A Gleason-type theorem for any dimension based on a gambling formulation of Quantum Mechanics
arXiv:1606.03615 · doi:10.1007/s10701-017-0097-0
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.
arXiv admin note: substantial text overlap with arXiv:1605.08177. The overlapping is due to the fact that the present paper uses definitions introduced in arXiv:1605.08177