Gleason’s Theorem

Full treatment. Read Gleason’s Theorem for the frame-function proof architecture, explicit qubit counterassignment, countable-additivity boundary, and POVM variants.

Statement

Let $\mathcal H$ be a separable complex Hilbert space with $\dim\mathcal H\geq3$. Every normalized nonnegative countably additive measure $\mu$ on its orthogonal projections has the unique form

$$\mu(P)=\operatorname{Tr}(\rho P),$$

where $\rho$ is a positive trace-class operator with $\operatorname{Tr}\rho=1$. In finite dimension, finite orthogonal additivity suffices.

Check before use

• One value $\mu(P)$ is assigned to a projector independently of which PVM contains it.
• Orthogonal alternatives are additive; infinite-dimensional versions use countable additivity.
• The original projection theorem requires dimension at least three.
• Qubit projector assignments can satisfy orthogonal additivity without having density-operator form.
• POVM or effect versions that cover qubits use a stronger premise and are Gleason-type extensions.

The result constrains sharp-event probabilities within Hilbert-space quantum mechanics. It does not derive projections, collapse, instruments, or dynamics from no assumptions.

References

• A. M. Gleason, “Measures on the closed subspaces of a Hilbert space,” Journal of Mathematics and Mechanics 6, 885–893, 1957, doi:10.1512/iumj.1957.6.56050.
• N. P. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Springer, 2017.