Gleason's theorem

Gleason's theorem (named after Andrew M. Gleason) is a mathematical result which is of particular importance for the field of quantum logic. It proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space. The theorem states:

The lattice of projections Q can be interpreted as the set of quantum propositions, each proposition having the form "a ≤ A ≤ b", where A is the measured value of some observable on H (given by a self-adjoint linear operator). The trace-class operator W can be interpreted as the density matrix of a quantum state. Effectively, the theorem says that any legitimate probability measure on the space of allowable propositions is generated by some quantum state. This implies that the Standard Quantum Logic can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain.

Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a "quantum logic" is a set of events that is closed under the operation of disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that such a logic forms a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product.

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