Kochen-Specker Theorem for a Single Qubit using Positive Operator-Valued Measures

A proof of the Kochen-Specker theorem for a single two-level system is presented. It employs five eight-element positive operator-valued measures and a simple algebraic reasoning based on the geometry of the dodecahedron.

Figure 1

Notation for the 20 vertices of the dodecahedron: $A+$ is the antipode of $A-$.

Figure 2

The cube $CEFG$ is one of the five inscribed (sharing vertices) in the dodecahedron. It corresponds to an eight-element POVM.