportation in general probabilistic theories, arXiv:0805.3553. (2008)Google Scholar
  13. 13.
    Dakic, B., Brukner, C.: Quantum theory and beyond: is entanglement special? arXiv:0911.0695. (2009)Google Scholar
  14. 14.
    D’Ariano, G.M.: Probabilistic theories: what is special about quantum mechanics? In: Bokulich, A., Jaeger G (eds.), Philosophy of Quantum Information and Entanglement, Cambridge University Press, Cambridge, 2010 (arXiv:0807.438, 2008)Google Scholar
  15. 15.
    Goyal, P.: An information-geometric reconstruction of quantum theory. Phys. Rev. A. 78 (2008) 052120Google Scholar
  16. 16.
    Hardy, L.: Quantum theory from five reasonable axioms, arXiv:quant-ph/00101012. (2001)Google Scholar
  17. 17.
    Mananes, L., Muller, M.P.: A derivation of quantum theory from physical requirements. New J. Phys. 13 (2011), 063001 (arXiv:1004.1483v1, 2010)Google Scholar
  18. 18.
    Rau, J.: On quantum vs. classical probability. Ann. Phys. 324, 2622–2637 (2009) (arXiv:0710.2119v1, 2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  19. 19.
    Koecher, M.: Die geoodätischen von positivitaätsbereichen. Mathematische Annalen 135, 192–202 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Vinberg, E.B.: Homogeneous cones. Doklady Academii Nauk SSR 141, 270–273 (1960); English translation, Soviet Mathematics–Doklady 2, 1416–1619 (1961)Google Scholar
  21. 21.
    Hanche-Olsen, H.: Jordan Algebras with Tensor Products are C* Algebras. Springer Lecture Notes in Mathematics 1132, pp. 223–229. Springer Verlag, Berlin (1985)Google Scholar
  22. 22.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75 (2007) 032304Google Scholar
  23. 23.
    Kläy, M., Randall, C.H., Foulis, D.J.: Tensor products and probability weights. Int. J. Theor. Phys. 26, 199–219 (1987)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wilce, A.: Symmetry and composition in probabilistic theories. Electron. Notes Theor. Comput. Sci. 270, 191–207 (2011) (arXiv:0910.1527)CrossRefGoogle Scholar
  25. 25.
    Kummer, H.: A constructive approach to the foundations of quantum mechanics. Found. Phys. 17, 1572–9516 (1987); The foundations of quantum theory and noncommutative spectral theory I, II. Found. Phys. 21, 1021–1069 (1991), 1183–1236Google Scholar
  26. 26.
    Bub, J., Pitowsky, I.: Two dogmas about quantum mechanics. In: Saunders, S., Barrett, J., Kent A., Wallace, C (eds.) Many Worlds? Everett, Quantum Theory and Reality, Oxford Universty press, Oxford, pp. 433–460 (2010)Google Scholar
  27. 27.
    Pitowsky, I.: Quantum mechanics as a theory of probability. In: Demopolous, W., Pitowsky, I. (eds.) Physical Theory and its Interpretation. Springer, Dordrecht (2006) (arXiv:quant-ph/0510095, 2005)Google Scholar
  28. 28.
    Foulis, D., Randall, C.H.: What are quantum logics, and what ought they to be? In: Betrametti, E., van Fraassen, B. (eds.) Current Issues in Quantum Logic. Plenum, New York (1981)Google Scholar
  29. 29.
    Wilce, A.: Quantum logic and probability, The Stanford Encyclopedia of Philosophy (Winter, 2002 Edition), E. Zalta (ed.), URL = <>
  30. 30.
    Wilce, A.: Test spaces. In: Gabbay, D., Engesser, K., Lehman, D. (eds.) Handbook of Quantum Logic, vol. II. Elsevier, Amsterdam (2009)Google Scholar
  31. 31.
    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, Berlin (1970)Google Scholar
  32. 32.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982)Google Scholar
  33. 33.
    Tran, Q., Wilce, A.: Covariance in quantum logic. Int. J. Theor. Phys. 47, 15–25 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Wilce, A.: Formalism and interpretation in quantum theory. Found. Phys. 40, 434–462 (2010)CrossRefzbMATHADSMathSciNetGoogle Scholar
  35. 35.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425. (2004) (arXiv: quant-ph/0402130)Google Scholar
  36. 36.
    Barnum, H., Barrett, J., Clark, L., Leifer, M., Spekkens, R., Stepanik, N., Wilce, A., Wilke, R.: Entropy and information causality in general probabilistic theories. New J. Phys. 12, 1367–2630 (2010) (arXiv:0909.5075. 2009)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Barnum, H., Gaebbler, C.P., Wilce, A.: Ensemble steering, weak self-duality, and the structure of probabilistic theories, arXiv:0912.5532. (2009)Google Scholar
  38. 38.
    Bellisard, J., Iochum, B.: Homogeneous self-dual cones, versus Jordan algebras. The theory revisited. Annales de l’Institut Fourier, Grenoble 28, 27–67 (1978)CrossRefGoogle Scholar
  39. 39.
    Vingerg, E.B.: Linear Representations of Groups. Birkhauser, Basel (1989)Google Scholar
  40. 40.
    Chiribella, G., D'Ariano G.M., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011) 012311-012350Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSusquehanna UniversitySelinsgroveUSA

Personalised recommendations