Theorem

Gleason's Theorem

Mathematical Physics

Gleason's Theorem shows that the Born rule, the rule used to calculate probabilities in quantum physics, can be derived from the ordinary mathematical representation of quantum measurements together with the assumption of non-contextuality, meaning a measurement's outcome probability does not depend on which other compatible measurements are performed alongside it. Andrew Gleason first proved the theorem in 1957, answering a question posed by George Mackey, and the result proved historically significant for showing that broad classes of hidden-variable theories are inconsistent with quantum physics; it remains especially important to the field of quantum logic and its search for a minimal set of axioms for quantum theory.

Facts

Statement

Every assignment of probabilities to unit vectors of a Hilbert space of dimension 3 or greater that sums to 1 over each orthonormal basis takes the form of applying the Born rule to some density operator. 1

Proof Year

1957 1

Classification

Statement Form

Characterization Theorem 1

Connections

Source Gleason's theorem (Wikipedia)

Sources

1. Gleason's theorem (Wikipedia)

Statement of the theorem, Conceptual background, final paragraph
take the form of applying the Born rule to some density operator
Introduction, sentence 2
Andrew M. Gleason first proved the theorem in 1957
Proved By: Andrew M. Gleason, Lead
Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.
In Group: Quantum Mechanics and Quantum Field Theory, Lead section
shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual

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