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

TL;DR

This paper extends Gleason's theorem to all quantum system dimensions using a gambling approach, showing that only trace-based probability measures are valid and explaining why dispersion-free probabilities are invalid.

Contribution

It introduces a Gleason-type theorem applicable to any dimension, including two, based on a gambling formulation of quantum mechanics, and clarifies the invalidity of dispersion-free probabilities.

Findings

Gleason-type theorem holds for all dimensions, including n=2

Only trace-based probability assignments are consistent with quantum mechanics

Dispersion-free probabilities are shown to be invalid in quantum theory

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.