Store:QLMen08

3.4. General theory (Davies–Lewis–Ozawa)

Finally, we formulate the general notion of quantum instrument. A superoperator acting in ${\textstyle {\mathcal {L}}({\mathcal {H}})}$ is called positive if it maps the set of positive semi-definite operators into itself. We remark that, for each $x,\Im _{A}(x)$  given by (13) can be considered as linear positive map.

Generally any map$x\rightarrow \Im _{A}(x)$ , where for each $x$, the map $\Im _{A}(x)$ is a positive superoperator is called Davies–Lewis (Davies and Lewis, 1970) quantum instrument.

Here index ${\textstyle A}$  denotes the observable coupled to this instrument. The probabilities of ${\textstyle A}$-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.

A superoperator is called completely positive if its natural extension ${\textstyle \jmath \otimes I}$ to the tensor product  ${\textstyle {\mathcal {L}}({\mathcal {H}})\otimes {\mathcal {L}}({\mathcal {H}})={\mathcal {L}}({\mathcal {H}}\otimes {\mathcal {H}})}$ is again a positive superoperator on ${\textstyle {\mathcal {L}}({\mathcal {H}})\otimes {\mathcal {L}}({\mathcal {H}})}$. A map $x\rightarrow \Im _{A}(x)$ , where for each ${\textstyle x}$, the map $\Im _{A}(x)$ 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) Ozawa M. Uncertainty relations for noise and disturbance in generalized quantum measurements Ann. Phys., NY, 311 (2004), pp. 350-416

Every observable  ${\textstyle A}$ of a system ${\textstyle S}$ is identified with the observable ${\textstyle A\otimes I}$ of a system ${\textstyle S+S'}$ with any system ${\textstyle S'}$ external to${\textstyle S}$ .10

Then, every physically realizable instrument  $\Im _{A}$ measuring ${\textstyle A}$ should be identified with the instrument  ${\textstyle \Im _{A}{_{\otimes }}_{I}}$ measuring ${\textstyle A{\otimes }I}$ such that ${\textstyle \Im _{A}{_{\otimes }}_{I}(x)=\Im _{A}(x)\otimes I}$. This implies that ${\textstyle \Im _{A}(x)\otimes I}$ is agin a positive superoperator, so that $\Im _{A}(x)$ is completely positive.

Similarly, any physically realizable instrument $\Im _{A}(x)$ measuring system ${\textstyle S}$ should have its extended instrument  ${\textstyle \Im _{A}(x)\otimes I}$ measuring system ${\textstyle S+S'}$ for any external system${\textstyle S'}$. This is fulfilled only if  $\Im _{A}(x)$ is completely positive. Thus, complete positivity is a necessary condition for $\Im _{A}$ to describe a physically realizable instrument.