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

@article{Caves2004GleasonTypeDO,
  title={Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements},
  author={Carlton M. Caves and Christopher A. Fuchs and Kiran K. Manne and Joseph M. Renes},
  journal={Foundations of Physics},
  year={2004},
  volume={34},
  pages={193-209}
}
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 projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields—settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original… Expand

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References

SHOWING 1-10 OF 26 REFERENCES
Quantum Mechanics as Quantum Information (and only a little more)
In this paper, I try once again to cause some good-natured trouble. The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicionExpand
Operational Quantum Physics
Quantum physics is certainly one of the greatest scientific achievements of the 20th century. Nevertheless, there has always been a gap between the formalistics approach of quantum theory and itsExpand
Finite-precision measurement does not nullify the Kochen-Specker theorem
It is proven that any hidden variable theory of the type proposed by Meyer [Phys. Rev. Lett. 83, 3751 (1999)], Kent [ibid. 83, 3755 (1999)], and Clifton and Kent [Proc. R. Soc. London, Ser. A 456,Expand
Finite Precision Measurement Nullifies the Kochen-Specker Theorem
Only finite precision measurements are experimentally reasonable, and they cannot distinguish a dense subset from its closure. We show that the rational vectors, which are dense in S 2 , can beExpand
Kochen-Specker theorem for a single qubit using positive operator-valued measures.
  • A. Cabello
  • Physics, Mathematics
  • Physical review letters
  • 2003
TLDR
A proof of the Kochen-Specker theorem for a single two-level system using five eight-element positive operator-valued measures and simple algebraic reasoning based on the geometry of the dodecahedron is presented. Expand
Quantum Theory: Concepts and Methods
Preface. Part I: Gathering the Tools. 1. Introduction to Quantum Physics. 2. Quantum Tests. 3. Complex Vector Space. 4. Continuous Variables. Part II: Cryptodeterminism and Quantum Inseparability. 5.Expand
Classical and quantum noise in measurements and transformations
This paper has been withdrawn. See quant-ph/0408115: G. M. D'Ariano, P. Perinotti and P. Lo Presti, "Classical randomness in quantum measurements"
States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin
States and effects.- Operations.- The first Representation theorem.- Composite systems.- The second representation theorem.- 6 Coexistent effects and observables.- References.
Group Theory in Physics
An introductory text book for graduates and advanced undergraduates on group representation theory. It emphasizes group theory's role as the mathematical framework for describing symmetry propertiesExpand
Transition to effect algebras
An account is given of the recent development of the theory of effect algebras, their connection with partially ordered abelian groups, and their use for the mathematical representation of fuzzy orExpand
...
1
2
3
...