A Gleason-type theorem for qubits based on mixtures of projective measurements

TL;DR

This paper extends Gleason's theorem to qubits by deriving Born's rule and the density operator formalism using weaker assumptions based on assigning probabilities to projective measurements and their mixtures.

Contribution

It provides a Gleason-type theorem for qubits that requires fewer assumptions than previous results, strengthening the foundations of quantum probability.

Findings

Derived Born's rule for qubits from minimal assumptions

Extended Gleason's theorem to two-dimensional Hilbert spaces

Established a weaker assumption framework for quantum probability assignments

Abstract

We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason's theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.