Gleason-Type Theorems from Cauchy’s Functional Equation

@article{Wright2019GleasonTypeTF,
  title={Gleason-Type Theorems from Cauchy’s Functional Equation},
  author={Victoria J Wright and Stefan Weigert},
  journal={Foundations of Physics},
  year={2019},
  volume={49},
  pages={594-606}
}
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. 

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References

SHOWING 1-10 OF 21 REFERENCES

A Gleason-type theorem for qubits based on mixtures of projective measurements

We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason’s theorem only relies upon the consistent

Quantum states and generalized observables: a simple proof of Gleason's theorem.

  • P. Busch
  • Physics
    Physical review letters
  • 2003
A simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator, and a von Neumann-type argument against noncontextual hidden variables is obtained.

Gleason's theorem and Cauchy's functional equation

We study measures on the effect algebras of the closed interval [0, 1] and we describe regular or bounded measures. Applying Gleason's theorem for measures on the system of all closed subspaces of a

An elementary proof of Gleason's theorem

Abstract Gleason's theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an

Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal

Unknown Quantum States: The Quantum de Finetti Representation

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the

Simulating Positive-Operator-Valued Measures with Projective Measurements.

This work proves that every measurement on a given quantum system can be realized by classical randomization of projective measurements on the system plus an ancilla of the same dimension, and shows that deciding whether it is PM simulable can be solved by means of semidefinite programming.

Measures on the Closed Subspaces of a Hilbert Space

In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on

On the functional equation f(x+y) =f(x) +f(y)

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