Stephen M. Barnett, James D. Cresser, John Jeffers, David T. Pegg
We show that Gleason's theorem, in the form recently generalised by Busch, may be further simplified by dropping one of the three properties from which it was derived. The result is a more general probability than that usually employed in quantum theory in that it shows that any set of positive operators can represent the probabilities for a set of possible events. Remarkably, our more general form seems to contain Bayes's rule for conditional probabilities so there is no need to add it as an additional element. There is no need, moreover, to postulate that the measurement operators sum to the identity; rather this condition follows from our more general rule when there is no prior measurement outcome information available. We show how the new and general probability law may be applied in quantum communications and in retrodictive quantum theory.
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http://arxiv.org/abs/1308.0946
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