Gleason’s theorem
E443148
Gleason’s theorem is a foundational result in the mathematical formulation of quantum mechanics that characterizes all probability measures on the lattice of projection operators in a Hilbert space, effectively justifying the Born rule.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ theorem in quantum foundations ⓘ |
| appliesTo | Hilbert spaces of dimension at least 3 ⓘ |
| assumes |
Hilbert space over the complex numbers
ⓘ
Hilbert space over the quaternions ⓘ Hilbert space over the real numbers ⓘ noncontextual probability measures on projections ⓘ σ-additivity of probability measures ⓘ |
| author | Andrew M. Gleason NERFINISHED ⓘ |
| characterizes |
frame functions on Hilbert spaces
ⓘ
probability measures on projection lattices of Hilbert spaces ⓘ |
| codomain | probability measures on projections ⓘ |
| concludes |
every probability measure on projections is given by a density operator
ⓘ
probabilities are given by the trace rule Tr(ρP) ⓘ probabilities have the form ⟨ψ|P|ψ⟩ for pure states ⓘ |
| doesNotApplyTo | 2-dimensional Hilbert spaces ⓘ |
| domain | lattice of projection operators on a Hilbert space ⓘ |
| field |
functional analysis
ⓘ
measure theory ⓘ operator algebras ⓘ quantum foundations ⓘ quantum mechanics ⓘ |
| generalizedBy | Gleason-type theorems for POVMs NERFINISHED ⓘ |
| implies | Born rule for quantum probabilities ⓘ |
| importance | justifies standard quantum probability assignments ⓘ |
| influenced | modern axiomatizations of quantum mechanics ⓘ |
| involves |
density operators
ⓘ
orthogonal projectors ⓘ orthonormal bases of Hilbert spaces ⓘ projection-valued measures ⓘ self-adjoint operators on Hilbert space ⓘ |
| mathematicalContext |
orthomodular lattices of projections
ⓘ
separable Hilbert spaces ⓘ |
| namedAfter | Andrew M. Gleason NERFINISHED ⓘ |
| publishedIn | Journal of Mathematics and Mechanics NERFINISHED ⓘ |
| relatedTo |
Born rule
NERFINISHED
ⓘ
C*-algebras NERFINISHED ⓘ Kochen–Specker theorem NERFINISHED ⓘ quantum logic ⓘ von Neumann’s no-hidden-variables argument ⓘ |
| requires | Hilbert space dimension ≥ 3 for nontrivial content ⓘ |
| status | foundational result in quantum theory ⓘ |
| strengthenedBy | Busch’s theorem NERFINISHED ⓘ |
| usedFor | derivation of the Born rule from structural assumptions ⓘ |
| usedIn | arguments against noncontextual hidden-variable theories ⓘ |
| yearProved | 1957 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.