The problem of dispersion-free probabilities in Gleason-type theorems for a two-dimensional Hilbert space

TL;DR

This paper explores weakening the lattice condition in Gleason's theorem to address the issue of dispersion-free probabilities in two-dimensional Hilbert spaces, specifically for qubits, offering an alternative to adding assumptions.

Contribution

It proposes a modified lattice structure based on commutability to resolve dispersion-free probabilities in Gleason's theorem for qubits.

Findings

Weakening the lattice condition can eliminate dispersion-free measures in 2D Hilbert spaces.

The approach applies specifically to qubits, demonstrating a potential solution.

The modified structure maintains consistency with quantum probability rules.

Abstract

As it is known, Gleason's theorem is not applicable for a two-dimensional Hilbert space since in this situation Gleason's axioms are not strong enough to imply Born's rule thus leaving room for a dispersion-free probability measure i.e., one that has only values 0 and 1. To strengthen Gleason's axioms one must add at least one more assumption. But, as it is argued in the present paper, alternatively one can give up the lattice condition lying in the foundation of Gleason's theorem. Particularly, the lattice structure based on the closed linear subspaces in the Hilbert space could be weakened by the requirement for the meet operation to exist only for the subspaces belonging to commutable projection operators. The paper demonstrates that this weakening can resolve the problem of the dispersion-free probability measure in the case of a qubit.