Gleason's Theorem

Gleason's theorem (1957) states that for a Hilbert space of dimension at least three, every countably additive probability measure on the lattice of closed subspaces is induced by a density operator. In plain language: if you want to assign probabilities to quantum propositions in a way that respects the lattice structure, you have no choice. The probabilities must come from a quantum state described by the standard formalism of quantum mechanics. This closes a logical circle. The Born rule, usually treated as a postulate, becomes a theorem once the logical structure of quantum propositions is accepted. Gleason's theorem is one of the deepest results connecting the geometry of Hilbert space to the epistemology of quantum measurement. It has been extended to various infinite-dimensional and algebraic settings, and its converse — showing that every density operator induces a valid probability measure — is nearly trivial. The theorem's power lies in its uniqueness: the standard quantum formalism is not merely consistent with the lattice structure. It is forced by it.