content_image/332748.jpg content_image/332747.jpg content_image/332745.jpg content_image/332744.jpg content_image/332742.jpg content_image/332743.jpg
set of operators { M † $_{i}$M$_{j}$ } n i,j =1 is linearly independent[22]. Therefore, part (b) of the theorem above gives a complete characterization of the HTOs of all extreme quantum operations. 

Proof: For (a), the use of the definition of the HTO given in Eq. (9) directly leads to Eq. (38) where 

To show (b), first write the isometry condition U † $_{AB}$U$_{AB}$ = 1$_{AB}$ as 

<img src='content_image/332748.jpg'>

<img src='content_image/332747.jpg'>

The linear independence of { M † $_{i}$M$_{j}$ } n i,j =1 then leads to the following relations 

<img src='content_image/332745.jpg'>

In other words, each L$_{i}$ is an isometry on the bath that maps H$_{B}$ onto mutually perpendicular subspaces of H$_{B}$ . The last set of relations is actually equivalent to the statement that U$_{AB}$ is an isometry. 

Consider now a hypothetical n -level system R. Let {| i 〉} n i =1 be an orthonormal basis for the Hilbert space of R. The following operator on the composite system RB, 

<img src='content_image/332744.jpg'>

is then an isometry. It can also be readily verified that W$_{RB}$ is an isometry if and only if U$_{AB}$ is also an isometry. The most important point in here is that W$_{RB}$ is a realization of a Landauer erasure on R. Furthermore, the HTO associated with this erasure is given by 

<img src='content_image/332742.jpg'>

<img src='content_image/332743.jpg'>

For n ≥ 2, the application of Theorem 3 proves the claim. The result for the special case of n = 1 also follows trivially from the discussion in this subsection. 

Part (c) follows from the converse of the argument used above for part (b). Suppose that q satisfies (39). In that case, Q ′ R given by Eq. (45) is an allowed HTO for a Landauer erasure. Hence one can construct the associated realization; let W$_{RB}$ be the isometry for this realization. We use Eq. (43) to define L$_{i}$ and then use Eq. (34) to define U$_{AB}$ . It is 

