A Gleason-type theorem for any dimension based on a gambling formulation of Quantum Mechanics

Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco (2017) A Gleason-type theorem for any dimension based on a gambling formulation of Quantum Mechanics. Foundations of Physics, 47. pp. 991-1002.

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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 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.

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