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Here index <math display="inline">A</math> denotes the observable coupled to this instrument. The probabilities of <math display="inline">A</math>-outcomes are given by Born’s rule in form (15) and the state-update by transformation (14). However, Yuen (1987)<ref>Yuen, H. P., 1987. Characterization and realization of general quantum measurements. M. Namiki and others (ed.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, pp. 360–363. | Here index <math display="inline">A</math> denotes the observable coupled to this instrument. The probabilities of <math display="inline">A</math>-outcomes are given by Born’s rule in form (15) and the state-update by transformation (14). However, Yuen (1987)<ref>Yuen, H. P., 1987. Characterization and realization of general quantum measurements. M. Namiki and others (ed.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, pp. 360–363. | ||
Google Scholar</ref> pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984)<ref name=":0">Ozawa M. | Google Scholar</ref> pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984)<ref name=":0">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref> introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity. | ||
Quantum measuring processes for continuous observables | |||
J. Math. Phys., 25 (1984), pp. 79-87 | |||
A superoperator is called ''completely positive'' if its natural extension <math display="inline">\jmath\otimes I</math> to the tensor product <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})=\mathcal{L}(\mathcal{H}\otimes\mathcal{H})</math> is again a positive superoperator on <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})</math>. A map <math>x\rightarrow\Im_A(x)</math> , where for each <math display="inline">x</math>, the map <math>\Im_A(x)</math> is a completely positive superoperator is called ''Davies–Lewis–Ozawa'' (Davies and Lewis 1970,<ref>Davies E.B., Lewis J.T. | A superoperator is called ''completely positive'' if its natural extension <math display="inline">\jmath\otimes I</math> to the tensor product <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})=\mathcal{L}(\mathcal{H}\otimes\mathcal{H})</math> is again a positive superoperator on <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})</math>. A map <math>x\rightarrow\Im_A(x)</math> , where for each <math display="inline">x</math>, the map <math>\Im_A(x)</math> is a completely positive superoperator is called ''Davies–Lewis–Ozawa'' (Davies and Lewis 1970,<ref>Davies E.B., Lewis J.T. | ||