Why qubits are exceptional in Gleason's theorem: a Bloch-space perspective

Abstract

Gleason's theorem is often cited as establishing the Born rule from the structure of Hilbert space, yet its original proof is mathematically sophisticated and rarely accessible to physicists. In this article we present a simple route to the dimensional distinction at the heart of Gleason's result, using the generalized Bloch representation of quantum states. More precisely, we consider a restricted but instructive class of probability assignments obtained by replacing the linear dependence of the Born expression on a Bloch-space scalar product with a real function f. For qubits, infinitely many continuous non-Born assignments of this form satisfy normalization for every projective measurement. In dimension greater than 3, by contrast, the geometry of the measurement simplex imposes additional relations among the relevant scalar products. Requiring normalization for arbitrary states then leads to a Cauchy-type functional equation, which, together with boundedness, forces f to be linear. Our argument provides a complementary geometric illustration of why the freedom present in dimension two disappears in higher dimensions, reaching this conclusion directly from the geometry of the measurement simplices and normalization, without explicitly invoking the intertwining of projectors across different orthogonal resolutions of the identity.