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

Foundations of Physics 34 (2):193-209 (2004)

Abstract
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures, 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 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.
Keywords quantum measurements  quantum probability rule  frame functions  POVM
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DOI 10.1023/B:FOOP.0000019581.00318.a5
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Interpreting the Quantum Mechanics of Cosmology.David Wallace - forthcoming - In A. Ijjas & B. Loewer (eds.), Philosophy of Cosmology: an Introduction. Oxford University Press.
Entropy - A Guide for the Perplexed.Roman Frigg & Charlotte Werndl - 2011 - In Claus Beisbart & Stephan Hartmann (eds.), Probabilities in Physics. Oxford University Press. pp. 115-142.

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