Probability theories in general and quantum theory in particular

We consider probability theories in general. In the first part of the paper, various constraints are imposed and classical probability and quantum theory are recovered as special cases. Quantum theory follows from a set of five reasonable axioms. The key axiom which gives us quantum theory rather than classical probability theory is the continuity axiom, which demands that there exists a continuous reversible transformation between any pair of pure states. In the second part of this paper, we consider in detail how the measurement process works in both the classical and the quantum case. The key differences and similarities are elucidated. It is shown how measurement in the classical case can be given a simple ontological interpretation which is not open to us in the quantum case. On the other hand, it is shown that the measurement process can be treated mathematically in the same way in both theories even to the extent that the equations governing the state update after measurement are identical. The difference between the two cases is seen to be due not to something intrinsic to the measurement process itself but, rather, to the nature of the set of allowed states and, therefore, ultimately to the continuity axiom.
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DOI 10.1016/S1355-2198(03)00034-0
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References found in this work BETA
A Foundational Principle for Quantum Mechanics.Anton Zeilinger - 1999 - Foundations of Physics 29 (4):631-643.
The Logic of Quantum Mechanics.Garrett Birkhoff & John von Neumann - 1937 - Journal of Symbolic Logic 2 (1):44-45.
Why Quantum Theory?Lucien Hardy - 2002 - In T. Placek & J. Butterfield (eds.), Non-Locality and Modality. Kluwer Academic Publishers. pp. 61--73.
Quantum Mechanics Without Probability Amplitudes.William K. Wootters - 1986 - Foundations of Physics 16 (4):391-405.

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Measurement-Based Quantum Foundations.Jochen Rau - 2011 - Foundations of Physics 41 (3):380-388.

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