Gleason-type Theorems from Cauchy's Functional Equation

Abstract

Gleason-type theorems derive the density operator and the Born rule formalism of quantum theory from the measurement postulate, by considering additive functions which assign probabilities to measurement outcomes. Additivity is also the defining property of solutions to Cauchy's functional equation. This observation suggests an alternative proof of the strongest known Gleason-type theorem, based on techniques used to solve functional equations.

arXiv commentA corrected proof of Theorem 1 is given which closes a gap in its previous (and published) version

References in corpus (5)

• Unknown Quantum States: The Quantum de Finetti Representation
• Quantum states and generalized observables: a simple proof of Gleason's theorem
• Simulating positive-operator-valued measures with projective measurements
• Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
• A Gleason-type theorem for qubits based on mixtures of projective measurements

Cited by in corpus (4)

• Ideas Abandoned en Route to QBism
• Extending Kolmogorov's axioms for a generalized probability theory on collections of contexts
• Quantum violation of the Suppes-Zanotti inequalities and "contextuality"
• Form of Contextuality Predicting Probabilistic Equivalence between Two Sets of Three Mutually Noncommuting Observables