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Finally, we formulate the general notion of quantum instrument. A superoperator acting in <math display="inline">\mathcal{L}(\mathcal{H})</math> is called positive if it maps the set of positive semi-definite operators into itself. We remark that, for each '''<u><math>x,\Im_A(x)</math></u>''' given by (13) can be considered as linear positive map. | Finally, we formulate the general notion of quantum instrument. A superoperator acting in <math display="inline">\mathcal{L}(\mathcal{H})</math> is called positive if it maps the set of positive semi-definite operators into itself. We remark that, for each '''<u><math>x,\Im_A(x)</math></u>''' given by (13) can be considered as linear positive map. | ||
Generally any map<math>x\rightarrow\Im_A(x)</math> , where for each <math>x</math>, the map <math>\Im_A(x)</math> is a positive superoperator is called ''Davies–Lewis'' (Davies and Lewis, 1970) quantum instrument | Generally any map<math>x\rightarrow\Im_A(x)</math> , where for each <math>x</math>, the map <math>\Im_A(x)</math> is a positive superoperator is called ''Davies–Lewis'' (Davies and Lewis, 1970) quantum instrument. | ||
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) pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984) introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity. | 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) pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984) introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity. | ||