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Gianfranco (talk | contribs) (Created page with "===3.4. General theory (Davies–Lewis–Ozawa)=== 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 <m...") |
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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) | 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)<ref>Davies E.B., Lewis J.T. | ||
An operational approach to quantum probability | |||
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, Ozawa, 1984) quantum instrument or simply quantum instrument. As we shall see in Section 4, complete positivity is a sufficient condition for an instrument to be physically realizable. On the other hand, necessity is derived as follows (Ozawa, 2004). | Comm. Math. Phys., 17 (1970), pp. 239-260 | ||
View Record in ScopusGoogle Scholar</ref> 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)<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. | |||
Quantum measuring processes for continuous observables | |||
J. Math. Phys., 25 (1984), pp. 79-87 | |||
View Record in ScopusGoogle Scholar</ref> introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity. | |||
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. | |||
An operational approach to quantum probability | |||
Comm. Math. Phys., 17 (1970), pp. 239-260 | |||
View Record in ScopusGoogle Scholar</ref> Ozawa, 1984<ref name=":0" />) quantum instrument or simply quantum instrument. As we shall see in Section 4, complete positivity is a sufficient condition for an instrument to be physically realizable. On the other hand, necessity is derived as follows (Ozawa, 2004).<ref>Ozawa M. | |||
Uncertainty relations for noise and disturbance in generalized quantum measurements | |||
Ann. Phys., NY, 311 (2004), pp. 350-416 | |||
Google Scholar</ref> | |||
Every observable <math display="inline">A</math> of a system <math display="inline">S</math> is identified with the observable <math display="inline">A\otimes I</math> of a system <math display="inline">S+S'</math> with any system <math display="inline">S'</math> external to<math display="inline">S</math> .10 | Every observable <math display="inline">A</math> of a system <math display="inline">S</math> is identified with the observable <math display="inline">A\otimes I</math> of a system <math display="inline">S+S'</math> with any system <math display="inline">S'</math> external to<math display="inline">S</math> .10 | ||