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Coexistent observables and effects in a convexity approach. (English) Zbl 0935.81010
From the introduction: Given two observables \(s\mapsto p(s,o_1)\) and \(s\mapsto p(s,o_2)\), the question arises under which conditions it is possible to collect the relevant experimental data under a single observable \(s \mapsto p(s,o)\). This leads to the notion of the coexistence of observables, and it is an important question in quantum mechanics to characterize such (pairs of or collections of) observables. In Sect. I of this paper the authors define the notion of an observable as an affine function from an abstract convex set to the set of probability measures on a measurable space, and introduce the notion of the coexistence of such observables. Theorem 2.1.5 characterizes an important class of functionally coexistent observables in terms of so-called biobservables and joint observables. In the remaining part of Sect. I they illustrate this result in the framework of a total effect algebra as well as in the Hilbert space formulation of quantum mechanics. Finally they regard an effect as a [0,1]-valued affine function on an abstract convex set of states. Adopting the usual notion of the coexistence of effects in this setting, they characterize coexistent sets of effects in terms of projective systems of simple observables.

MSC:
81P15 Quantum measurement theory, state operations, state preparations
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