Difference between revisions of "Store:QLMen08"

===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 <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.

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=":Ozawa M (1984)">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.   

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.

View Record in ScopusGoogle Scholar</ref> Ozawa, 1984<ref name=":Ozawa M (1984)">Ozawa M. Quantum measuring processes for continuous observables. J. Math. Phys., 25 (1984), pp. 79-87. Google Scholar</ref> 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.

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   

Then, every physically realizable instrument  <math>\Im_A</math> measuring <math display="inline">A</math> should be identified with the instrument  <math display="inline">\Im_A{_\otimes}_I

</math> measuring <math display="inline">A{\otimes}I

</math> such that <math display="inline">\Im_A{_\otimes}_I(x)=\Im_A(x)\otimes I

</math>. This implies that <math display="inline">\Im_A(x)\otimes I

</math> is agin a positive superoperator, so that <math>\Im_A(x)</math> is completely positive.  

Similarly, any physically realizable instrument <math>\Im_A(x)</math> measuring system <math display="inline">S</math> should have its extended instrument  <math display="inline">\Im_A(x)\otimes I

</math> measuring system <math display="inline">S+S'</math> for any external system<math display="inline">S'</math>. This is fulfilled only if  <math>\Im_A(x)</math> is completely positive. Thus, complete positivity is a necessary condition for <math>\Im_A</math> to describe a physically realizable instrument.