bibkey: busch2003effectprobabilities
authors: Paul Busch
year: 2003
title: Quantum States and Generalized Observables — A Simple Proof of Gleason’s Theorem
doi: 10.1103/PhysRevLett.91.120403
url: https://arxiv.org/html/quant-ph/9909073v3
claim: A normalized additive probability on quantum effects is represented by a density operator; in finite dimension binary additivity suffices.
strata_touched:
  - D5/S3/Quantum/FiniteDimensional
  - D5/S3/Quantum/Foundation/EffectProbabilityExtension
license: citation-only
triage: anchor

Effect probabilities and finite-dimensional trace representation

The theorem and its proof treat all positive contractions, rather than only projections. The probability is a function of the effect itself, with normalization and additivity for admissible sums. The proof obtains monotonicity, rational and real homogeneity, a positive-cone extension and a positive linear functional. The result includes two-dimensional Hilbert spaces.

In finite dimension, binary additivity suffices for these steps. A positive matrix can be scaled into the effect interval. Independence of scale follows from homogeneity; additivity follows by choosing a common scale. Differences of positive matrices give the Hermitian extension, and complexification gives a functional on the whole matrix algebra. The nondegenerate trace pairing represents it uniquely. Rank-one positivity tests and evaluation at the identity give positivity and trace one of the representative. Countable additivity in the paper supplies normality in infinite dimension; finite-dimensional trace duality requires no additional countable-additivity hypothesis.

The matrix representation is an additional quantum probability assumption. It does not follow from a static event structure, a history grading, or a choice of clock. For pure density and rank-one projection the trace pairing is the squared modulus of the corresponding vector inner product.

Verified locator

• Full text: https://arxiv.org/html/quant-ph/9909073v3 , theorem and following proof; the two-dimensional consequence follows immediately after the proof.
• DOI: https://doi.org/10.1103/PhysRevLett.91.120403 .