SWhat changed
Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions.
Scientific scope
Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions. The measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
- R. V. Kadison · SCIENCE: Helpful remarks acknowledged in the original; not a byline coauthor.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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