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Keywords:
lattice effect algebra; MV algebra; observable; state; Markov kernel; weak Markov kernel; smearing; generalized observable
lattice effect algebra; MV algebra; observable; state; Markov kernel; weak Markov kernel; smearing; generalized observable
Summary:
Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $\sigma$-effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $\sigma$-MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for $\sigma$-MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.
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