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

Foundations of Physics 47 (7):991-1002 (2017)
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Abstract

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 \. 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.

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References found in this work

The Problem of Hidden Variables in Quantum Mechanics.Simon Kochen & E. P. Specker - 1967 - Journal of Mathematics and Mechanics 17:59--87.
La Prévision: Ses Lois Logiques, Ses Sources Subjectives.Bruno de Finetti - 1937 - Annales de l'Institut Henri Poincaré 7 (1):1-68.
Betting on the outcomes of measurements: A bayesian theory of quantum probability.Itamar Pitowsky - 2002 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34 (3):395-414.

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