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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gleason’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4461422 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gleason’s theorem Context triple: [Wigner’s theorem on symmetry transformations, relatedConcept, Gleason’s theorem]
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A.
Mathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics is John von Neumann’s landmark 1932 treatise that rigorously formulates quantum theory using functional analysis and operator theory on Hilbert spaces.
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B.
Wigner’s theorem on symmetry transformations
Wigner’s theorem on symmetry transformations is a fundamental result in quantum mechanics stating that any symmetry of transition probabilities is represented by either a unitary or antiunitary operator on the system’s Hilbert space.
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C.
Born rule in quantum mechanics
The Born rule in quantum mechanics is the fundamental postulate that connects a system’s wavefunction to experimentally observed probabilities by stating that measurement outcomes occur with probabilities given by the squared magnitude of the wavefunction’s amplitudes.
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D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gleason’s theorem Target entity description: 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.
-
A.
Mathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics is John von Neumann’s landmark 1932 treatise that rigorously formulates quantum theory using functional analysis and operator theory on Hilbert spaces.
-
B.
Wigner’s theorem on symmetry transformations
Wigner’s theorem on symmetry transformations is a fundamental result in quantum mechanics stating that any symmetry of transition probabilities is represented by either a unitary or antiunitary operator on the system’s Hilbert space.
-
C.
Born rule in quantum mechanics
The Born rule in quantum mechanics is the fundamental postulate that connects a system’s wavefunction to experimentally observed probabilities by stating that measurement outcomes occur with probabilities given by the squared magnitude of the wavefunction’s amplitudes.
-
D.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
E.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
- F. None of above. chosen
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 ⓘ |
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Input
Subject: Gleason’s theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.