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Foundations of Physics

, Volume 34, Issue 2, pp 193–209 | Cite as

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

  • Carlton M. Caves
  • Christopher A. Fuchs
  • Kiran K. Manne
  • Joseph M. Renes
Article

Abstract

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 result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.

quantum measurements quantum probability rule frame functions POVM 

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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Carlton M. Caves
    • 1
  • Christopher A. Fuchs
    • 2
    • 3
  • Kiran K. Manne
    • 1
  • Joseph M. Renes
    • 1
  1. 1.Department of Physics and AstronomyUniversity of New MexicoAlbuquerque
  2. 2.Quantum Information and Optics Research, Bell LabsLucent TechnologiesMurray Hill
  3. 3.Communication Networks Research InstituteDublin Institute of TechnologyDublin 6Ireland

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