Sharp and unsharp quantum effects (Q1271888)

This paper presents a survey of the work done on classical and quantum measurement theory over the past thirty years, distinguishing between the so-called ``sharp'' theory based upon repeatable operations and the so-called ``fuzzy'' theory based upon non-repeatable operations. Elementary operations on a physical system can be modelled by contractive positive linear operators \(T\) on a GL-space \(V\) [cf. \textit{E. B. Davies} and \textit{J. T. Lewis}, Commun. Math. Phys. 17, 239-260 (1970; Zbl 0194.58304)]. When the adjoint \(T^*\) of \(T\) is applied to the order unit \(e\) in the dual unital GM-space \(V^*\), an element \(T^*e\) is created. Such an element is said to be an effect (of the operation \(T\)) and the set \(E\) of all such effects forms a subset of the order unit interval \([0,e]\) in \(V^*\). Such an object \(E\) has been algebraically axiomatized, and is known as an effect algebra. In this paper, the author remarks that the various measurement theories provide examples of effect algebras, and goes on to consider when tensor products of effect algebras may be formed, thereby relating his work to that of \textit{D. J. Foulis} and \textit{C. H. Randall} [`Interpretations and foundations of quantum theory', Proc. Conf., Marburg 1979, 9-20 (1981; Zbl 0495.03041)].