Gleason's theorem made simple: a Bloch-space perspective

TL;DR

This paper simplifies Gleason's theorem using the Bloch-space perspective, clarifying why the Born rule is fundamental in quantum mechanics for systems of dimension three and above, and explaining the special case of qubits.

Contribution

It offers an accessible, simplified explanation of Gleason's theorem through the Bloch representation, highlighting the dimensional dependence of probability rules.

Findings

Non-Born probability rules exist for two-dimensional systems.

Born rule becomes unavoidable in systems of dimension three or higher.

Qubits are a special, exceptional case in quantum probability rules.

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 essence of Gleason's result, using the generalized Bloch representation of quantum states. We show explicitly why non-Born probability rules exist for two-dimensional systems, and why they become impossible in dimension three and higher. Our argument does not reproduce Gleason's full mathematical theorem, but it clarifies why the Born rule is unavoidable in higher dimension and why qubits are truly exceptional.