Foundations of Physics

, Volume 47, Issue 7, pp 991–1002 | Cite as

A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics

  • Alessio BenavoliEmail author
  • Alessandro Facchini
  • Marco Zaffalon


Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for \(n=2\). The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.


Gleason’s type theorem Dispersion-free probabilities Desirable gambles 


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)LuganoSwitzerland

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